A study on quantitatively pricing various convertible bonds

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A study on quantitatively pricing various convertible bonds A study on quantitatively pricing various convertible bonds

Sha Lin University of Wollongong

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A study on quantitatively pricing various convertible bonds

Sha Lin

This thesis is presented as part of the requirements for the conferral of the degree:

Doctor of Philosophy

Supervisor:Prof. Song-Ping Zhu & Dr. Xiaoping Lu

The University of WollongongSMAS School of Mathematics and Applied Statistics

March 19, 2019

This work © copyright by Sha Lin, 2019. All Rights Reserved.

No part of this work may be reproduced, stored in a retrieval system, transmitted, in any form orby any means, electronic, mechanical, photocopying, recording, or otherwise, without the priorpermission of the author or the University of Wollongong.

This research has been conducted with the support of an Australian Government ResearchTraining Program Scholarship.

Declaration

I, Sha Lin, declare that this thesis is submitted in partial fulfilment of the require-ments for the conferral of the degree Doctor of Philosophy, from the University ofWollongong, is wholly my own work unless otherwise referenced or acknowledged.This document has not been submitted for qualifications at any other academicinstitution.

Sha Lin

March 19, 2019

Abstract

Financial derivatives are becoming increasingly popular among investors as well asacademic researchers. Among these, convertible bonds have received a large amountof attention since they are beneficial to both the issuer and holder in the sense thatthey help reduce the cash interest payment for the issuer while enabling the holderto reduce the risk of directly holding the underlying stocks. Despite the popularityof convertible bonds in practice, their pricing problems still remain challenging notonly because most of them traded in real markets are of American-style, but alsodue to many additional features that could be introduced into vanilla convertiblebonds to cater for different demands. Thus, convertible bonds are often priced withvarious numerical approaches because the predominant complexity arises from thedetermination of the bond price together with different free boundaries, which areintroduced due to the incorporation of various additional features.

This thesis contributes to the literature significantly by pricing various types ofconvertible bonds under different models. In particular, Chapters 3-5 focus on deriv-ing integral equation formulations for the prices of puttable, callable-puttable andresettable convertible bonds under the Black-Scholes model. The pricing of con-vertible bonds under stochastic volatility and interest rate models is then discussedin Chapter 6. Due to the additional stochastic source, analytical pricing formulaeare no longer available, and an efficient predictor-corrector scheme is established toobtain the convertible bond prices as well as the optimal conversion boundary.

iv

Acknowledgments

During my Ph.D. study, there are many people to whom I would like to express mysincere gratitude. Without them, this thesis could not be finished.

First of all, my supervisor, Prof. Song-Ping Zhu, has given me a lot of help.It is he who has guided me into the world of financial mathematics. It is he whoencourages me whenever I feel frustrated. It is he who helps me find the right waywhen I am confused. It is also he who corrects various mistakes I make. All in all,Prof. Song-Ping Zhu is a responsible and qualified supervisor, and I am so proud tobe one of his Ph.D. students.

My gratitude also goes to my co-supervisor, Dr. Xiaoping Lu, for her help andpatient guidance. I would also like to thank Dr. Heather Jamieson and Ms. AshleeDavis for their help in polishing my thesis. I am also grateful for my dear friends,Dr. Xinjiang He, Dr. Guiyuan Ma, Dr. Xiangchen Zeng, Dr. Ziwei Ke, Ms. DongYan, Dr. Chengbo Yang and Dr. Chi Chung Siu, who have made my studies at theUniversity of Wollongong full of fun.

I am also grateful for the International Postgraduate Tuition Award from theUniversity of Wollongong and the Chinese Government Scholarship from the ChinaScholarship Council which have provided me with financial support so that I canrealize my dream of doing a Ph.D. here. I would like to thank the IMIA and theAustralian Mathematical Society for their financial support that made it possiblefor me to attend academic conferences, including ANZIAM 2018 and QMF 2018.

Finally, I would like to thank my parents, sister and boyfriend for their endlesssupport, understanding and encouragement.

v

Contents

Abstract iv

1 Introduction and Literature review 11.1 Convertible bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Vanilla convertible bonds . . . . . . . . . . . . . . . . . . . . 31.1.2 Additional features . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Pricing models . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Convertible bonds pricing . . . . . . . . . . . . . . . . . . . . 8

1.3 Structure of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Background of Mathematics 122.1 The Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Derivation of the PDE . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Stochastic volatility or/and interest rate model . . . . . . . . . . . . 172.2.1 PDE for the Heston model . . . . . . . . . . . . . . . . . . . . 172.2.2 PDE for the CIR interest rate model . . . . . . . . . . . . . . 202.2.3 PDE for the hybrid stochastic volatility and interest rate model 222.2.4 Boundary conditions along the direction of the volatility and

interest rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Monte-Carlo method . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Finite difference method . . . . . . . . . . . . . . . . . . . . . 282.3.3 Binomial tree pricing method . . . . . . . . . . . . . . . . . . 302.3.4 Predictor-corrector method . . . . . . . . . . . . . . . . . . . 322.3.5 Alternating direction implicit method . . . . . . . . . . . . . . 33

2.4 Integral equation method and Fourier transform . . . . . . . . . . . . 36

3 Pricing puttable convertible bonds with integral equation approaches 403.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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CONTENTS vii

3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Integral equation formulations of puttable convertible bond . . . . . . 45

3.3.1 First integral equation formulation of puttable convertible bond 453.3.2 Second integral equation formulation for puttable convertible

bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4 The numerical implementation . . . . . . . . . . . . . . . . . . . . . . 513.5 Examples and discussions . . . . . . . . . . . . . . . . . . . . . . . . 553.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Pricing callable-puttable convertible bonds with an integral equation ap-proach 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Models and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 The numerical implementation . . . . . . . . . . . . . . . . . . . . . . 824.4 Numerical results and discussions . . . . . . . . . . . . . . . . . . . . 854.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 Pricing resettable convertible bonds with an integral equation approach 965.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2 Model set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Integral equation representation . . . . . . . . . . . . . . . . . . . . . 1015.4 Numerical schemes and the results . . . . . . . . . . . . . . . . . . . 1055.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Pricing convertible bonds under stochastic volatility or interest rate 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Pricing convertible bonds with stochastic volatility . . . . . . . . . . 117

6.2.1 The PDE system under the Heston model . . . . . . . . . . . 1186.2.2 Discretize the PDE system . . . . . . . . . . . . . . . . . . . . 1196.2.3 Numerical scheme for the prediction step . . . . . . . . . . . . 1246.2.4 Numerical scheme for the correction step . . . . . . . . . . . . 1256.2.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 126

6.3 Pricing convertible bonds with stochastic interest rate . . . . . . . . . 1326.3.1 The PDE system and its numerical scheme . . . . . . . . . . . 1336.3.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 139

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

CONTENTS viii

7 Summary and Conclusion 143

Bibliography 145

A Appendix for Chapter 3 154A.1 Appendix A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154A.2 Appendix A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157A.3 Appendix A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A.4 Appendix A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

B Appendix for Chapter 4 170B.1 Appendix B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170B.2 Appendix B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171B.3 Appendix B.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173B.4 Appendix B.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.5 Appendix B.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178B.6 Appendix B.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180B.7 Appendix B.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C Appendix for Chapter 5 187C.1 Appendix C.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187C.2 Appendix C.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188C.3 Appendix C.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191C.4 Appendix C.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194C.5 Appendix C.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

D Appendix for Chapter 6 198D.1 Appendix D.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198D.2 Appendix D.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200D.3 Appendix D.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200D.4 Appendix D.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201D.5 Appendix D.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202D.6 Appendix D.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

List of Figures

2.1 A simple example of the binomial tree . . . . . . . . . . . . . . . . . 32

3.1 The value of the optimal boundaries for three different conversion ratios 553.2 The price of the puttable CB at four different time moments . . . . . 563.3 The price of the puttable and vanilla CBs at the same time . . . . . . 573.4 The price of puttable CBs for three different volatilities . . . . . . . . 583.5 Optimal boundaries prices for three different volatilities . . . . . . . . 583.6 The price of puttable CBs for three different risk-free interest rates . 593.7 Optimal boundaries prices for three different risk-free interest rates . 59

4.1 The value of the optimal exercise boundaries for three different valuesof the conversion ratios. . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 The optimal exercise prices for three cases . . . . . . . . . . . . . . . 874.3 The bond price for different time moments. . . . . . . . . . . . . . . . 894.4 The bond prices of three cases. . . . . . . . . . . . . . . . . . . . . . 904.5 The prices of CBs, PCBs and CPCBs. . . . . . . . . . . . . . . . . . 914.6 Comparison by three different volatilities. . . . . . . . . . . . . . . . 924.7 Comparison by three different interest rates.. . . . . . . . . . . . . . . 94

5.1 The optimal conversion price of RCB and also two vanilla CBs. . . . 1085.2 The bond price of RCB and also two CBs at τ=0.05. . . . . . . . . . 1095.3 The bond price of RCB and also two CBs at τ=0.50. . . . . . . . . . 1105.4 The bond price of RCB and also two CBs at τ=0.95. . . . . . . . . . 1105.5 The bond price of RCB at three moments. . . . . . . . . . . . . . . . 1115.6 The bond price of RCB. . . . . . . . . . . . . . . . . . . . . . . . . . 1125.7 The conversion boundary price of RCB. . . . . . . . . . . . . . . . . 1125.8 The bond price of RCB. . . . . . . . . . . . . . . . . . . . . . . . . . 1135.9 The conversion boundary price of RCB. . . . . . . . . . . . . . . . . 114

6.1 The comparison of the optimal conversion boundary obtained by ourmethod and that from the integral equation method [109], at v = 0.1. . 128

ix

LIST OF FIGURES x

6.2 The comparison of the bond prices obtained by our method and thosefrom the Monte Carlo method, at t = 0 and v = 0.4. . . . . . . . . . . 129

6.3 The optimal conversion price. . . . . . . . . . . . . . . . . . . . . . . 1306.4 The optimal conversion price. . . . . . . . . . . . . . . . . . . . . . . 1306.5 The optimal conversion price. . . . . . . . . . . . . . . . . . . . . . . 1316.6 The bond price at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 1316.7 The bond price at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 1326.8 The optimal conversion price. . . . . . . . . . . . . . . . . . . . . . . 1396.9 The optimal conversion price. . . . . . . . . . . . . . . . . . . . . . . 1406.10 The optimal conversion price. . . . . . . . . . . . . . . . . . . . . . . 1406.11 The bond price at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 1416.12 The bond price at τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 142

List of Tables

2.1 Stochastic interest rate models . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Convergency test of the Binomial tree method . . . . . . . . . . . . . 533.2 Accuracy and efficiency test of IE method . . . . . . . . . . . . . . . 543.3 Accuracy and efficiency test of IE method . . . . . . . . . . . . . . . 54

4.1 Accuracy and efficiency test of IE method . . . . . . . . . . . . . . . 85

5.1 MC method vs IE method . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1 Convergence test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xi

Chapter 1

Introduction and Literature review

1.1 Convertible bondsIn finance practice, a derivative is a contract whose value is dependent on the priceof an underlying asset; this underlying asset could be a stock, index, or interest rate.The aims of issuing financial derivatives are to insure against price movements forhedging, increasing exposure to price movements for speculation or getting accessto otherwise hard-to-trade assets or markets. Among these, the use for hedgingpurposes is the most important one, as the management of different types of riskcaused by the price movement is always an ongoing topic for investors. Due to this,financial derivatives are becoming increasingly popular, ever since the establishmentof the Chicago Board Options Exchange in 1973.

Financial derivatives can be classified into two main types according to how theyare traded, i.e., exchange-traded and over-the-counter (OTC) derivatives. Whileexchange-traded derivatives, that is those that are traded on exchange markets, arestandardized financial instruments and traded through a recognized intermediary,the OTC derivatives, representing those traded directly between two parties withoutgoing through any intermediary, provide more flexibility as these contracts can becustomized according to the demand of the two parties. In fact, the OTC derivativemarket is actually the largest market for derivatives.

Among all the financial derivatives, convertible bonds have attracted a lot of at-tention and their trading volumes have experienced rapid growth. A convertiblebond is actually one of the hybrid financial instruments, combining the attributes offixed-income securities and equities. For the simplest convertible bond, the holderreceives fixed-rate coupon payments as if holding a classical bond when the conver-sion has not been exercised, and he/she is also entitled to convert the bond intoa predetermined number of stocks to maximize his/her benefit. Of course, moresophisticated convertible bond contracts with different embedded options and trig-

1

CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW 2

gering conditions have been established to cater for the different kinds of demandfrom issuers as well as investors. This has contributed to the rapid development ofthe convertible bond market.

This hybrid nature of convertible bonds actually favours a vast class of investorswho want to gain more return than that provided by the classical bond while at thesame time avoiding the high risk involved in directly holding the underlying stocks.The convertible bond also provides an opportunity for the investors to participatein both the fixed-income and equity market. On the other hand, the issuer canalso take advantage of convertible bonds in the sense that selling convertible bondsenables the issuer to receive more payments compared with selling classical bonds.More importantly, this actually helps reduce cash interest payments, since once thebonds are converted into the stocks, the issuers no longer need to pay anything.

Although this particular hybrid nature of convertible bonds has made them ex-tremely useful to both the issuer and investor, it has also led to a much more com-plicated pricing problem. Even with the simplest American-style convertible bonds,there is an additional optimal conversion boundary. This arises from the right ofthe holder to choose whether the bond is to be converted or not, which needs tobe determined simultaneously with the bond price. When additional clauses areintroduced into convertible bonds, the corresponding pricing problem can becomeeven more complex.

Because of the tremendous complexity and diversity of convertible bonds, findingsuitable mathematical tools to derive the solution of the pricing problem is very im-portant and crucial. Although a large amount of research interest has been directedinto this area, most of the developed approaches for the pricing of convertible bondswith additional features are numerical ones, even when the simplest Black-Scholesmodel is adopted. There are few existing results on how to price convertible bondswhen the Black-Scholes model is unable to capture the features of the underlyingasset. Thus, a more sophisticated model, such as stochastic volatility and stochasticinterest rate models, must be adopted.

The aim of this thesis can be summarized from two aspects. On one hand, analyt-ical pricing formulae for convertible bonds with different additional features underthe Black-Scholes are presented, using the integral equation approach. This integralequation approach is superior to numerical methods because errors are usually in-troduced at very early stage of computation when numerical methods are adoptedand these methods often suffer from inefficiency problems, making them difficultto apply in practice. On the other hand, when stochastic volatility and stochasticinterest rate models are adopted, analytical approaches are no longer possible dueto the additional dimension introduced by the newly added stochastic source. Toovercome these difficulties we establish an efficient and accurate numerical approach


for the pricing of convertible bonds.Before we move to conduct a review of the literature, several important and

common types of convertible bonds need to be introduced, the details of which arepresented in the next two subsections.

1.1.1 Vanilla convertible bonds

The most classical and simplest type of convertible bonds is the so-called standard(vanilla) convertible bond, which forms the basis of other more complex convertiblebonds. The vanilla convertible bond can be treated as a straight bond plus a calloption, and it gives the right to the holder to convert the bond into a preset number,which is named as the conversion ratio, of the stocksa. This means that the holderof the bond can choose to receive the face value of the bond or a certain number ofthe stocks at expiry.

It should be pointed out that similar to the option contract, there are also twodifferent styles, according to whether the early exercise of the conversion is allowedor not. While a European-style convertible bond only allows the holder to convertthe bond at expiry, the holder of an American-style one can convert the bond atany time during the lifetime of the bond. The latter case is much more complicatedthan the former one, as the pricing of the American-style convertible bond is actuallya free boundary problem, where an unknown optimal conversion boundary of theholder always needs to be determined together with the bond price. This addsanother degree of complexity to the corresponding pricing problem.

1.1.2 Additional features

As mentioned above, some additional features can be incorporated to formulatenon-standard convertible bonds according to practical demand. In the following,some important and widely used features are illustrated. It should be noted thatsimilar to the conversion feature, all the features introduced below can also be setas European-style or American-style, and this will be omitted when discussing thesefeatures.

Call feature

This is the right that enables the issuer to recall (repurchase) the bond at the pre-determined call price. In practice, there are two styles of the call feature. The firstallows the issuer to recall the bond at the call price after a certain date without

aIt should be remarked here that in the real market, the contract often specifies the conversionprice instead of the conversion ratio, and there is also the relationship between these two values,i.e., Face value = Conversion ratio × Conversion price.


imposing any other conditions, the so-called “hard-call” feature. The second one,named as “soft-call” feature, entitles the issuer to recall the bond when the un-derlying asset price satisfies a typical condition specified in the contract, a typicalexample of which is that the underlying price exceeds 120% of the conversion pricefor 15 days out of past consecutive 30 days. One can clearly observe that such a callfeature actually protects the benefit of the issuer, and thus the formulated callableconvertible bond would be worth less than the corresponding vanilla convertiblebond.

It should be remarked here that although the soft-call convertible bonds have beenpopular in the Chinese market in recent years, very few researchers have consideredthe corresponding pricing problems. A very recent work produced by Ma et. al. [81]

evaluated the contract using the two-factor willow tree method. In fact, this soft-call feature can be treated as the moving window Parisian feature (see [47]), whichis much more complicated than the Parisian options considered in [108]. This is notonly because of the early exercise feature but also the moving window, which isintroduced by the fact that we always need to consider past consecutive days to seeif the call feature is activated or not. Such a complicated problem is left for futureresearch.

Put feature

This is the right that allows the holder to sell the bond back to the issuer at theput price listed in the contract. Similar to the call feature introduced above, theput feature can also be classified into two different types, i.e., the hard-put and thesoft-put feature. While the former gives the right to the holder to sell the bond atthe put price before a certain date without any other conditions, the latter can onlybe activated if the underlying asset price satisfies a typical condition, such as theunderlying price staying below 80% of the conversion price for 20 days out of pastconsecutive 30 days. Since this feature is in favor of the holder, it is expected thatthe value of the formulated puttable convertible bond would be higher than that ofthe corresponding vanilla convertible bond.

It should be pointed out that the call/put feature embedded in the convertiblebond and the call/put options are very different. In fact, all the convertible bondscan be treated as different types of call options, and thus there is no put-call parityfor convertible bonds.

Reset feature

With the reset feature, the conversion ratio/price can be reset to a new value de-pending on the evolution of the underlying asset price, and this is usually used in


the case that the conversion price will be decreased if the underlying asset does notperform well for a certain period. For example, the conversion price will be replacedby the current underlying price when the underlying price is below 60% of the con-version price after a year. This is not a right given to the holder, but it is beneficialto the holder as it reduces the cash flow of the issuer.

Contingent conversion

As is well known, there is a conversion price associated with all the convertiblebonds, which is the amount that the holder needs to pay for each share of the stockwhen they chose to convert the bond into stocks. Being different from most of theconvertible bonds, contingent convertible bonds, proposed by Flannery [45], do notspecify the conversion price in the contract, and instead the actual stock price atconversion acts as the conversion price. Moreover, the conversion is no longer theright of the holder, but will take place automatically, once a pre-specified eventleading to the conversion process called trigger happens. For example, if the currentstock price of the issuer is $100, the contingent clause would be that the bond willbe directly converted into stocks when the stock price falls below $50. It has beenwidely acknowledged that contingent convertible bonds have the potential to preventsystematic collapse of important financial institutions [1], and a bankruptcy can befully prevented because of fast input of capital coming from the conversion.

Non-dilutive feature

The non-dilutive feature can be realized by the issuer through selling a standardconvertible bond and hedging it through purchasing call options on its own stockwith the same notional amount and maturity as specified in the convertible bondcontract. This will cancel out the dilution in the case of a conversion taking place.The contract would often restrict the possibility of early conversion to fully preventdilution. This has been introduced in the environment of lower interest rates, fromwhich the issuers already benefit in the straight bond market, to encourage theissuance of convertible bonds.

1.2 Literature review

1.2.1 Pricing models

In 1973, Black & Scholes [9] and Merton [85] made a great contribution by propos-ing the so-called Black-Scholes model (B-S model) or the Black-Scholes-Mertonmodel, assuming that the underlying asset price follows a geometric Brownian mo-


tion (GBM). This model is very popular as it could lead to simple pricing formulaefor various important financial derivatives, such as European and barrier options,and thus it is still widely used in today’s finance markets.

However, some simplified assumptions, such as the constant interest rate andvolatility, made in the B-S model are not consistent with real market observations.In particular, the distribution of the underlying asset price is usually asymmetricand exhibits the features like skewness [92] and fat-tails [94], which are at odds withthe assumption of the GBM. On the other hand, the implied volatility extractedfrom the real market option prices always shows a “smile” curve [40], which violatesthe assumption of the constant volatility in the B-S model.

There are two different types of modifications to the B-S model; the first is toreplace the standard Brownian motion with another stochastic process, while theother is to add the stochastic interest rate or/and the stochastic volatility into theB-S model. Levy processes are a typical example included in the former category,and they are very popular because they also possess the property of the independentand stationary increments as the Brownian motion does. For example, Merton [86]

considered a jump-diffusion model by utilizing a Guassian distribution to modeljumps of log-returns, while the Variance-Gamma model and CGMY model wererespectively proposed by Madan [82] and Carr, et. al. [19], both of which are infiniteactivity Levy processes. Of course, apart from the Levy processes, there are alsomany other stochastic processes that have been applied to replace the Brownianmotion, such as the fractional Brownian motion used by Necula [87] to capture thelong range dependence in asset prices [101].

On the other hand, the relaxation of the constant interest rate or volatility as-sumptions in the B-S model has also received a lot of attention. Thus, these twoapproaches will be elaborated in the following.

Stochastic interest rate models

In the context of the stochastic interest rate, several short-rate models have beenestablished to describe the future evolution of the short rate. These models can bemainly divided into two main categories, depending on how many stochastic factorsare used to determine the process of the interest rate.

The first category is the so-called one-factor short-rate model, where the interestrate is controlled by a single Brownian motion. For example, the Merton model [85]

assumes that the stochastic interest rate follows a Gauss-Wiener process, while theVasicek model [100] adopts an Ornstein-Uhlenbeck stochastic process for the interestrate. Other well-known models include the Rendleman-Bartter model [95], usinganother GBM for the interest rate, the Cox-Ingersoll-Ross model (CIR model) [29],which is defined as a sum of squared Ornstein-Uhlenbeck processes, the Ho-Lee


model [53], defining the stochastic interest rate as a normal process, the Hull-Whitemodel [56], which extends the Vasicek model by allowing the model parameters tobe time dependent, the Black-Karasinski model [8], a log-normal version of the Hull-White model, and the Kalotay-Williams-Fabozzi model [66], which is a log-normalanalogue to the Ho-Lee model.

On the other hand, multi-factor short-rate models, under which the stochasticinterest rate is controlled by two or more Brownian motions, have also been pro-posed to provide more flexibility in fitting real market data. The Longstaff-Schwartzmodel [78] and the Chen model [24] are examples for the two-factor model and thethree-factor model, respectively. A general framework of multi-factor interest ratemodels are established in [72], based on the assumption that the term structure ofinterest rates is “embedded in a large macroeconomic system”.

Apart from the short-rate models, another framework to incorporate the stochasticinterest rate is the Heath-Jarrow-Morton framework (HJM) [51]. In fact, the modelunder the HJM framework is very different from the short-rate models mentionedabove, since the HJM-type models are able to capture the full forward rate curve,while the short-rate ones only yield a point on the rate curve.

Local and stochastic volatility models

Due to the phenomenon of the “volatility smile”, another popular approach in mod-ifying the B-S model is to add a non-constant volatility. A natural choice for thisis to make the volatility a deterministic function of the underlying asset price andtime, formulating the local volatility model [35,41]. The constant elasticity of variance(CEV) model [27] can be treated as a local volatility model, which is used by markettraders in the finance practice, especially for modeling equities and commodities.

On the other hand, an alternative approach belonging to this category is to makethe volatility of the underlying asset price another random variable, formulating thestochastic volatility (SV) models. In particular, the volatility is assumed to follow alog-normal distribution under the Hull-White model [55], while the SABR volatilitymodel proposed in [49] proves to be able to reproduce the smile effect of the volatilitysmile [57]. Moreover, the popular Heston model [52] adopts the CIR process for thevariance of the underlying asset price, while a similar 3/2 model has recently gained alot of attention due to its attractive features [20], which assumes that the randomnessof the variance process varies with v3/2

t . Being different from the Heston and 3/2models, the Generalized Autoregressive Conditional Heteroskedasticity (GARCH)model [10] assumes that the randomness of the variance process varies with vt .


1.2.2 Convertible bonds pricing

It should be pointed out that the pricing of American-style financial derivatives isusually a very challenging problem [54,79] as a result of the inherent characteristic ofthese contracts, that they can be exercised at any time before the expiry. Thus, foran American-style convertible bond, the buyer has an additional right to convertthe bond into stocks earlier during the life of the contract, which formulates thecorresponding pricing problem as a free boundary problem. This means that theoptimal conversion price should be determined together with the bond price in thesolution procedure. Mathematically, the unknown domain of the solution, resultedfrom this additional right, makes the pricing problem highly nonlinear and thusdifficult to solve.

One of the earliest work for pricing convertible bonds was produced by Inger-soll [62], who took the firm value as the underlying variable under the B-S frame-work and derived a closed-form pricing formula for some special cases using theno-arbitrage theory. This approach was further extended in [15,73] to consider morecomplex situations. However, these results were very restrictive as they did not in-corporate the possibility of early conversion, which is a common feature of convert-ible bonds in practice. Brennan & Schwartz [12] also worked under a firm-value basedB-S model, but adopted a finite difference method for the pricing of American-Styleconvertible bonds. However, considering that firm values are usually not observablein real markets, this model is not suitable to be applied in practice as far as modelcalibration is concerned. Thus, just a few year later, McConnel & Schwartz [84]

modified the approach to propose a single-factor pricing model for a zero-couponconvertible bond with the stock price, which is available in real markets, beingselected as the underlying variable.

Since then, a large amount of research interest has been led into the pricingof American-style vanilla convertible bonds under the stock-value based B-S frame-work. Of course, numerical approaches, such as the binomial tree method [31], MonteCarlo simulation [79], finite difference method [98] and finite volume method [112], canbe adopted. However, numerical methods often suffer from inaccuracy and in-efficiency problems, which hinder their potential applications in finance practice.Therefore, a large number of authors have been focusing on searching for analyticalsolutions. In particular, Zhu [106] presented an analytical solution in the form of aTaylor series expansion for the American-style vanilla convertible bond, using theHomotopy Analysis Method (HAM) [74], and this approach was further extended byChan & Zhu [23], who successfully derived an approximation solution for the price of aconvertible bond under the regime-switching model. An alternative semi-analyticalapproach that is often employed for financial derivative pricing is the integral equa-


tion method, and it has already been applied by Zhu & Zhang [111] to formulate anintegral equation representation for vanilla convertible bond prices, following Kim’sapproach [68].

As mentioned above, the B-S model is sometimes inadequate to capture the maincharacteristics exhibited by the underlying asset price, and thus various more sophis-ticated models have been proposed to modify the B-S model. In terms of pricing theconvertible bonds, stochastic interest rate models are among the most popular onessimply due to the nature of long lifetime for most of the convertible bonds in realmarkets. However, due to the additional stochastic source introduced, the analyti-cal solution is only available in very special cases, a typical example of which is theclosed-form pricing formula for a simple European-style convertible bond presentedby Nyborg [88], who assumed that the holder can only convert the bond at matu-rity. When it comes to pricing American-style convertible bonds under stochasticinterest rate models, numerical methods must be resorted to. For example, thefinite difference method has been adopted by Brennan & Schwartz [12], Ayache et,al. [5] and Andersen & Buffum [4] among many others to value convertible bonds,while Barone-Adesi et, al. [6] priced these contracts using the method of character-istics together with finite elements. Binomial/trinomial trees are also very popularin pricing convertible bonds with stochastic interest rates, and they have been dis-cussed by a number of different authors, including Takahashi et, al. [97], David &Lischka [34], Hung & Wang [59], Carayannopoulos & Kalimipalli [18] and Chambers &Lu [21]. Recently, Lin & Zhu [77] established a predictor-corrector method embeddedwith an ADI scheme for the pricing of convertible bonds when the interest rate orvolatility is assumed to be stochastic, the results of which are presented in Chapter6.

Nowadays, due to the different kinds of demands in practice, many useful clauseshave been introduced into the vanilla convertible bond, examples of which includethe call feature, put feature and reset feature, formulating different types of convert-ible bonds. For example, incorporating the call feature yields the so-called callableconvertible bonds. Brennan & Schwartz [12] was believed to be the first to discussthe pricing problem of these contracts in theory, while the corresponding solutionswere derived in their later article using the finite difference method [13]. Later on,the binomial tree method was also applied to obtain the price of a callable convert-ible bond by Bernini [7], while Yagi & Sawaki [103] considered the pricing problem ofcallable convertible bonds using the game option defined in [67]. On the other hand,when the put feature is taken into consideration, Nyborg [88] presented the boundaryconditions for the formulated puttable convertible bonds, while Lvov et al. [80] andAmmann et. al. [3] numerically solved the pricing problem by using Monte Carlosimulations. Recently, an integral equation formulation for the puttable convertible


bond price was presented by Zhu et, al. [109], which forms the main context of Chap-ter 3. Lin & Zhu [76] went even further to apply the integral equation approach forconvertible bond pricing when both call and put features are available, with detailsprovided in Chapter 4. Finally, the pricing of resettable convertible bonds has notbeen investigated until very recently, and there are only few numerical results on thistopic [43,70]. However, the resettable convertible bonds are gaining attention fromboth market practitioners and academic researchers, because they are now widelyused in the finance industry. To cope with the demand for the accurate and efficientpricing of resettable convertible bonds, Lin & Zhu [75] developed an integral equationrepresentation for the price of this relatively new type of convertible bonds, and thiswill be illustrated in Chapter 5.

1.3 Structure of thesisIn Chapter 2, we introduce some basic mathematical knowledge needed for pricingdifferent kinds of American-style convertible bonds under various models. We startby recalling different models used to describe the underlying dynamics, and derivethe PDEs accompanied by appropriate boundary conditions for convertible bondpricing. We also review some numerical methods and the integral equation approach,with some examples presented to make these approaches better understood. Someof these techniques will be extended to price different types of convertible bondsunder various models in later chapters.

In Chapter 3, we adopt an integral equation approach to price American-styleputtable convertible bonds. Upon applying an incomplete Fourier transform, anintegral equation representation for the bond price is derived under the B-S model.To avoid numerical difficulty caused by the discontinuity along both free boundariesas well as the involvement of two first-order derivatives of the unknown optimalexercise prices in this formulation, a second integral equation formulation is furtherpresented, after some manipulations of the first form. The effect of the put featureis investigated through numerical examples.

In Chapter 4, the call feature is further introduced into the American-style put-table convertible bonds to formulate the so-called callable-puttable convertible bonds,which makes the corresponding pricing problem even more complicated due to thetangled presence of callability, puttability, as well as conversion. Such complexitycan be further shown when mathematically solving this problem, which involves thediscussion of various different scenarios. Different integral equation formulationsare presented by solving different PDE systems, and various properties of callable-puttable convertible bonds are numerically demonstrated.

In Chapter 5, we propose an integral equation approach for pricing American-style


resettable convertible bonds under the B-S model. This is a challenging problembecause an unknown optimal conversion price needs to be determined together withthe bond price. There is also the additional complexity that the value of the con-version ratio will change when the underlying price touches the reset price. Despitethese difficulties, we still manage to present an integral equation formulation for thebond price, after successfully establishing the governing PDE system. The bondprice turns out to be a non-monotonic increasing function of the underlying price,which is a unique feature that distinguishes it from other types of convertible bonds.

In Chapter 6, we move out of the B-S framework by allowing the volatility or inter-est rate to be stochastic when pricing American-style convertible bonds. The newlyintroduced stochastic source leads to a two-dimensional free boundary problem, forwhich a predictor-corrector scheme with the Douglas-Rachford method embeddedin the correction step has been developed to solve the corresponding pricing PDEsystem. Numerical results are used to validate our approach as well as to show theinfluence of stochastic volatility and interest rate on the bond prices and the optimalconversion boundary.

Some concluding remarks are provided in Chapter 7 to summarize the main resultspresented in this thesis.

Chapter 2

Background of Mathematics

In this chapter, we introduce some mathematical models and techniques, which arethe foundation of and will be utilized in this thesis.

In particular, some popular models for the underlying asset price when pricing theconvertible bond are firstly discussed. The details on how to obtain the PDE gov-erning the convertible bond prices from the stochastic differential equation (SDE) ofthe underlying asset price will then be illustrated, after which appropriate boundaryand terminal conditions are given to formulate closed PDE systems. In addition,some mathematical techniques, including different numerical and analytical meth-ods, are further investigated, which are the main tools in obtaining the value of theconsidered bond. It should be pointed out that these techniques can of course beused for pricing other financial derivatives, as long as their prices solve the similarPDE system as we discuss here.

2.1 The Black-Scholes modelOne of the most popular and classical mathematical models to price the convert-ible bond is the so-called Black-Scholes model, which was established by Black &Scholes [9] and Merton [85] in 1973. It was initially proposed for evaluating options,and it can be regarded as the foundation of the financial derivative pricing theory.In particular, they assumed that the underlying price, St , should satisfy a GBM:

dSt = µStdt +σStdWt , (2.1)

where µ is the drift term, σ is the constant volatility and Wt is a standard Brownianmotion. The B-S model was firstly applied for the pricing of convertible bondsby Ingersoll [61] and Brennan & Schwartz [12] in 1977, by taking the firm value asthe underlying asset. However, this is not appropriate as the value of the firmis very difficult to obtain in the real market, and thus McConnel & Schwartz [84]

12

CHAPTER 2. BACKGROUND OF MATHEMATICS 13

improved this method, by replacing the firm value with the stock price, which is moreobservable. Before we are able to derive the PDE governing the convertible bondprices, several important assumptions made under the B-S model should be pointedout. Firstly, there exists a risk-free asset, such as a bank account, whose valueaccumulates with a continuously compounded risk-free interest rate r. Secondly,the financial market is perfect in the sense that the market trading is continuouswithout transaction costs. Thirdly, short selling is permitted and all securities areperfectly divisible. Fourthly, there are no arbitrage opportunities, and all derivativescan be perfectly hedged with the underlying price and bank deposit. Under theseassumptions, two methods that are equivalent to each other will be presented toderive the PDE for the bond prices from the SDE for the underlying asset price.

2.1.1 Derivation of the PDE

Let us begin with the Ito’s Lemma.

Lemma 2.1.1 (Ito’s Lemma). Assume Xt is a random variable, and satisfies thefollowing SDE

dXt = A(X , t)dt +B(X , t)dWt , (2.2)

then any twice differentiable scalar function F(X , t) follows the stochastic dynamicsas follows

dF = B(X , t)∂F∂X

dWt +(∂F∂ t

+A(X , t)∂F∂X

+12

B2(X , t)∂ 2F∂X2 )dt. (2.3)

If V (St , t) denotes the value of the convertible bond, and the underlying asset priceSt satisfies the SDE (2.1), applying Ito’s Lemma yields

dV (S, t) = σS∂V∂S

dW +(∂V∂ t

+µS∂V∂S

+12

σ2S2 ∂ 2V∂S2 )dt. (2.4)

The first method is the so-called martingale method (or the risk neutral pricingprinciple), which states that the discounted asset price is a martingale, i.e.,

E[e−rT ST |St ] = e−rtSt , ∀ T > t, (2.5)

andE[e−rTV (ST ,T )|St ] = e−rtV (St , t), ∀ T > t. (2.6)

If one applies (2.5), it is not difficult to find that µ in (2.1) should be replaced byr. Equation (2.6) further implies

E[d(e−rtV (St , t))|St ] = 0, ∀ t > 0. (2.7)


and thus we can obtain

E[−re−rtV (St , t)dt + e−rtdV (St , t)|St ] = 0

⇒ ∂V∂ t

+12

σ2S2 ∂ 2V∂S2 + rS

∂V∂S

− rV = 0. (2.8)

The other method is the hedging method, where we construct a self-financingportfolio consisting of a bond and −∆ shares of the underlying asset. In this case,the value of this portfolio is

Π =V −∆S, (2.9)

from which we havedΠ = dV −∆dS. (2.10)

Again, using Ito’s Lemma leads to

dΠ = σS∂V∂S

dW +(∂V∂ t

+µS∂V∂S

+12

σ2S2 ∂ 2V∂S2 )dt −∆(µSdt +σSdW )

= σS(∂V∂S

−∆)dW +(∂V∂ t

+µS∂V∂S

+12

σ2S2 ∂ 2V∂S2 −∆µS)dt. (2.11)

To make the portfolio a risk-free asset, any stochastic term should be eliminated.Therefore, setting

∆ =∂V∂S

, (2.12)

yields

dΠ = (∂V∂ t

+µS∂V∂S

+12

σ2S2 ∂ 2V∂S2 − ∂V

∂SµS)dt

= (∂V∂ t

+12

σ2S2 ∂ 2V∂S2 )dt. (2.13)

On the other hand, since Π is a risk-free asset, we have

dΠ = rΠdt = r(V −S∂V∂S

)dt. (2.14)

Then, we obtain

∂V∂ t

+12

σ2S2 ∂ 2V∂S2 = r(V −S

∂V∂S

)

⇒ ∂V∂ t

+12

σ2S2 ∂ 2V∂S2 + rS

∂V∂S

− rV = 0. (2.15)

Actually, there is a famous theorem, the Feynman-Kac theorem, that establishesthe relationship between the SDE of the underlying asset price and the PDE of thefinancial derivative price. The content of this theorem is provided below.


Theorem 2.1.1 (Feynman-Kac theorem) If the underlying asset price St satisfies theSDE

dSt = µ(S, t)Stdt +σ(S, t)StdWt , (2.16)

and the financial derivative price V (S, t) solves the PDE∂V∂ t

(S, t)+µ(S, t)∂V∂S

(S, t)+12

σ2(S, t)∂ 2V∂S2 (S, t)−u(S, t)V (S, t)+ f (S, t) = 0,

V (S,T ) = ϕ(S),(2.17)

for all S ≥ 0 and t ∈ [0,T ], the solution of V (S, t) can be written as a conditionalexpectation

V (S, t) = EQ[∫ T

te−

∫ Tt u(Sτ ,τ)dτ f (Sr,r)dr+ e−

∫ Tt u(Sτ ,τ)dτϕ(ST )|St = S]. (2.18)

Remark: The known functions, µ(S, t), σ(S, t) and u(S, t), should satisfy the followingconditions, respectively.

µ(S, t) : PQ[∫ t

0|µ(Sτ ,τ)|dτ] = 1, ∀ 0 ≤ t ≤ ∞, (2.19)

σ(S, t) : PQ[∫ t

0|σ2(Sτ ,τ)|dτ] = 1, ∀ 0 ≤ t ≤ ∞, (2.20)

u(S, t) : R× [0,T ]→ R. (2.21)

2.1.2 Boundary conditions

In this subsection, the boundaries conditions as well as the terminal condition areset up to close the pricing PDE system. Firstly, the terminal condition can be easilygiven with the payoff of the convertible bond:

V (S,T ) = maxnS,Z, (2.22)

where n is the conversion ratio, which is the amount of stocks the holder can obtainwhen he/she converts the bond, and Z is the face value of the bond. When the priceof the stock, S, approaches zero, it is almost impossible for the holder to convert theconvertible bond into stocks with such a low price. Thus, in this case, the holderwill choose to hold the bond until the expiry and receive the face value, implyingthat the bond price is actually the discounted face value, i.e.,

V (0, t) = Ze−r(T−t). (2.23)

Other boundary conditions should be considered separately, as they can be differ-ent and should be determined according to their style, the European-style or the


American-style. For the European-style convertible bond, the second boundary con-dition should be specified when S approaches infinity, and its value is equal to thatof the bond if it was converted into the stocks right now. This is because if the un-derlying asset price is high enough, the convertible bond will definitely be convertedat expiry, which means that the holder will be receiving n shares of stocks. In thiscase, we have

limS→∞

V (S, t) = nS. (2.24)

In contrast, when an American-style convertible bond is taken into consideration,there will be an unknown optimal conversion boundary, S f (t), that needs to beconsidered, as a result of the holder being entitled to the right to convert the bondbefore the expiry time. In particular, when the underlying asset price is higherthan the optimal conversion boundary, the bond should be converted immediately,otherwise the holder is willing to wait until the time to expiry to receive the facevalue, since in this case the value of the obtained stocks after early conversion is lessthan the value of the contract. Thus, we should have

V (S f (t), t) = nS f (t), (2.25)

which is accompanied by a smooth pasting condition

∂V∂S

(S f (t), t) = n. (2.26)

In this case, the PDE systems for the vanilla European-style and American-styleconvertible bond prices under the B-S model can be written as

∂V∂ t

+12

σ2S2 ∂ 2V∂S2 +(r−D0)S

∂V∂S

− rV = 0,

V (S,T ) = maxnS,Z,V (0, t) = Ze−r(T−t),

limS→∞

V (S, t) = nS,

(2.27)

and

∂V∂ t

+12

σ2S2 ∂ 2V∂S2 +(r−D0)S

∂V∂S

− rV = 0,

V (S,T ) = maxnS,Z,V (0, t) = Ze−r(T−t),

V (S f (t), t) = nS f (t),∂V∂S

(S f (t), t) = n,

(2.28)

respectively.


2.2 Stochastic volatility or/and interest rate modelSince the B-S model is too simple to capture the main characteristics exhibited byreal market data, various modifications have been proposed, among which stochasticvolatility and stochastic interest rate models have received a lot of attention. Inparticular, a list of popular stochastic interest rate models [22] is presented in Table2.1.

Table 2.1: Stochastic interest rate models

Merton dr = αdt +σdWVasicek dr = (α +β r)dt +σdWCIR SR dr = (α +β r)dt +σr1/2dWDothan dr = σrdWGBM dr = β rdt +σrdW

CIR VR dr = σr3/2dWCEV dr = β rdt +σrγdW

In summary, all of the models in this table can be written as: dr = κ(η + r)dt +

σrαdW ; choosing different values for these parameters, κ , η and α , would yielddifferent models. Similarly, popular stochastic volatility models can also be generallyexpressed as dv= a(β +v)dt+θvλ dW (here, for illustration purposes, we deliberatelychoose different parameters for volatility and interest rate processes). Furthermore,combing these two classes will give rise to hybrid stochastic volatility and interestrate models. Having the knowledge of the SDE under stochastic volatility or/andinterest rate models, we are now ready to derive the PDE for the bond prices underthese models. For simplicity, we will pick one model in each class to show the detailson how to derive the PDE.

2.2.1 PDE for the Heston model

Under the Heston model, the underlying asset price is assumed to follow the dynamic

dSt = µStdt +√

vtStdWt , (2.29)

where µ is the drift term and Wt is a standard Brownian motion. vt representsstochastic volatility, satisfying

dvt = θ(ω − vt)dt +ξ√

vtdBt , (2.30)

where θ , ω and ξ are the mean reversion speed, the long-term mean and the volatil-ity of the volatility, respectively. Bt is also a standard Brownian motion, and it iscorrelated with Wt with the correlation ρ . To derive the PDE, the hedging method


is used, involving the construction of a self-financing portfolio. However, it shouldbe noted that unlike the B-S model, where only one random variable is involved,there are two stochastic sources in this model. Thus, the self-financing portfolio,Π, should not only contain the bond, V (S,v, t), and −∆1 shares of the underlyingassets, it should also have −∆2 shares of another bond U(S,v, t). In other words,

Π =V −∆1S−∆2U. (2.31)

According to Ito’s lemma,

dΠ = dV −∆1dS−∆2dU

=∂V∂S

dS+∂V∂v

dv+∂V∂ t

dt +12(∂ 2V∂S2 d2S+

∂ 2V∂v2 d2v+2

∂ 2V∂v∂S

dvdS)

−∆1dS−∆2[∂U∂S

dS+∂U∂v

dv+∂U∂ t

dt +12(∂ 2U∂S2 d2S+

∂ 2U∂v2 d2v+2

∂ 2U∂v∂S

dvdS)]

= (∂V∂S

−∆2∂U∂S

−∆1)dS+(∂V∂v

−∆2∂U∂v

)dv+(∂V∂ t

−∆2∂U∂ t

)dt

+12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )d

2S+12(∂ 2V∂v2 −∆2

∂ 2U∂v2 )d2v+(

∂ 2V∂v∂S

−∆2∂ 2U∂v∂S

)dvdS

= (∂V∂S

−∆2∂U∂S

−∆1) · (µSdt +√

vSdWt)+(∂V∂v

−∆2∂U∂v

) · (θ(ω − v)dt +ξ√

vdBt)

+(∂V∂ t

−∆2∂U∂ t

)dt +12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )vS2dt

+12(∂ 2V∂v2 −∆2

∂ 2U∂v2 )ξ 2vdt +(

∂ 2V∂v∂S

−∆2∂ 2U∂v∂S

)ρξ vSdt

= [(∂V∂S

−∆2∂U∂S

−∆1)µS+(∂V∂v

−∆2∂U∂v

)θ(ω − v)+(∂V∂ t

−∆2∂U∂ t

)

+12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )vS2 +

12(∂ 2V∂v2 −∆2

∂ 2U∂v2 )ξ 2v+(

∂ 2V∂v∂S

−∆2∂ 2U∂v∂S

)ρξ vS]dt

+(∂V∂S

−∆2∂U∂S

−∆1)√

vSdWt +(∂V∂v

−∆2∂U∂v

)ξ√

vdBt . (2.32)

Applying the strategy of dynamic hedging, the term dWt and dBt should be elim-inated, which implies that we need to set

∆2 =∂V∂v

/∂U∂v

, (2.33)

∆1 =∂V∂S

−∆2∂U∂S

. (2.34)

On the other hand, since this portfolio is risk-free, it should satisfy the followingrisk-free condition

dΠ = rΠdt = r(V −∆1S−∆2U)dt. (2.35)


These eventually yield

r(V −∆1S−∆2U) = (∂V∂S

−∆2∂U∂S

−∆1)µS+(∂V∂v

−∆2∂U∂v

)θ(ω − v)+(∂V∂ t

−∆2∂U∂ t

)

+12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )vS2 +

12(∂ 2V∂v2 −∆2

∂ 2U∂v2 )ξ 2v+(

∂ 2V∂v∂S

−∆2∂ 2U∂v∂S

)ρξ vS

⇒ ∂V∂ t

+12

vS2 ∂ 2V∂S2 + rS

∂V∂S

+12

ξ 2v∂ 2V∂v2 +θ(ω − v)

∂V∂v

+ρξ vS∂ 2V∂v∂S

− rV

= ∆2(∂U∂ t

+12

vS2 ∂ 2U∂S2 + rS

∂U∂S

+12

ξ 2v∂ 2U∂v2 +θ(ω − v)

∂U∂v

+ρξ vS∂ 2U∂v∂S

− rU)

⇒ (∂V∂ t

+12

vS2 ∂ 2V∂S2 + rS

∂V∂S

+12

ξ 2v∂ 2V∂v2 +θ(ω − v)

∂V∂v


− rV )/∂V∂v

= (∂U∂ t

+12

vS2 ∂ 2U∂S2 + rS

∂U∂S

+12

ξ 2v∂ 2U∂v2 +θ(ω − v)

∂U∂v


− rU)/∂U∂v

.

(2.36)

Clearly, the left hand side of the above equation only involves the function V ,while the right hand side contains U only. Thus, both sides of the above equationshould only depend on the variables S, v and t, i.e.,

[∂V∂ t

+12

vS2 ∂ 2V∂S2 + rS

∂V∂S

+12

ξ 2v∂ 2V∂v2 +θ(ω − v)

∂V∂v


− rV ]/∂V∂v

= [∂U∂ t

+12

vS2 ∂ 2U∂S2 + rS

∂U∂S

+12

ξ 2v∂ 2U∂v2 +θ(ω − v)

∂U∂v


− rU ]/∂U∂v

= λ (S,v, t). (2.37)

As a result, we obtain

∂V∂ t

+12

vS2 ∂ 2V∂S2 + rS

∂V∂S

+12

ξ 2v∂ 2V∂v2 +θ(ω − v)

∂V∂v


− rV = λ∂V∂v

⇒ ∂V∂ t

+12

vS2 ∂ 2V∂S2 + rS

∂V∂S

+12

ξ 2v∂ 2V∂v2 +[θ(ω − v)−λ ]

∂V∂v


− rV = 0.

(2.38)

By now, the PDE for pricing the convertible bond under the Heston model has beenderived, which involves an arbitrary function λ (S,v, t), which is consistent with theargument that a market with stochastic volatility is incomplete and there existsdifferent risk-neutral measures. Actually, it is the market price of the volatility risk,and it can be selected according to different financially meaningful arguments. Forsimplicity, λ (S,v, t)≡ 0 is a common choice.


2.2.2 PDE for the CIR interest rate model

We now consider a well-known stochastic interest rate model, the CIR model, underwhich the dynamic of the underlying asset price can still be formulated as

dSt = (rt −D0)Stdt +σStdWt , (2.39)

where D0 is the continuous dividend yield, σ is the constant volatility of the under-lying asset, and Wt is a standard Brownian motion. However, the interest rate is nolonger a constant, but a random variable, following the CIR process

drt = κ(η − rt)dt +ζ√

rtdBt , (2.40)

where κ , η and ζ are the mean reversion speed, the long-term mean and the volatilityof the interest rate, respectively. The correlation between Bt , another standardBrownian motion, andWt is ρ . Similar to the previous subsection, we also construct aself-financing portfolio, consisting of the bond, V (S,r, t), −∆1 shares of the underlyingassets and −∆2 shares of another bond U(S,r, t), leading to

Π =V −∆1S−∆2U. (2.41)

Using the Ito’s lemma, we can obtain

dΠ = dV −∆1dS−∆2dU

=∂V∂S

dS+∂V∂ r

dr+∂V∂ t

dt +12(∂ 2V∂S2 d2S+

∂ 2V∂ r2 d2r+2

∂ 2V∂S∂ r

dSdr)

−∆1dS−∆2[∂U∂S

dS+∂U∂ r

dr+∂U∂ t

dt +12(∂ 2U∂S2 d2S+

∂ 2U∂ r2 d2r+2

∂ 2U∂S∂ r

dSdr)]

= (∂V∂S

−∆1 −∆2∂U∂S

)dS+(∂V∂ r

−∆2∂U∂ r

)dr+(∂V∂ t

−∆2∂U∂ t

)dt

+12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )d

2S+12(∂ 2V∂ r2 −∆2

∂ 2U∂ r2 )d2r+(

∂ 2V∂S∂ r

−∆2∂ 2U∂S∂ r

)dSdr

= (∂V∂S

−∆1 −∆2∂U∂S

) · [(r−D0)Sdt +σSdW ]+ (∂V∂ r

−∆2∂U∂ r

) · [κ(η − r)dt +ζ√

rdB]

+(∂V∂ t

−∆2∂U∂ t

)dt +12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )σ

2S2dt +12(∂ 2V∂ r2 −∆2

∂ 2U∂ r2 )ζ 2rdt

+(∂ 2V∂S∂ r

−∆2∂ 2U∂S∂ r

)ρσζ S√

rdt

= [(∂V∂S

−∆1 −∆2∂U∂S

)(r−D0)S+(∂V∂ r

−∆2∂U∂ r

)κ(η − r)+(∂V∂ t

−∆2∂U∂ t

)

+12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )σ

2S2 +12(∂ 2V∂ r2 −∆2

∂ 2U∂ r2 )ζ 2r+(

∂ 2V∂S∂ r

−∆2∂ 2U∂S∂ r

)ρσζ S√

r]dt

+(∂V∂S

−∆1 −∆2∂U∂S

)σSdW +(∂V∂ r

−∆2∂U∂ r

)ζ√

rdB. (2.42)


The strategy of dynamic hedging requires that the stochastic term dWt and dBt

should be removed, as a result of which we need

∆2 =∂V∂ r

/∂U∂ r

, (2.43)

∆1 =∂V∂S

−∆2∂U∂S

. (2.44)

Furthermore, by making use of the fact that the self-financing portfolio is risk free,we can obtain

dΠ = rΠdt = r(V −∆1S−∆2U)dt. (2.45)

Thus,

r(V −∆1S−∆2U) = (∂V∂S

−∆1 −∆2∂U∂S

)(r−D0)S+(∂V∂ r

−∆2∂U∂ r

)κ(η − r)+(∂V∂ t

−∆2∂U∂ t

)

+12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )σ

2S2 +12(∂ 2V∂ r2 −∆2

∂ 2U∂ r2 )ζ 2r+(

∂ 2V∂S∂ r

−∆2∂ 2U∂S∂ r

)ρσζ S√

r

⇒ r[V −∆2U − (∂V∂S

−∆2∂U∂S

)S] = (∂V∂ r

−∆2∂U∂ r

)κ(η − r)+(∂V∂ t

−∆2∂U∂ t

)

+12(∂ 2V∂S2 −∆2

∂ 2U∂S2 )σ

2S2 +12(∂ 2V∂ r2 −∆2

∂ 2U∂ r2 )ζ 2r+(

∂ 2V∂S∂ r

−∆2∂ 2U∂S∂ r

)ρσζ S√

r

⇒ rV − rS∂V∂S

−κ(η − r)∂V∂ r

− ∂V∂ t

− 12

σ2S2 ∂ 2V∂S2 − 1

2ζ 2r

∂ 2V∂ r2 −ρσζ S

√r

∂ 2V∂S∂ r

= ∆2[rU − rS∂U∂S

−κ(η − r)∂U∂ r

− ∂U∂ t

− 12

σ2S2 ∂ 2U∂S2 − 1

2ζ 2r

∂ 2U∂ r2 −ρσζ S

√r

∂ 2U∂S∂ r

]

⇒ [∂V∂ t

+12

σ2S2 ∂ 2V∂S2 + rS

∂V∂S

+12

ζ 2r∂ 2V∂ r2 +κ(η − r)

∂V∂ r

+ρσζ S√

r∂ 2V∂S∂ r

− rV ]/∂V∂ r

= [∂U∂ t

+12

σ2S2 ∂ 2U∂S2 + rS

∂U∂S

+12

ζ 2r∂ 2U∂ r2 +κ(η − r)

∂U∂ r

+ρσζ S√

r∂ 2U∂S∂ r

− rU ]/∂U∂ r

.

(2.46)

It is clearly shown that the left hand side and the right hand side of the aboveequation are only dependent on V and U , respectively. Therefore, both sides of theabove equation should be equal to a certain function λ (S,r, t) with the variables S,r and t, which implies

[∂V∂ t

+12

σ2S2 ∂ 2V∂S2 + rS

∂V∂S

+12

ζ 2r∂ 2V∂ r2 +κ(η − r)

∂V∂ r

+ρσζ S√

r∂ 2V∂S∂ r

− rV ]/∂V∂ r

= [∂U∂ t

+12

σ2S2 ∂ 2U∂S2 + rS

∂U∂S

+12

ζ 2r∂ 2U∂ r2 +κ(η − r)

∂U∂ r

+ρσζ S√

r∂ 2U∂S∂ r

− rU ]/∂U∂ r

= λ (S,r, t). (2.47)


Thus, we can finally arrive at

∂V∂ t

+12

σ2S2 ∂ 2V∂S2 + rS

∂V∂S

+12

ζ 2r∂ 2V∂ r2 +[κ(η − r)−λ ]

∂V∂ r

+ρσζ S√

r∂ 2V∂S∂ r

− rV = 0.

(2.48)

Similar to the Heston model, we still need to choose a function for λ (S,r, t), whichis expected as the market here is also incomplete.

2.2.3 PDE for the hybrid stochastic volatility and interest rate model

In this subsection, we consider a particular model obtained by combining the Hestonmodel and the CIR model together. If the underling asset price, stochastic interestrate and stochastic volatility are denoted by St , rt and vt , respectively, this hybridmodel can be specified as

dSt = (rt −D0)Stdt +√

vtStdWt , (2.49)drt = κ(η − rt)dt +ζ

√rtdB1

t , (2.50)dvt = θ(ω − vt)dt +ξ

√vtdB2

t , (2.51)

where D0 is the continuous dividend yield, κ and θ are the mean reversion speed ofthe interest rate and that of the volatility, respectively, η and ω are the long termmean of the interest rate and the corresponding one of the volatility, respectively, andζ and ξ are the volatility of the interest rate and that of the volatility, respectively.Wt , B1

t and B2t are all standard Brownian motions, and the correlations between each

two areCor(Wt ,B1

t ) = ρ1, Cor(Wt ,B2t ) = ρ2, Cor(B1

t ,B2t ) = ρ3. (2.52)

We now construct a self-financing portfolio whose value can be expressed as

Π =V −∆1S−∆2U1 −∆3U2, (2.53)

where V (S,v,r, t) represents the target bond price, and U1(S,v,r, t) and U2(S,v,r, t)

are the prices of another two bonds, the introduction of which are due to the factthat there are three stochastic variables in this model. Using Ito’s lemma, we obtain

dΠ = dV −∆1dS−∆2dU1 −∆3dU2

=∂V∂S

dS+∂V∂ r

dr+∂V∂v

dv+∂V∂ t

dt

+12(∂ 2V∂S2 d2S+

∂ 2V∂ r2 d2r+

∂ 2V∂v2 d2v+2

∂ 2V∂S∂ r

dSdr+2∂ 2V∂S∂v

dSdv+2∂ 2V∂ r∂v

drdv)

−∆1dS−∆2[∂U1

∂SdS+

∂U1

∂ rdr+

∂U1

∂vdv+

∂U1

∂ tdt


+12(∂ 2U1

∂S2 d2S+∂ 2U1

∂ r2 d2r+∂ 2U1

∂v2 d2v+2∂ 2U1

∂S∂ rdSdr+2

∂ 2U1

∂S∂vdSdv+2

∂ 2U1

∂ r∂vdrdv)]

−∆3[∂U2

∂SdS+

∂U2

∂ rdr+

∂U2

∂vdv+

∂U2

∂ tdt

+12(∂ 2U2

∂S2 d2S+∂ 2U2

∂ r2 d2r+∂ 2U2

∂v2 d2v+2∂ 2U2

∂S∂ rdSdr+2

∂ 2U2

∂S∂vdSdv+2

∂ 2U2

∂ r∂vdrdv)]

= (∂V∂S

−∆1 −∆2∂U1

∂S−∆3

∂U2

∂S)dS+(

∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r)dr

+(∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v)dv+(

∂V∂ t

−∆2∂U1

∂ t−∆3

∂U2

∂ t)dt

+12(∂ 2V∂S2 −∆2

∂ 2U1

∂S2 −∆3∂ 2U2

∂S2 )d2S+12(∂ 2V∂ r2 −∆2

∂ 2U1

∂ r2 −∆3∂ 2U2

∂ r2 )d2r

+12(∂ 2V∂v2 −∆2

∂ 2U1

∂v2 −∆3∂ 2U2

∂v2 )d2v+(∂ 2V∂S∂ r

−∆2∂ 2U1

∂S∂ r−∆3

∂ 2U2

∂S∂ r)dSdr

+(∂ 2V∂S∂v

−∆2∂ 2U1

∂S∂v−∆3

∂ 2U2

∂S∂v)dSdv+(

∂ 2V∂ r∂v

−∆2∂ 2U1

∂ r∂v−∆3

∂ 2U2

∂ r∂v)drdv

= (∂V∂S

−∆1 −∆2∂U1

∂S−∆3

∂U2

∂S) · [(r−D0)Sdt +

√vSdW ]

+(∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r) · [κ(η − r)dt +ζ

√rdB1]

+(∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v) · [θ(ω − v)dt +ξ

√vdB2]+ (

∂V∂ t

−∆2∂U1

∂ t−∆3

∂U2

∂ t)dt

+12(∂ 2V∂S2 −∆2

∂ 2U1

∂S2 −∆3∂ 2U2

∂S2 )vS2dt +12(∂ 2V∂ r2 −∆2

∂ 2U1

∂ r2 −∆3∂ 2U2

∂ r2 )ζ 2rdt

+12(∂ 2V∂v2 −∆2

∂ 2U1

∂v2 −∆3∂ 2U2

∂v2 )ξ 2vdt +(∂ 2V∂S∂ r

−∆2∂ 2U1

∂S∂ r−∆3

∂ 2U2

∂S∂ r)ρ1ζ

√rvSdt

+(∂ 2V∂S∂v

−∆2∂ 2U1

∂S∂v−∆3

∂ 2U2

∂S∂v)ρ2ξ vSdt +(

∂ 2V∂ r∂v

−∆2∂ 2U1

∂ r∂v−∆3

∂ 2U2

∂ r∂v)ρ3ζξ

√rvdt

= [(∂V∂S

−∆1 −∆2∂U1

∂S−∆3

∂U2

∂S)(r−D0)S+(

∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r)κ(η − r)

+(∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v)θ(ω − v)+(

∂V∂ t

−∆2∂U1

∂ t−∆3

∂U2

∂ t)

+12(∂ 2V∂S2 −∆2

∂ 2U1

∂S2 −∆3∂ 2U2

∂S2 )vS2 +12(∂ 2V∂ r2 −∆2

∂ 2U1

∂ r2 −∆3∂ 2U2

∂ r2 )ζ 2r

+12(∂ 2V∂v2 −∆2

∂ 2U1

∂v2 −∆3∂ 2U2

∂v2 )ξ 2v+(∂ 2V∂S∂ r

−∆2∂ 2U1

∂S∂ r−∆3

∂ 2U2

∂S∂ r)ρ1ζ

√rvS

+(∂ 2V∂S∂v

−∆2∂ 2U1

∂S∂v−∆3

∂ 2U2

∂S∂v)ρ2ξ vS+(

∂ 2V∂ r∂v

−∆2∂ 2U1

∂ r∂v−∆3

∂ 2U2

∂ r∂v)ρ3ζξ

√rv]dt

+(∂V∂S

−∆1 −∆2∂U1

∂S−∆3

∂U2

∂S)√

vSdW +(∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r)ζ

√rdB1

+(∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v)ξ

√vdB2. (2.54)


Again, we need to eliminate the stochastic terms, dWt , dB1t and dB2

t , to make theportfolio risk free, i.e.,

∂V∂S

−∆1 −∆2∂U1

∂S−∆3

∂U2

∂S= 0, (2.55)

∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r= 0, (2.56)

∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v= 0. (2.57)

Furthermore, since we have already made the self-financing portfolio risk free, wealso have

dΠ = rΠdt = r(V −∆1S−∆2U1 −∆3U2)dt. (2.58)

Thus,

r(V −∆1S−∆2U1 −∆3U2) = (∂V∂S

−∆1 −∆2∂U1

∂S−∆3

∂U2

∂S)(r−D0)S

+(∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r)κ(η − r)+(

∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v)θ(ω − v)

+(∂V∂ t

−∆2∂U1

∂ t−∆3

∂U2

∂ t)+

12(∂ 2V∂S2 −∆2

∂ 2U1

∂S2 −∆3∂ 2U2

∂S2 )vS2

+12(∂ 2V∂ r2 −∆2

∂ 2U1

∂ r2 −∆3∂ 2U2

∂ r2 )ζ 2r+12(∂ 2V∂v2 −∆2

∂ 2U1

∂v2 −∆3∂ 2U2

∂v2 )ξ 2v

+(∂ 2V∂S∂ r

−∆2∂ 2U1

∂S∂ r−∆3

∂ 2U2

∂S∂ r)ρ1ζ

√rvS+(

∂ 2V∂S∂v

−∆2∂ 2U1

∂S∂v−∆3

∂ 2U2

∂S∂v)ρ2ξ vS

+(∂ 2V∂ r∂v

−∆2∂ 2U1

∂ r∂v−∆3

∂ 2U2

∂ r∂v)ρ3ζξ

√rv

⇒ r[V − (∂V∂S

−∆2∂U1

∂S−∆3

∂U2

∂S)S−∆2U1 −∆3U2] = (

∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r)κ(η − r)

+(∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v)θ(ω − v)+(

∂V∂ t

−∆2∂U1

∂ t−∆3

∂U2

∂ t)

+12(∂ 2V∂S2 −∆2

∂ 2U1

∂S2 −∆3∂ 2U2

∂S2 )vS2 +12(∂ 2V∂ r2 −∆2

∂ 2U1

∂ r2 −∆3∂ 2U2

∂ r2 )ζ 2r

+12(∂ 2V∂v2 −∆2

∂ 2U1

∂v2 −∆3∂ 2U2

∂v2 )ξ 2v+(∂ 2V∂S∂ r

−∆2∂ 2U1

∂S∂ r−∆3

∂ 2U2

∂S∂ r)ρ1ζ

√rvS

+(∂ 2V∂S∂v

−∆2∂ 2U1

∂S∂v−∆3

∂ 2U2

∂S∂v)ρ2ξ vS+(

∂ 2V∂ r∂v

−∆2∂ 2U1

∂ r∂v−∆3

∂ 2U2

∂ r∂v)ρ3ζξ

√rv

⇒ ∂V∂ t

+12

vS2 ∂ 2V∂S2 + r

∂V∂S

+12

ζ 2r∂ 2V∂ r2 +κ(η − r)

∂V∂ r

+12

ξ 2v∂ 2V∂v2 +θ(ω − v)

∂V∂v

+ρ1ζ√

rvS∂ 2V∂S∂ r

+ρ2ξ vS∂ 2V∂S∂v

+ρ3ζξ√

rv∂ 2V∂ r∂v

− rV

−∆2[∂U1

∂ t+

12

vS2 ∂ 2U1

∂S2 + r∂U1

∂S+

12

ζ 2r∂ 2U1

∂ r2 +κ(η − r)∂U1

∂ r+

12

ξ 2v∂ 2U1

∂v2 +θ(ω − v)∂U1

∂v

+ρ1ζ√

rvS∂ 2U1

∂S∂ r+ρ2ξ vS

∂ 2U1

∂S∂v+ρ3ζξ

√rv

∂ 2U1

∂ r∂v− rU1]


−∆3[∂U2

∂ t+

12

vS2 ∂ 2U2

∂S2 + r∂U2

∂S+

12

ζ 2r∂ 2U2

∂ r2 +κ(η − r)∂U2

∂ r+

12

ξ 2v∂ 2U2

∂v2 +θ(ω − v)∂U2

∂v

+ρ1ζ√

rvS∂ 2U2

∂S∂ r+ρ2ξ vS

∂ 2U2

∂S∂v+ρ3ζξ

√rv

∂ 2U2

∂ r∂v− rU2] = 0. (2.59)

If we define an operator as

L =∂∂ t

+12

vS2 ∂ 2

∂S2 + r∂

∂S+

12

ζ 2r∂ 2

∂ r2 +κ(η − r)∂∂ r

+12

ξ 2v∂ 2V∂v2 +θ(ω − v)

∂∂v

+ρ1ζ√

rvS∂ 2

∂S∂ r+ρ2ξ vS

∂ 2

∂S∂v+ρ3ζξ

√rv

∂ 2

∂ r∂v− rI, (2.60)

then the above equation can be rewritten as

LV −∆2LU1 −∆3LU2 = 0. (2.61)

With the utilization of

∂V∂ r

−∆2∂U1

∂ r−∆3

∂U2

∂ r= 0, (2.62)

∂V∂v

−∆2∂U1

∂v−∆3

∂U2

∂v= 0, (2.63)

it is not difficult to obtain

LV = λ1(S,v,r, t)∂V∂ r

+λ2(S,v,r, t)∂V∂v

, (2.64)

for arbitrary functions λ1(S,v,r, t) and λ2(S,v,r, t). Thus, we can finally arrive at

∂V∂ t

+12

vS2 ∂ 2V∂S2 + r

∂V∂S

+12

ζ 2r∂ 2V∂ r2 +[κ(η − r)−λ1(S,v,r, t)]

∂V∂ r

+12

ξ 2v∂ 2V∂v2

+ [θ(ω − v)−λ2(S,v,r, t)]∂V∂v

+ρ1ζ√

rvS∂ 2V∂S∂ r

+ρ2ξ vS∂ 2V∂S∂v

+ρ3ζξ√

rv∂ 2V∂ r∂v

− rV = 0,

(2.65)

which is the PDE for the bond price under the hybrid stochastic volatility andinterest rate model.

2.2.4 Boundary conditions along the direction of the volatility andinterest rate

In this section, some boundary conditions in the direction of the volatility andinterest rate are given, while those with respect to the underlying asset price andtime are omitted since these are the same as what have been specified for the B-Smodel in the previous section.


One of the choices for the boundary conditions in the volatility direction can begiven as

limv→0

U(S,v, t) = maxnS,Ze−r(T−t), (2.66)

limv→∞

∂U∂v

(S,v, t) = 0, (2.67)

and the boundary conditions in the interest rate direction can be selected as

limr→0

U(S,r, t) =UBS(S, t)|r=0, (2.68)

limr→∞

U(S,r, t) = 0. (2.69)

The reason for such kind of choices is elaborated in Chapter 5, and is thus omittedhere.

Before we end this section, it should also be pointed out that sometimes a certainboundary condition is not necessary to close a PDE system, and we introduce thefamous Fichera’s result [44,90] below.

Fichera’s result

Consider the linear second-order differential equation

−m

∑i, j=1

ai j(x)uxix j −m

∑i=1

bi(x)uxi + c(x)u = f (x), (2.70)

where x = (xi,x2, . . . ,xm) ∈ Ω ⊂ Rm, ai j = a ji, i, j = 1,2, . . . ,m, ∀ x. We assume theequation is the non-negative type in Ω, i.e.,

m

∑i, j=1

ai j(x)ξiξ j ≥ 0, ∀ x ∈ Ω, ξ = (ξ1,ξ2, . . . ,ξm) ∈ Rm, (2.71)

and denote the unit outer-normal direction of ∂Ω by n = (n1,n2, . . . ,nm). In thiscase, the Fichera’s function can be defined as

B(x) =m

∑i=1

[bi(x)−m

∑j=1

∂∂x j

ai j(x)]ni. (2.72)

If the whole boundary domain is separated into ∂Ω = Γ0∪Γ1∪Γ2∪Γ3, and for eachdomain, we have

m

∑i, j=1

ai j(x)nin j > 0, x ∈ Γ3, (2.73)


m

∑i, j=1

ai j(x)nin j = 0, x ∈ Γ0 ∪Γ1 ∪Γ2, (2.74)

with

B(x) = 0, x ∈ Γ0, (2.75)B(x)< 0, x ∈ Γ1, (2.76)B(x)> 0, x ∈ Γ2, (2.77)

then, we only need the boundary conditions on Γ2 and Γ3 to close the PDE system,while the boundary conditions on Γ0 and Γ1 are not necessary.

It should be remarked here that whether a certain boundary condition is neededor not is dependent on the choices of parameter values, and one always needs tolook into the case once parameters have been determined.

2.3 Numerical methodsIt should be pointed out that it is often very difficult to price even a simple financialcontract, especially when the adopted model captures the main characteristics of theunderlying asset price, such as incorporating stochastic volatility and/or stochasticinterest rate. Therefore, numerical methods must be resorted to in most cases, andseveral basic numerical approaches are illustrated below.

2.3.1 Monte-Carlo method

The Monte-Carlo method is a classical but useful method to price financial deriva-tives. The main idea of it is to generate a set of sample paths satisfying the givenSDE, and all the computation/approximation is dependent the generated samplepaths. The main advantages of the Monte-Carlo method are its generality, relativeease of use, and flexibility, and it is useful in pricing many complex financial deriva-tives, especially when the lattice and PDE framework cannot be applied. However,it also suffers from one main drawback that it is very time intensive if one wants toensure that the approximation error is small. In the following, a simple approach isto be illustrated to generate N sample paths for the B-S model

dSt = µStdt +σStdWt , (2.78)

when t ∈ [0,T ].We first need to uniformly discretize the domain [0,T] into 0 = t0 < t1 < t2 < · · ·<

tJ = T with dt = T/J and t j = j ∗ dt, j = 0,1, ...,J. We start by rewriting the B-S


model asSt j+1 = St j +µSt jdt +σSt jdWt j . (2.79)

According to the property of the Brownian motion, dWt j =Wt j+1 −Wt j , j = 0,1, ...,J−1are independent of each other, following a normal distribution with 0 and dt as themean and variance, respectively. Therefore, for the n-th sample path where n =

1,2, ...,N, we only need to independently generate J standard normally distributednumbers, M j, j = 0,2, ...,J−1, so that for each j = 0,2, ...,J−1, we can compute

Snt j+1

= Snt j+Sn

t j∗µ ∗dt +Sn

t j∗σ ∗ sqrt(dt)∗M j, (2.80)

to yield a complete sample path of the underlying price.Once we obtain all the sample paths, the task left is straightforward; for each

path, we compute the derivative price, which is conditional upon the information ofthe underlying asset, according to the payoff function, and the target price is justthe average of all these obtained conditional prices.

2.3.2 Finite difference method

Another popular approach that is often applied in derivative pricing is the finitedifference method, which is mainly used to solve differential equations numerically.There are in fact different kinds of the finite difference method, which depend onthe choices of differences defined below.

Definition 2.3.1 (Finite differences) For the first-order derivative function, d f/ds,there are mainly three types of differences:

(i) Forward difference: d fn

ds=

fn+1 − fn

∆s;

(ii) Backward difference: d fn

ds=

fn − fn−1

∆s;

(ii) Central difference: d fn

ds=

fn+1 − fn−1

2∆s.

On the other hand, for the second-order derivative function, d2 f/ds2, the most

popular operator is the half-central difference, d2 fn

ds2 =fn+1 −2 fn + fn−1

(∆s)2 , which isalso named as the second-order central difference.

Here, an example using a vanilla heat PDE, which is a degenerated equation ofthe considered problems in this thesis, is presented to show how to establish threedifferent kinds of the finite difference method to solve this PDE.

Example 2.3.1 Consider the following PDE

∂U∂ t

=∂ 2U∂x2 , (2.81)


with the initial conditionU(x,0) = f (x), (2.82)

and the boundary conditions

U(0, t) = 0 =U(1, t), (2.83)

where x and t are both defined on [0,1]. To apply the finite difference method tosolve this PDE system, we first need to divide the time domain and the space domainas

[t0, t1], [t1, t2], · · · , [tN−1, tN ], with ti = i∗dt,

[x0,x1], [x1,x2], · · · , [xL−1,xL], with x j = j ∗dx,

where dt = 1/N and dx = 1/L. If we denote the value U ji as the numerical approxi-

mation of U(x j, ti), the explicit method is defined below.

Definition 2.3.2 (Explicit method) Using the forward difference scheme and thesecond-order central difference scheme to replace the time derivative and the second-order space derivative, respectively, at point (ti,x j), yields:

U ji+1 −U j

i

dt=

U j+1i −2U j

i +U j−1i

(dx)2 , (2.84)

a rearrangement of which leads to

U ji+1 =U j

i +dt[U j+1

i −2U ji +U j−1

i(dx)2 ]

⇒ U ji+1 = (1−2λ )U j

i +λU j+1i +λU j−1

i , (2.85)

where λ = dt/(dx)2. Clearly, once we know the function values at time ti, thecorresponding values at time ti+1 can be calculated directly.

Remark: Although the Explicit method is very easy to implement, it is not alwaysstable and a stability condition for this case is 0 < λ ≤ 1/2.

Definition 2.3.3 (Implicit method) Using the backward difference scheme and thesecond-order central difference scheme to replace the time derivative and the second-order space derivative, respectively, at point (ti+1,x j), yields:

U ji+1 −U j

i

dt=

U j+1i+1 −2U j

i+1 +U j−1i+1

(dx)2 , (2.86)


which can be simplified as

U ji+1 −dt[

U j+1i+1 −2U j

i+1 +U j−1i+1

(dx)2 ] =U ji

⇒ (1+2λ )U ji+1 −λU j+1

i+1 −λU j−1i+1 =U j

i , (2.87)

where λ = dt/(dx)2.Remark: The Implicit method has overcome the stability problem of the Explicit

method.

Definition 2.3.4 (Crank-Nicolson method) A summation of the Explicit and Implicitmethod, or adding Equation (2.84) and (2.86), yields

U ji+1 −U j

i

dt=

12[U j+1

i+1 −2U ji+1 +U j−1

i+1

(dx)2 +U j+1

i −2U ji +U j−1

i(dx)2 ], (2.88)

from which it is easy to obtain

2U ji+1 −dt[

U j+1i+1 −2U j

i+1 +U j−1i+1

(dx)2 ] = 2U ji +dt[

U j+1i −2U j

i +U j−1i

(dx)2 ]

⇒ (1+2λ )U ji+1 −λU j+1

i+1 −λU j−1i+1 = (1−2λ )U j

i +λU j+1i +λU j−1

i , (2.89)

where λ = dt/(dx)2.Remark: This method is also unconditionally stable, In fact, we can use p ∗

(2.84)+q∗ (2.86) to obtain different schemes, as long as p,q ∈ (0,1) and p+q = 1.

2.3.3 Binomial tree pricing method

In this subsection, a simple but useful numerical approach to price financial deriva-tives is introduced, and this is applicable when the model of the underlying asset isdiscrete.

As an example, we consider a single period where the underlying price starts atS0. At the next time instant, dt, we assume that the underlying price can onlybecome either uS0 or dS0 with probabilities Pu and Pd, respectively, where u > d,Pu, Pd ∈ [0,1] and Pu +Pd = 1. Clearly, if we respectively denote Vu and Vd as thepayoff corresponding to the two cases, and assume Pu and Pd are given, then theprice of the derivative at the current time, V0, can be directly computed as

V0 = e−rdt(PuVu +PdVd), (2.90)

using the risk neutral pricing principle. However, Pu and Pd are usually unknown,and an alternative approach is needed to find the derivative price.


We now construct a portfolio consisting of the risk-free asset valued at Φ, and ∆shares of the underlying asset, which implies that the initial value of this portfoliois

Π0 = ∆S0 +Φ, (2.91)

and its possible future value at dt will be either

Πu = ∆uS0 +Φerdt or Πd = ∆dS0 +Φerdt . (2.92)

If we try to use this portfolio to replicate the payoff of the target derivative, i.e.,

∆uS0 +Φerdt =Vu, (2.93)∆dS0 +Φerdt =Vd, (2.94)

we can easily obtain

∆ =Vu −Vd

S0(u−d), (2.95)

Φ = e−rdt uVd −dVu

u−d. (2.96)

In this case, in order to avoid arbitrage opportunities, the initial value of this port-folio must be equal to V0, yielding

V0 = ∆S0 +Φ

=Vu −Vd

u−d+ e−rdt uVd −dVu

u−d

= e−rdt(erdt −du−d

Vu +u− erdt

u−dVd). (2.97)

From this, it is not difficult to deduce that

Pu =erdt −du−d

, Pd =u− erdt

u−d. (2.98)

For the easiness of understanding, we have also provided a figure below to illustratethe main idea of the binomial tree method.

Although the binomial method is a numerical approximation approach, the de-rived result can be treated as the true value of this financial derivative if the numberof the time steps is large enough. This is especially useful when American-stylederivatives are taken into consideration, as for such kind of derivatives, it is usuallyimpossible to find analytical solutions, and we would always need a benchmark tocheck the accuracy of a certain approach.


Figure 2.1: A simple example of the binomial tree

2.3.4 Predictor-corrector method

A predictor-corrector method is adopted in this thesis to solve the pricing PDE ofthe convertible bonds. In fact, it refers to a class of algorithms, designed to solveODEs. Although it can give rise to different algorithms depending on the differentforms of ODEs and different numerical schemes used, the main idea behind it isactually the same, and represents a two-step solution process as

Step 1: (Predictor) Given the function-values and derivative-values at a precedingset of points, extrapolation is used to obtain the value of the target function at asubsequent new point. For this step, the numerical scheme should be an explicitone.

Step 2: (Corrector) Refine the prediction obtained in Step 1 by using anothermethod to interpolate the unknown function’s value at the same subsequent point.For this step, the numerical scheme should be an implicit one.

In the following, some simple and classical examples are given to describe thismethod more clearly.

Example 2.3.2 Consider the ODE

y′ = f (x,y), y(x0) = y0. (2.99)

The step size here is denoted as dx such that xi = x0 + i ∗ dt, and the value of thefunction at each point, y(xi), is represented by yi.

Firstly, a classical algorithm is considered. For the predictor step, if yi is known,applying the Euler scheme yields

yi+1 = yi + f (xi,yi)dx. (2.100)

Once we obtain the predicted value, the corrector step implements the trapezoidal


rule such that the prediction can be refined as

yi+1 = yi +12( f (xi,yi)+ f (xi+1, yi+1))dx, (2.101)

which is the corrected value of the function at xi+1. It should be pointed out that thisis the so-called Predict-Evaluate-Correct-Evaluate (PECE) mode, where we alwaysupdate the function values f according to the update of the value y. However, it isalso possible to evaluate the function f only once per step by using the method ofthe Predict-Evaluate-Correct (PEC) mode:

yi+1 = yi + f (xi, yi)dx, (2.102)

yi+1 = yi +12( f (xi, yi)+ f (xi+1, yi+1))dx. (2.103)

In addition, the corrector step can be repeated in the hope that this achieves an evenbetter approximation to the true solution. For example, if the corrector method isrun twice, this yields the PECECE mode:

yi+1 = yi + f (xi,yi)dx, (2.104)

yi+1 = yi +12( f (xi,yi)+ f (xi+1, yi+1))dx, (2.105)

yi+1 = yi +12( f (xi,yi)+ f (xi+1, yi+1))dx. (2.106)

Remark: When this method is applied to solve PDEs, one should be very carefulas we need to fix other variables when we are working on one particular variable.

2.3.5 Alternating direction implicit method

The Alternating Direction Implicit (ADI) method is a very useful method to solvethe parabolic equations on rectangular domains. Let us consider a standard form ofthe parabolic equations

Ut = b1Uxx +b2Uyy, (2.107)

defined on a rectangular domain. Let A1 and A2 be two linear operators defined as

A1U = b1Uxx, (2.108)A2U = b2Uyy. (2.109)

Then, the problem can be rewritten as

Ut = A1U +A2U. (2.110)


The ADI method is able to transform the initial two-dimensional problem into aset of simple one-dimensional ones. In the following, two different schemes areintroduced.

The Peaceman-Rachford Method

Using the central-time finite difference scheme at t = (n+1/2)dt, Equation (2.110)becomes

Un+1 −Un

dt=

12(A1Un+1 +A1Un)+

12(A2Un+1 +A2Un), (2.111)

which can be rewritten as

(I− dt2

A1 −dt2

A2)Un+1 = (I+dt2

A1 +dt2

A2)Un. (2.112)

To solve this equation, a term dt2A1A2Un+1/4 is added to both sides of the aboveequation, i.e.,

(I− dt2

A1 −dt2

A2)Un+1 +dt2

4A1A2Un+1 = (I+

dt2

A1 +dt2

A2)Un +dt2

4A1A2Un+1.

(2.113)A simple rearrangement leads to

(I− dt2

A1 −dt2

A2 +dt2

4A1A2)Un+1 = (I+

dt2

A1 +dt2

A2)Un +dt2

4A1A2Un+1

⇒ (I− dt2

A1)(I−dt2

A2)Un+1 = (I+dt2

A1)(I+dt2

A2)Un +dt2

4A1A2(Un+1 −Un).

(2.114)

Since Un+1 =Un +O(dt), we can further obtain

(I− dt2

A1)(I−dt2

A2)Un+1 = (I+dt2

A1)(I+dt2

A2)Un +O(dt3). (2.115)

If A1h and A2h are the second-order approximations of A1 and A2, respectively, theabove equation can be expressed as

(I− dt2

A1h)(I−dt2

A2h)Un+1 = (I+dt2

A1h)(I+dt2

A2h)Un+O(dt3)+O(dth2). (2.116)

According to this formulation, the ADI scheme can be finally derived as

(I− dt2

A1h)(I−dt2

A2h)vn+1 = (I+dt2

A1h)(I+dt2

A2h)vn. (2.117)

To solve the above equation, Peaceman & Rachford [91] designed a two-step process


as

(I− dt2

A1h)vn+1/2 = (I+dt2

A2h)vn, (2.118)

(I− dt2

A2h)vn+1 = (I+dt2

A1h)vn+1/2, (2.119)

which is the original form of the ADI method. One can clearly see that this particularADI method is constructed with two steps, each dealing with a one-dimensionalproblem.

The Douglas-Rachord Method

Another well-known ADI scheme that is widely adopted in derivative pricing is theDouglas-Rachord method [38], since its accuracy is first-order in time and second-order in space. We still start with Equation (2.110), but apply the backward-timecentral-space scheme so that

(I−dtA1 −dtA2)Un+1 =Un. (2.120)

Adding a term dt2A1A2Un+1 on both sides yields

(I−dtA1 −dtA2 +dt2A1A2)Un+1 =Un +dt2A1A2Un+1

⇒ (I−dtA1 −dtA2 +dt2A1A2)Un+1 =Un +dt2A1A2Un+1 −dt2A1A2Un +dt2A1A2Un

⇒ (I−dtA1 −dtA2 +dt2A1A2)Un+1 =Un +dt2A1A2Un +dt2A1A2(Un+1 −Un),

(2.121)

which implies

(I−dtA1 −dtA2 +dt2A1A2)Un+1 =Un +dt2A1A2Un +O(dt3). (2.122)

Omitting the term O(dt3), the scheme we can obtain is

(I−dtA1 −dtA2 +dt2A1A2)vn+1 = (I+dt2A1A2)vn. (2.123)

for which the Douglas-Rachord method is

(I−dtA1h)vn+1/2 = (I+dtA2h)vn, (2.124)(I−dtA2h)vn+1 = vn+1/2 −dtA2hvn. (2.125)


2.4 Integral equation method and Fourier transformA main disadvantage of purely numerical approaches mentioned in the previous sec-tion is that errors are always introduced at a very early stage of computation, whichcould sometimes severely affect the accuracy of the obtained results. One possibleway to overcome such a disadvantage is to use semi-analytical approaches, in whichanalytical analysis is performed until a point beyond which numerical calculationsmust be resorted to. Belonging to this category, the integral equation method isone of the most popular approaches that have wide applications in derivative pric-ing. A key step in realizing the integral equation approach is to derive an integralequation representation for the target derivative price, involving the utilization ofseveral useful techniques. In particular, the Fourier transform is one well-knownmethod that can be applied to derive the integral equation representationsa, and itsdefinition as well as that of the Fourier inversion transform are presented below.

Definition 2.4.1 (Fourier Transform) The Fourier transform of a smooth functionU(x) is defined as

FU(x)=∫ ∞

−∞U(x)eiωxdx, (2.126)

for any real number ω , and it is often denoted as U(ω).

Definition 2.4.2 (Fourier Inversion Transform) The Fourier inversion transform ofthe function U(ω) in the Fourier space is defined as

U(x) = F−1U(ω)= 12π

∫ ∞

−∞U(ω)e−iωxdω. (2.127)

The Fourier transform possesses some useful properties that are often used inthe process of applying this particular transform, and these are illustrated in thefollowing proposition and theorem.

Proposition 2.4.1 (Linearity) If

P(x) = aU(x)+bV (x), (2.128)

thenP(ω) = aU(ω)+bV (ω), (2.129)

for any complex numbers a and b.

Theorem 2.4.1 (Convolution theorem) If

P(x) = (U ∗V )(x) =∫ ∞

−∞U(u)V (x−u)du, (2.130)

aFourier transform also has wide applications in the area of financial mathematics.


where ∗ is the convolution operator, then

P(ω) = U(ω) ·V (ω). (2.131)

It also means that ifP(ω) = U(ω) ·V (ω), (2.132)

then

P(x) = F−1P(ω)

= F−1U(ω) ·V (ω)

= (U ∗V )(x). (2.133)

Now, a simple example is presented to explain how to apply the Fourier transformto solve PDE systems.

Example 2.4.1 Consider a classical heat equation

∂U∂ t

=∂ 2U∂x2 , (2.134)

with the initial conditionU(x,0) = f (x), (2.135)

and the boundary conditions

U(−∞, t) = 0, (2.136)U(∞, t) = 0, (2.137)

where the domains of x and t are (−∞,∞) and [0,∞), respectively.To solve the target system, applying the Fourier transform with respect to x on

Equation (2.134) yields

F∂U∂ t

= F∂ 2U∂x2 . (2.138)

The left hand side of the equation can be computed through

F∂U∂ t

=∫ ∞

−∞

∂U∂ t

(x, t)eiωxdx

=∂∂ t

∫ ∞

−∞U(x, t)eiωxdx

=∂U∂ t

(ω, t), (2.139)


and the right hand side can be calculated as

F∂ 2U∂x2 =

∫ ∞

−∞

∂ 2U∂x2 (x, t)eiωxdx

=∂U∂x

(x, t)eiωx|∞−∞ − iω∫ ∞

−∞

∂U∂x

(x, t)eiωxdx

= −iω∫ ∞

−∞

∂U∂x

(x, t)eiωxdx

= −iωU(x, t)eiωx|∞−∞ −ω2∫ ∞

−∞U(x, t)eiωxdx

= −ω2U(ω, t). (2.140)

A combination of the two equations leads to∂U∂ t

(ω, t)+ω2U(ω, t) = 0

U(ω,0) =∫ ∞

−∞f (x)eiωxdx.

(2.141)

Clearly, this is a first-order linear ODE with an initial condition, the solution towhich can be easily formulated as

U(ω, t) = U(ω,0)e−ω2t . (2.142)

By now, the solution to Equation (2.134) has been derived in the Fourier space, andto obtain the solution in the original space, the Fourier inversion transform needsto be applied, which yields

U(x, t) = F−1U(ω, t)

=1

2π

∫ ∞

−∞U(ω, t)e−iωxdω

=1

2π

∫ ∞

−∞U(ω,0)e−ω2te−iωxdω. (2.143)

Using the Convolution theorem, we further obtain

U(x, t) = U(x,0)∗F−1e−ω2t

= f (x)∗ 12π

∫ ∞

−∞e−ω2te−iωxdω

= f (x)∗ 12π

∫ ∞

−∞e−ω2t−iωxdω

= f (x)∗ 12π

√πt

e−x24t

= f (x)∗√

14πt

e−x24t


=

√1

4πt

∫ ∞

−∞f (u) · e−

(x−u)24t du, (2.144)

which is the desired result.

Chapter 3

Pricing puttable convertible bonds withintegral equation approaches

3.1 IntroductionA convertible bond (CB) is one of the widely-used hybrid financial instruments. Itgives the holder the right to convert a bond into a predetermined number of stocks atany time during the life of the bond, or to hold the bond until maturity to receive theprincipal payment. Such a conversion right gives the holder the possibility to gaina maximum benefit. But, this particular conversion feature has made the valuationproblem more complicated because the optimal conversion boundary needs to bedetermined as part of the solution of the problem.

The theoretical framework for pricing CBs under the Black-Scholes model wasinitially proposed by Ingersoll [62] and Brennan & Schwartz [12]. They priced a con-vertible bond by using contingent claims, in which they took the firm value as theunderlying variable. However, the model is not practical since the firm value is notobservable in market. In 1986, McConnel & Schwartz [84] proposed a single-factorpricing model for a zero-coupon convertible bond, using stock price as the underlyingvariable.

Since then, various approaches have been proposed to price convertible bonds.Analytical solutions are only available for CBs with very simply exercises clauses.For example, Nyborg [88] obtained a closed-form solution for a simple convertiblebond, which can only be converted at maturity, while Zhu [106] presented a closed-form analytical solution for a convertible bond, which can be converted at any timeon or before maturity, using the homotopy analysis method. Recently, Chan &Zhu [23] provided an approximate solution for the price of a convertible bond underthe regime-switching model.

On the other hand, numerical approaches are resorted to when CBs with more

40

CHAPTER 3. PRICING PUTTABLE CONVERTIBLE BONDS 41

complex exercise clauses need to be priced. Among them, the finite element ap-proach [6], the finite difference approach [98] and the finite volume approach [112] havebeen adopted by various authors. In terms of integral equation formulations forpricing CBs, Zhu & Zhang [111] used a decomposition approach to obtain an integralequation formulation for pricing a vanilla convertible bond without any additionalfeature such as the puttability discussed in this chapter.

Apart from the Black-Scholes model, there are other models having been adoptedfor the evaluation of CBs. For example, Brennan & Schwartz [14] proposed a stochas-tic interest rate model to price convertible bonds, taking the value of the issuing firmas the underlying state variable. Carayannopoulos [17] priced convertible bonds witha different stochastic interest rate model (the so-called CIR model (Cox-Ingersoll-Ross) [29]), while David & Lischka [34] adopted the Vasicek’s model [100]. All thesemodels are based on an assumption that CBs are usually designed for a long timeperiod, during which interest rate itself may be subject to changes. However, suchan addition of stochastic nature of interest rate would not be necessary if one onlyneeds to price a CB with short time to expiry. It is certainly not necessary if oneaims to develop numerical approaches as their first step. Furthermore, Hung &Wang [59] used the binomial tree model to value the convertible bond, taking therisk of interest rate change as well as the default risk of the issuer into considera-tion, while Chambers & Lu [21] further extended Hung & Wang’s work by allowingcorrelations among those two stochastic processes.

In addition to model complexity contributing to the pricing of CBs, various addedadditional rights to either or both the bond issuer and/or the bond holder, mayalso make the pricing problem more complicated, which demands better numericalsolution approaches. For example, call and put features can be added to convert-ible bonds to form the so-called callable convertible bonds and puttable convertiblebonds [2], respectively. A callable convertible bond is a bond in which the issuer hasthe right to call (repurchase) the bond from the investor for a predetermined callprice within a predetermined callable period. The call feature in a convertible bondis in favor to the issuer, as if the underlying price increases significantly beyond thecall price, the issuer can call back the bond. As a result, a callable convertible bondshould be worth less than that of a vanilla convertible bond. A puttable convertiblebond, on the other hand, allows the holder to sell the bond back to the issuer, priorto maturity, at a price that is specified at the time that the bond is issued. Thisprice is commonly referred to as the put price [80], which is also called the strike orexercise price [84]. Obviously, the put feature benefits the holder of the bond, andhence, a puttable convertible bond trades at a higher price than that of a vanillaconvertible bond.

The pricing problem of callable convertible bonds has been studied for many


years. For example, Brennan & Schwartz [12] explained in theory how to price suchcontracts, and provided numerical solutions in their later article [13], Bernini [7] useda binomial tree method to obtain their numerical solution. It is interesting to notethat Kifer [67] presented a new derivative security called game options, similar to thecallable convertible bond, which was used by Yagi & Sawaki [103] to study callableconvertible bonds. There are also many references on puttable convertible bonds inthe literature. For example, Nyborg [88] presented the boundary condition of puttableCBs, and checked if the boundary condition is reasonable and correct, while Lvov etal. [80] obtained the numerical solution by using Monte Carlo simulations. However,there has not been any integral equation formulation for puttable convertible bonds,which forms the base of the current research.

In this chapter, we present two integral equation formulations to analyze a put-table convertible bond under the Black-Scholes model. It should be pointed outthat although it is more practical to adopt a stochastic interest rate for convertiblebond pricing, we assume a constant interest rate in our formulation. This is becauseit is more feasible to start with a simpler model when introducing a new solutionapproach to an already complicated problem with two free boundaries. There aretwo partial differential equation (PDE) systems governing the price of a puttableconvertible bond, as the lifetime of a puttable CB is divided into two intervals bythe time when the face value of the bond discounted by the time to expiry equals thepredetermined put price. From this critical time, only convertible bond boundaryconditions need to be considered since the price of a puttable CB is always greaterthan the put price during this time period there is no financial incentive to exercisethe put feature. Thus, the PDE system for this part should be the same as thatfor the vanilla CB presented in [106]. On the other hand, from the beginning of thecontract until the critical time, the minimal price of puttable CB would be flooredbelow by the put price, forming a second free boundary. Financially, the boundprice is bounded below is because the holder would otherwise sell the bond back tothe issuer at the put price with the warranted puttability. As a result, the puttableCB can no longer be treated as a vanilla CB and another PDE system is neededwith two free boundary conditions associated with the conversion and put feature,respectively.

In order to obtain the first integral equation formulation, we apply the methodof incomplete Fourier transform [26] to both of the two PDE systems. However,the resulting integral equations possess a discontinuity at both of two free bound-aries and they contain the first-order derivatives of the unknown free boundaries.These problems could lead to computational difficulties when the numerical resultsare calculated. To overcome the problems, we derive a second integral equationrepresentation from the first integral representation.


The chapter is organized as follows. In Section 2, the PDE systems governing theprice of a puttable CB are established to reflect all the unique features associatedwith conversion and puttability at any time prior to expiry. In Section 3, the firstform of integral equation is derived by using the incomplete Fourier transform, whichserves as a base to obtain another integral equation representation. In Section 4,we compared our results with the known benchmarks such as the convergent resultsobtained with the binomial tree method. Numerical examples are presented inSection 5, followed by some concluding remarks given in the last section.

3.2 The modelIn this section, we will establish the PDE systems to price a puttable convertiblebond.

Let S be an underlying asset price and we assume that its dynamics follows thestochastic different equation:

dS = (r−D0)Sdt +σSdWt , (3.1)

where Wt is a Brownian motion, σ is the volatility of the underlying asset, r is therisk-free interest rate, and D0 is the rate of continuous dividend.

Now, consider a puttable convertible bond of maturity T , with face value Z,conversion ratio n and put price M. Let the time to expiry be τ = T − t, thereexists a critical value of τ = τM when the minimum value of the puttable CB (facevalue discounted by time to expiry) equals the put price, that is, Ze−rτM = M, orτM =−1

rlog

MZ.

Let V1(S,τ) be the value of the puttable CB in the interval τ ∈ [0,τM]. The priceof the CB is always greater than the put price in this time interval, and thus theoptimal put exercise price is always equal to zero. As a result, there is no differencebetween the price of a vanilla CB and that of a puttable CB in this time interval,and it should satisfy the following PDE system (cf. Zhu [106]).

−∂V1

∂τ+

12

σ2S2 ∂ 2V1

∂S2 +(r−D0)S∂V1

∂S− rV1 = 0,

V1(S,0) = maxnS,Z,V1(Sc(τ),τ) = nSc(τ),∂V1

∂S(Sc(τ),τ) = n,

V1(0,τ) = Ze−rτ ,

(3.2)

where Sc(τ) is the optimal conversion boundary, S ∈ [0,Sc(τ)] and τ ∈ [0,τM].In the interval τ ∈ [τM,T ], the price of bond should not fall below the put price, as


the holder would otherwise sell the bond back to the issuer at the predetermined putprice. Therefore, when τ > τM, the value of the bond is bounded below by the putprice, which is an important feature of puttable CBs. In fact there exist an optimalput price Sp(τ) associated with the puttability, as well as an optimal conversionprice Sc(τ) in this time interval. The price of the puttable CB, V2(S,τ), satisfies thefollowing PDE system:

−∂V2

∂τ+

12

σ2S2 ∂ 2V2

∂S2 +(r−D0)S∂V2

∂S− rV2 = 0,

V2(Sc(τ),τ) = nSc(τ),∂V2

∂S(Sc(τ),τ) = n,

V2(Sp(τ),τ) = M,∂V2

∂S(Sp(τ),τ) = 0,

V2(S,τ+M) =V1(S,τ−M),

(3.3)

The value of the puttable convertible bond for the lifetime τ ∈ [0,T ] can be foundby solving the two PDE systems (3.2) and (3.3). We start the solution process ofthe systems by making the following variable transforms

x = log(S), v1(x,τ) =V1(S,τ), v2(x,τ) =V2(S,τ).

The PDE systems (3.2) and (3.3) become

−∂v1

∂τ+

12

σ2 ∂ 2v1

∂x2 +(r−D0 −12

σ2)∂v1

∂x− rv1 = 0,

v1(x,0) = maxnex,Z,v1(ln(Sc(τ)),τ) = nSc(τ),∂v1

∂x(ln(Sc(τ)),τ) = nSc(τ),

v1(−∞,τ) = Ze−rτ ,

(3.4)

with the domain x ∈ (−∞, ln(Sc(τ))] and τ ∈ [0,τM], and

−∂v2

∂τ+

12

σ2 ∂ 2v2

∂x2 +(r−D0 −12

σ2)∂v2

∂x− rv2 = 0,

v2(ln(Sc(τ)),τ) = nSc(τ),∂v2


v2(ln(Sp(τ)),τ) = M,∂v2

∂x(ln(Sp(τ)),τ) = 0,

v2(x,τ+M) = v1(x,τ−M),

(3.5)


with the domain of x and τ being [ln(Sp(τ)), ln(Sc(τ))] and [τM,T ], respectively. Bynow, we have derived two dimensionless PDE systems. In next section, the solutiontechniques to obtain integral equation formulations for Systems (3.4) and (3.5) willbe discussed.

3.3 Integral equation formulations of puttable convertible bondIn this section, two forms of integral equations will be presented for pricing a puttableconvertible bond. One is obtained by applying the so-called incomplete Fouriertransform to the PDE systems directly, and the second one is a further extension ofthe first one, in order to avoid some potential numerical problems.

3.3.1 First integral equation formulation of puttable convertible bond

In this subsection, we use the incomplete Fourier transform method to derive an in-tegral equation representation to price a puttable convertible bond. The incompleteFourier transform is adopted as a result of the presence of free boundaries, whichhave limited the domain of x to a semi-infinite domain, rather than an infinite domainfrom −∞ to ∞, on which the classical Fourier transform can be applied [26]. Beforeapplying the incomplete Fourier transform to System (3.4), it should be noted thatthe boundary condition at infinity is non-zero, which can cause problems. Therefore,a simple transform

U(x,τ) = v1(x,τ)−Ze−rτ , (3.6)

is introduced, so that System (3.4) can be rewritten as

−∂U∂τ

+12

σ2 ∂ 2U∂x2 +(r−D0 −

12

σ2)∂U∂x

− rU = 0,

U(x,0) = maxnex −Z,0,U(ln(Sc(τ)),τ) = nSc(τ)−Ze−rτ ,∂U∂x

(ln(Sc(τ)),τ) = nSc(τ),

U(−∞,τ) = 0.

(3.7)

Now, define the following incomplete Fourier transform

FU(x,τ)=∫ ln(Sc(τ))

−∞U(x,τ)eiωxdx , U(ω,τ), (3.8)


and by applying (3.8) to System (3.7), we can obtain the following ordinary differ-ential equation (ODE) system

∂U∂τ

(ω,τ)+B(ω)U(ω,τ) = f (ω,τ),

U(ω,0) =∫ ln(Sc(0))

−∞maxnex −Z,0eiωxdx,

(3.9)

where

B(ω) =12

σ2ω2 +(r−D0 −12

σ2)iω + r,

f (ω,τ) = (nSc(τ)−Ze−rτ)eiω ln(Sc(τ))[S′c(τ)

Sc(τ)+ r−D0 −

12

σ2 − 12

σ2iω]+12

σ2nSc(τ)eiω ln(Sc(τ)).

System (3.9) is a non-homogeneous first-order linear ODE system with an initialcondition. The solution of this system is as follows

U(ω,τ) = U(ω,0)e−B(ω)τ +∫ τ

0f (ω,ξ )e−B(ω)(τ−ξ )dξ . (3.10)

By now, the integral equation formulation in the Fourier space has been derived.In order to obtain the formulation in the original space, we define the followingincomplete Fourier inversion transform

F−1U(ω,τ)=∫ ∞

−∞U(ω,τ)e−iωxdω. (3.11)

The incomplete Fourier inversion transform appears to be the same as the classicalone, but there is a difference in the domain of x, in our case, the domain of x isreplaced by (−∞, ln(Sc(τ))]. Applying this new definition to (3.10) and after sometedious algebraic manipulations (see Appendix A.1), we obtain

U(x,τ) =∫ ln(Sc(0))

−∞

e−rτ

σ√

2πτe−

[(r−D0−12 σ2)τ+x−u]2

2σ2τ ·maxneu −Z,0du

+∫ τ

0

e−r(τ−ξ )

σ√

2π(τ −ξ )e− [(r−D0−

12 σ2)(τ−ξ )−ln(Sc(ξ ))+x]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12

σ2 +ln(Sc(ξ ))− x

τ −ξ)]

12

nσ2Sc(ξ )dξ . (3.12)

Rewriting the integral equation using the original parameters, we derive an inte-gral equation formulation of System (3.2) as follows

V1(S,τ) =∫ ln(Sc(0))

−∞

e−rτ

σ√

2πτe−

[(r−D0−12 σ2)τ+ln(S)−u]2



+∫ τ

0

e−r(τ−ξ )

σ√

2π(τ −ξ )e− [(r−D0−

12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(S)]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12

σ2 +ln(Sc(ξ ))− ln(S)

τ −ξ)]+

12

nσ2Sc(ξ )dξ +Ze−rτ .

(3.13)

We have just presented the first part of pricing a puttable convertible bond, andthe next step is to solve System (3.5). It should be emphasized that there areactually two free boundaries that need to be determined at the same time when wetry to find the solution of this particular system. Therefore, we need to introduceanother definition of the incomplete Fourier transform as

Fv2(x,τ)=∫ ln(Sc(τ))

ln(Sp(τ))v2(x,τ)eiωxdx , v2(ω,τ). (3.14)

Substituting Equation (3.14) into System (3.5) yields the following ODE system∂ v2

∂τ(ω,τ)+B(ω)v2(ω,τ) = g(ω,τ),

v2(ω,τM) =∫ ln(Sc(τM))

ln(Sp(τM))v1(x,τM)eiωxdx,

(3.15)

where

g(ω,τ) = nSc(τ)eiω ln(Sc(τ))

[S′c(τ)

Sc(τ)− 1

2σ2iω + r−D0

]

− Meiω ln(Sp(τ))

[S′p(τ)

Sp(τ)− 1

2σ2iω + r−D0 −

12

σ2

]. (3.16)

System (3.15) is again a non-homogeneous first-order linear ODE system, the solu-tion of which can be derived as

v2(ω,τ) = v2(ω,τM)e−B(ω)(τ−τM)+∫ τ−τM

0g(ω,τM +ξ )e−B(ω)(τ−τM−ξ )dξ . (3.17)

To obtain the integral equation formulation in the original space, we apply theFourier inversion transform to Equation (3.17) (the details are left in Appendix A.2).Finally, we arrive at

V2(S,τ) =∫ ln(Sc(τM))

ln(Sp(τM))V1(eu,τM)

e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0

nSc(τM +ξ )e−r(τ−τM−ξ )

σ√

2π(τ − τM −ξ )e− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(S)−ln(Sc(τM+ξ ))]2

2σ2(τ−τM−ξ )


·S′c(τM +ξ )

Sc(τM +ξ )+

12[r−D0 +

12

σ2 +ln(Sc(τM +ξ ))− ln(S)

τ − τM −ξ]dξ

−∫ τ−τM

0

Me−r(τ−τM−ξ )

σ√

2π(τ − τM −ξ )e− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(S)−ln(Sp(τM+ξ ))]2


·S′p(τM +ξ )

Sp(τM +ξ )+

12[r−D0 −

12

σ2 +ln(Sp(τM +ξ ))− ln(S)

τ − τM −ξ]dξ . (3.18)

Equation (3.13) and Equation (3.18) could be used, for τ ∈ [0,τM] and τ ∈ [τM,T ],respectively, to determine the value of puttable CBs. However, both (3.13) and(3.18) involve the optimal conversion price and the optimal put price, Sc(τ) andSp(τ), which still remain unknown. Fortunately, we can derive three integral equa-tions for the boundary using the free boundaries conditions

nSc(τ)2

=∫ ln(Sc(0))

−∞

e−rτ

σ√

2πτe−

[(r−D0−12 σ2)τ+ln(Sc(τ))−u]2


+∫ τ

0

e−r(τ−ξ )

σ√

2π(τ −ξ )e− [(r−D0−

12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(Sc(τ))]2

2σ2(τ−ξ )

·(nSc(ξ )−Ze−rξ )[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12

σ2 +ln(Sc(ξ ))− ln(Sc(τ))

τ −ξ)]

+12

nσ2Sc(ξ )dξ +12

Ze−rτ , (3.19)

nSc(τ)2

=∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(Sc(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(Sc(τ))−ln(Sc(τM+ξ ))]2


·S′c(τM +ξ )

Sc(τM +ξ )+

12[r−D0 +

12

σ2 +ln(Sc(τM +ξ ))− ln(Sc(τ))


−∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(Sc(τ))−ln(Sp(τM+ξ ))]2


·S′p(τM +ξ )

Sp(τM +ξ )+

12[r−D0 −

12

σ2 +ln(Sp(τM +ξ ))− ln(Sc(τ))

τ − τM −ξ]dξ ,(3.20)

M2

=∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(Sp(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(Sp(τ))−ln(Sc(τM+ξ ))]2


·S′c(τM +ξ )

Sc(τM +ξ )+

12[r−D0 +

12

σ2 +ln(Sc(τM +ξ ))− ln(Sp(τ))


−∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(Sp(τ))−ln(Sp(τM+ξ ))]2



·S′p(τM +ξ )

Sp(τM +ξ )+

12[r−D0 −

12

σ2 +ln(Sp(τM +ξ ))− ln(Sp(τ))

τ − τM −ξ]dξ .(3.21)

It should be noted that there is a factor of 1/2 on the left hand side of Equa-tions (3.19), (3.20) and (3.21), which arises by performing the incomplete Fouriertransform. Actually, it can be viewed as the complete Fourier transform of a dis-continuous function, and thus the corresponding Fourier inversion converges to themidpoint of the discontinuity [36]. Sometimes such discontinuity can lead to problemswhen numerical experiments are conducted. An even worse problem is that bothof these two integral equation formulations contain the first-order derivative of theoptimal exercise prices which can lead to large numerical errors due to the infiniteslope associated with these derivative functions at expiry for the optimal conversionprice and at threshold value of the time to expiry for the optimal put price. Toovercome these shortfalls, we propose another integral equation formulation in thenext subsection.

3.3.2 Second integral equation formulation for puttable convertiblebond

As pointed out in the previous subsection, the integral representations (3.13) and(3.18) and the integral equations (3.19), (3.20) and (3.21) are not ideal to be usedfor computing the value of a puttable convertible bond and its optimal boundaries,since they all contain first-order derivatives of the optimal exercise prices. So wederive the second integral representation as an extension from the first one. Whilewe shall leave the details of the derivation in Appendix A.3 and Appendix A.4, themain results are summarized here

V1(S,τ) =∫ τ

0nSD0e−D0(τ−ξ )N (

(r−D0 +12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))

σ√

τ −ξ)dξ

+ nSe−D0τN ((r−D0 +

12σ2)τ + ln(S)− ln(Sc(0))

σ√

τ)

− Ze−rτN ((r−D0 − 1

2σ2)τ + ln(S)− ln(Sc(0))σ√

τ)+Ze−rτ , (3.22)

and



e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0nD0Se−D0(τ−τM−ξ )


·N ((r−D0 +

12σ2)(τ − τM −ξ )+ ln(S)− ln(Sc(τM +ξ ))

σ√

τ − τM −ξ)dξ

−∫ τ−τM

0rMe−r(τ−τM−ξ )

·N ((r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Sp(τM +ξ ))σ√


+ nSe−D0(τ−τM)N ((r−D0 +

12σ2)(τ − τM)+ ln(S)− ln(Sc(τM))

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1

2σ2)(τ − τM)+ ln(S)− ln(Sp(τM))

σ√

τ − τM)+M.

(3.23)

To determine the price of a puttable convertible bond, V1(S,τ) and V2(S,τ), thetwo free boundaries, Sp and Sc, need to be computed first from the following three in-tegral equations constructed from substituting (3.22) and (3.23) into the boundariesconditions of Systems (3.2) and (3.3):

nSc(τ) =∫ τ

0nSc(τ)D0e−D0(τ−ξ )N (

(r−D0 +12σ2)(τ −ξ )+ ln(Sc(τ))− ln(Sc(ξ ))

σ√

τ −ξ)dξ

+ nSc(τ)e−D0τN ((r−D0 +

12σ2)τ + ln(Sc(τ))− ln(Sc(0))

σ√

τ)

− Ze−rτN ((r−D0 − 1

2σ2)τ + ln(Sc(τ))− ln(Sc(0))σ√

τ)+Ze−rτ , (3.24)

nSc(τ) =∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(Sc(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0nD0Sc(τ)e−D0(τ−τM−ξ )

·N ((r−D0 +

12σ2)(τ − τM −ξ )+ ln(Sc(τ))− ln(Sc(τM +ξ ))

σ√


−∫ τ−τM


·N ((r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(Sc(τ))− ln(Sp(τM +ξ ))σ√


+ nSc(τ)e−D0(τ−τM)N ((r−D0 +

12σ2)(τ − τM)+ ln(Sc(τ))− ln(Sc(τM))

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1

2σ2)(τ − τM)+ ln(Sc(τ))− ln(Sp(τM))

σ√

τ − τM)+M,(3.25)

M =∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(Sp(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0nD0Sp(τ)e−D0(τ−τM−ξ )


·N ((r−D0 +

12σ2)(τ − τM −ξ )+ ln(Sp(τ))− ln(Sc(τM +ξ ))

σ√


−∫ τ−τM


·N ((r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(Sp(τ))− ln(Sp(τM +ξ ))σ√


+ nSp(τ)e−D0(τ−τM)N ((r−D0 +

12σ2)(τ − τM)+ ln(Sp(τ))− ln(Sc(τM))

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1

2σ2)(τ − τM)+ ln(Sp(τ))− ln(Sp(τM))

σ√

τ − τM)+M.(3.26)

By now, the integral equation formulations for pricing a puttable convertiblebond have been presented. The solutions of the integral equations (3.24), (3.25)and (3.26) would give rise to the optimal boundaries, which can be plugged into theintegral representations (3.22) and (3.23) to calculate the bond price. However, theintegral equations are highly non-linear that a numerical method is needed to obtaintheir solutions. Therefore, in the next section, the numerical implementation of thesolution procedure for pricing a puttable convertible bond will be presented.

3.4 The numerical implementationIn the following, we will provide an outline of our numerical scheme and its valida-tion.

The major task in obtaining the numerical solutions of the integral equations isto find the values of free boundaries from Equations (3.24), (3.25) and (3.26). Oncethe free boundaries are known, we only need to numerically integrate (3.22) and(3.23) to obtain the bond prices. Our solution procedure for the free boundaries isas follows:

First, Equation (3.24) is used to obtain the values of the function Sc(τ) whenτ ∈ [0,τM]. In this process, we discretize uniformly the time interval

0 = s1 < s2 < · · ·< sN = τM where si = (i−1)∗ τM/(N −1),

and thus we obtain a set of non-linear algebraic equations for Sc(si) (denoted byS(i)c ) for i = 1,2, ...,N,

nS(i)c = Σi−1k=1nS(i)c D0e−D0(si−sk)N (

(r−D0 +12σ2)(si − sk)+ ln(S(i)c )− ln(S(k)c )

σ√

si − sk)(sk+1 − sk)

− nS(i)c D0e−D0(si−s1)N ((r−D0 +

12σ2)(si − s1)+ ln(S(i)c )− ln(S(1)c )

σ√

si − s1) · s2 − s1

2


+ nS(i)c D0 ·si − si−1

4+nS(i)c e−D0siN (

(r−D0 +12σ2)si + ln(S(i)c )− ln(S(1)c )

σ√si

)

− Ze−rsiN ((r−D0 − 1

2σ2)si + ln(S(i)c )− ln(S(1)c )

σ√si

)+Ze−rsi. (3.27)

Since the terminal value of the free boundary, S(1)c =Zn, is known, we can calculate S(i)c

for i= 2,3, ...,N recursively with a MATLAB built-in root finding function (fsolve).It should be noted that the integral term here and those in other places are replacedby summations using a standard quadrature rule, the trapezoidal rule. The aboveprocedure is similar to the one used in [68].

Equations (3.25) and (3.26) are discretized and solved simultaneously to obtainthe values of functions Sc(τ) and Sp(τ) in the interval τ ∈ [τM,T ] by using the samemethod mentioned above. Instead of providing lengthy discretized equations, herewe give brief outlines only. The time interval τ ∈ [τM,T ] is again divided uniformlyinto L−1 time intervals: [h1,h2], [h2,h3] · · · [hL−1,hL], where h1 = τM and hL = T . Thediscretized free boundaries Sc(hi) and Sp(hi) are denoted by S(i)c and S(i)p , respectively,for i = 1,2, ...,L. Since the value of the function Sp(τ) at τ = τM is equal to 0, wehave S(1)p = 0. In addition, S(1)c should be the same as that of S(N)

c obtained in[0,τM]. Knowing S(1)c and S(1)p , we calculate S(i)c and S(i)p for i = 2,3, ...,L, recursivelyusing another MATLAB built-in root finding function (lsqnonlin) as this is a twodimensional problem.

Once the values of the functions Sc(τ) and Sp(τ) are obtained, the value of thebond can be straightforwardly computed through Equations (3.22) and (3.23).

We are now ready to validate our numerical scheme. Unless otherwise stated,parameters used are listed below (the same parameter setting will be used in thenext section):

• Face value Z = 100,

• Conversion ratio n = 1,

• Maturity T = 1 (year),

• Risk-free annual interest rate r = 0.1,

• Rate of continuous dividend payment D0 = 0.07,

• Volatility σ = 0.4,

• The put price M = 95.

Under these parameters, the critical value of time to expiry τM = 0.5129 (year).


We choose to use the results calculated from the binomial tree method as thebenchmark to validate our numerical scheme. Prior to benchmarking the numer-ical results obtained from the integral equation approach against the benchmark,numerical experiments are conducted in order to make sure that the benchmarkitself obtained from the binomial tree method displays convergency. This is indeedverified as our numerical test results show that the convergence of the binomialtree method match with those reported in the literature, i.e., convergence has beenestablished, as shown in Table 3.1, when the length of time interval is reduced to1/10000 [65,69]. Since the binomial tree method will always converge according to [64],it can be used as a benchmark, and unless otherwise stated, 1,000 time steps will beused to produce the binomial-tree results in the remaining numerical experimentsfor comparison purposes, so that the number of time steps used in both methodsmatch with each other.

Table 3.1: Convergency test of the Binomial tree method

Puttable Convertible bond price at t = 0S N=1,000 N=5,000 N=8,000 N=10,000

100 107.6894 107.6905 107.6906 107.6906110 114.4379 114.4371 114.4369 114.4366120 122.1746 122.1736 122.1736 122.1736130 130.7636 130.7629 130.7629 130.7629

Now, we are ready to carry out numerical experiments to benchmark the accuracyand efficiency of our integral equation approach against the binomial tree method.From Table 3.2, one can see that all of the results from the integral equation approachat different N (the number of time intervals) agree very well with the benchmarkresults with maximum relative error within the order of 10−4. It is observed, fromthe CPU time listed in Table 3.2, that the integral equation approach is slightlymore efficient than the binomial tree methoda In addition, it should be noted thatthe time consumed in the integral equation approach includes the computation ofthe free boundaries, whereas the much longer time spent in the binomial tree methodis only limited to producing the bond price, which makes the computational speedof the integral equation approach even more impressive. To further illustrate theaccuracy of the integral equation method, the optimal boundaries obtained by theintegral equation method are compared with those obtained by the binomial treemethod in Table 3.3. It should be remarked that optimal boundaries produced by

aFollowing a similar procedure presented by Goswami & Saini [46], it is not difficult to show thatthe computational complexity of our integral equation approach is O(N2), which is the same as thatof the binomial tree method. This theoretical result is indeed consistent with what is displayedhere.


the binomial tree method with 1,000 time steps are not accurate enough, and thusthe number of time steps is further increased to 10,000. In this case, the resultsfrom the two approaches agree very well with the maximum relative error beingless than 0.5%. This again shows the superiority of our integral equation method.Overall, the benchmark tests clearly demonstrated the accuracy and efficiency ofour integral equation method.

Table 3.2: Accuracy and efficiency test of IE method

Puttable convertible bond price at t = 0S Benchmark IE IE IE

N=1,000 N=1,000 N=2,000 N=3,000100 107.6894 107.6895 107.6873 107.6866105 110.9297 110.9307 110.9291 110.9286110 114.4379 114.4381 114.4367 114.4363115 118.1927 118.1923 118.1911 118.1907120 122.1746 122.1756 122.1745 122.1741

max. relative error - 8.80×10−5 1.95×10−5 2.60×10−5

Time (second) 13.8836 10.4098 21.8943 38.6281


Optimal boundariesSc Sc Sc Sp Sp Sp

τ Benchmark Benchmark IE Benchmark Benchmark IEN=1,000 N=10,000 N=2,000 N=1,000 N=10,000 N=2,000

0.1021 123.5048 123.9848 124.2248 - - -0.2560 132.5256 133.0856 133.3356 - - -0.4098 137.4900 138.0200 138.2800 - - -0.5612 140.5231 141.0831 141.3731 61.0431 58.4431 58.39310.7073 142.4547 143.0447 143.3247 65.6605 63.4205 63.83050.8534 143.7388 144.3288 144.6088 66.6848 66.6847 67.2047

max. RE 6.11×10−3 2.06×10−3 - 4.34×10−2 7.80×10−3 -

In the following section, the number of time intervals in solving our integral equa-tions is set to be 2000 to achieve a balance between accuracy and efficiency. Inaddition, all of our calculations in this chapter are done on a PC with the followingspecifications: Intel(R) Xeon(R), CPU E5-1640 v4 @3.60GHz 3.60 GHz, and 32.0GB of RAM.


3.5 Examples and discussionsIn this section, numerical examples are provided to illustrate various properties ofputtable convertible bonds, and the difference between vanilla and puttable CBs isalso demonstrated.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiry

0

50

100

150

200

250

300

Valu

e o

f op

tim

al bo

unda

rie

s

τMo

n=0.5

n=1

n=2

Figure 3.1: The value of the optimal boundaries for three different conversionratios

Figure 3.1 shows both the optimal conversion price and the optimal put price withrespect to the time to expiry. It can be seen that both the optimal conversion priceand the optimal put price are the monotonically increasing functions of the time toexpiry, τ = T − t. And as the conversion ratio becomes larger, the optimal exercisecurves become flatter. Naturally, both the optimal conversion price and the optimalput price vary inversely with the conversion ratio, and the optimal conversion pricesat expiry are the strike price divided by the conversion ratio. In fact, these propertiesare the same as those of the vanilla convertible bond. For puttable CBs, it shouldbe observed that there is only one free boundary, the optimal conversion boundary,during [0,τM], since the value of the optimal put boundary is equal to zero in thistime interval. When the time to expiry is greater than the critical value of thetime to expiry, τM, the optimal put boundary “appears” due to the existence of the“put” feature. It is observed that when time to expiry is closer to zero, the optimalconversion price decreases quickly to the value of the strike price divided by theconversion ratio, and that as time to expiry approaches τM, the optimal put price


drops rapidly to zero. The large slope of Sc(τ) and that of Sp(τ) near τ = 0 andτ = τM, respectively, are similar to the behavior of the optimal exercise price nearexpiry [42].

0 20 40 60 80 100 120 140 160 180 200

Stock Price

80

100

120

140

160

180

200

Valu

e o

f th

e b

ond

t=0.7435

t=0.4871

t=0.2435

t=0.0000

Figure 3.2: The price of the puttable CB at four different time moments

Depicted in Figure 3.2 are the price curves of the puttable CB versus the un-derlying asset value, S, at times t = 0.0000, t = 0.2435, t = 0.4871, t = 0.7435. Weobserve that the slope of the price curves is zero when the underlying asset is worth-less, increasing slowly at first at lower underlying asset price, and eventually allcurves become tangent to the payoff line. This observation indicates that the first-order partial derivative of the bond price with respect to the underlying asset priceis between 0 and the conversion ratio n, that is 0 ≤ ∂V

∂S≤ n. It can be seen that

the bond price remains almost unchanged when the underlying asset price is low,the greater the time, the higher the bond price. However, when the underlyingasset price increases to a certain extend, a completely different phenomenon can beobserved: the bond price becomes lower as time increases.

Figure 3.3 displays the value of a puttable CB and its vanilla counterpart. It canbe seen that the value of the puttable convertible bond is higher than that of thevanilla one. This certainly makes sense since the holder of a puttable convertiblebond has an additional right to sell the bond back to the issuer, and thus theholder should be expected to pay an extra amount as a “premium”. It is interestingto observe that such a premium decreases as the underlying asset price becomes


0 20 40 60 80 100 120 140 160 180 200

Stock price

0

20

40

60

80

100

120

140

160

180

200

Va

lue

of

the

bo

nd

Price of puttable CBs

Price of vanilla CBs

Figure 3.3: The price of the puttable and vanilla CBs at the same time

higher, and when the price of the underlying asset is very high, this premium isalmost equal to zero. In other words, the price of the puttable convertible bond andthat of the vanilla counterpart are almost equivalent to each other when the price ofthe underlying asset is very high. This can be easily explained since when the priceof the underlying asset is high, there is no financial incentive for the holder to sellthe bond back to the issuer. In this case, the puttable convertible bond can almostbe replaced by the vanilla convertible bond. In contrast, the puttable convertiblebond is worth more when the price of the underlying asset is low.

Figures 3.4 and 3.5 show the effects of the volatility on the bond price as well as itsoptimal conversion price and optimal put price. In particular, exhibited in Figure 3.4are the bond prices corresponding to three different volatility values. When the stockprice is very low, the bond prices are insensitive to the variation of volatility, sincewhen the stock price is low, the bond price remains almost unchanged and are equalwhen S is zero. Moreover, the value of the puttable CB is a monotonically increasingfunction of volatility. This is reasonable because when the volatility becomes larger,there is a higher risk, which will lead to a higher price. On the other hand, fromFigure 3.5, it is easy to note that both the optimal conversion price and the optimalput price are the increasing functions of the time to expiry. Another interestingphenomenon is that a higher volatility will lead to a higher optimal conversion pricewhile it will lead to a less optimal put price.


0 20 40 60 80 100 120 140 160 180 200

Stock price

0

20

40

60

80

100

120

140

160

180

200

Valu

e o

f th

e b

ond

σ=0.3

σ=0.4

σ=0.5

Figure 3.4: The price of puttable CBs for three different volatilities

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiry

0

20

40

60

80

100

120

140

160

180

Valu

e o

f o

ptim

al bonudaries

τMo

σ=0.3

σ=0.4

σ=0.5

Figure 3.5: Optimal boundaries prices for three different volatilities

In Figures 3.6 and 3.7, we show how the price of a puttable convertible bond andboth its optimal conversion price and optimal put price change with the risk-free


0 20 40 60 80 100 120 140 160 180 200

Stock price

0

20

40

60

80

100

120

140

160

180

200

Valu

e o

f th

e b

ond

r=0.08

r=0.1

r=0.15

Figure 3.6: The price of puttable CBs for three different risk-free interest rates

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiry

0

50

100

150

Valu

e o

f o

ptim

al bonudaries

r=0.08

r=0.1

r=0.15

Figure 3.7: Optimal boundaries prices for three different risk-free interest rates

interest rate. Comparing the bond price as well as its two free boundaries shown inthis figure with those shown in Figures 3.4 and 3.5, respectively, one can observe that


the risk-free interest rate has quite a different influence than volatility does. If weincrease the risk-free interest rate, the bond price will decrease, this can be easilyexplained since when the risk-free interest rate is higher, it gives more incentivefor investors to leave their money in a risk-free environment than buying a riskybond, resulting a lower CB price as displayed in Figure 3.6. On the other hand,in Figure 3.7 opposite trends for the two sets of free boundaries are observed, theoptimal conversion price is a decreasing function of the risk-free interest rate whilethe optimal put price is an increasing function.

3.6 ConclusionIn this chapter, the pricing problem of a puttable convertible bond on a single under-lying asset with constant dividend is considered, two integral equation formulationsare presented for the first time. The integral equations are solved numerically to ob-tain the two free boundaries, and the bond price is then calculated from the integralrepresentations in their respective domains. Numerical examples are provided toshow some interesting properties of puttable convertible bonds, subject to differentvalues of the volatility and the interest rate.

It should be remarked here that the current approach can be extended to solvingthe pricing problem for callable convertible bonds, in which case there exists at mostone free boundary.

Chapter 4

Pricing callable-puttable convertible bondswith an integral equation approach

4.1 IntroductionConvertible bonds (CBs) are widely used financial instruments, which are differentfrom bonds and stocks. However, CBs can be treated as a combination of bondsand options, since they possess the essential characteristics of these two. A CB givesits holders a right to convert the bond into a predetermined number of underlyingstocks either only at the expiry (the so-called European-style) or during the entirelife of the bond (the so-called American-style). Although such a right enables theholders to benefit from both the security of a bond as well as a possible higherreturn through a more risky underlying asset such as stocks, it also results in amuch more complex pricing problem, especially for those of American-style sincethey are allowed to be converted at any time.

Various models have been used to price CBs. A simple choice was the Black-Scholes model [63]. Ingersoll [62] and Brennan & Schwartz [12] were the first to workon the problem under this model. In their approach, the firm value was utilized asthe underlying asset. However, firm values are not observable in real markets andthus their approach has some drawbacks in practice as far as model calibration isconcerned. Later on, McConnel & Schwartz [84] proposed to adopt the stock priceinstead of the firm value as the underlying variable to price CBs.

Since then, research activities in the area of pricing CBs intensified. Among alarge number of papers published in the past 30 years, numerical approaches, suchas the finite element method [6], the finite difference method [98] and the finite volumemethod [112], are often adopted. However, two main drawbacks, i.e. the accuracyproblem and the time-consuming feature that exist in most of the numerical meth-ods, prompted researchers to seek analytical solution approaches for their simplicity

61

CHAPTER 4. PRICING CALLABLE-PUTTABLE CONVERTIBLE BONDS 62

and analytical elegancy, though they are quite often restricted to some relativelysimple cases. For example, a closed-form solution for a simple CB, which can onlybe converted at maturity, was obtained by Nyborg [88], while Zhu [106] presented ananalytical solution in the form of a Taylor series expansion for the simplest Ameri-can style CB without any other clauses being added, using the Homotopy AnalysisMethod [74].

As one of the most popularly used financial derivatives in financial practice forfirms to raise needed capital, CBs today, stemming from the very basic original con-cept, have many variations with some quite involved terms, clauses and conditions.Among them, callable CBs and puttable CBs are two kinds of the most popularCBs [2]. The former is a bond that allows the issuer to call (repurchase) the bondfrom the holder for a predetermined call price, which is used to protect the issueragainst the risk of the underlying running unreasonably much higher than initiallyexpected. When the underlying asset price increases beyond a preset critical valuethat is related to the conversion ratio and the call price, the issuer can call back thebond at the call price. Therefore, the price of the callable CB should be less thanthat of the vanilla counterpart, as a result of the holder’s potential return is cappedfrom the above. On the other hand, puttability permits the holder to sell the bondback to the issuer at a predetermined put price. Obviously, the put feature benefitsthe holder of the bond, and thus a puttable CB is traded at a higher price than thatof its vanilla counterpart.

Regarding solving the pricing problem of callable CBs, there are many referencematerials. While Brennan & Schwartz [12] explained in theory on how to price thecallable CB, and provided solutions using the finite difference method in their laterarticle [13], Bernini [7] used the binomial tree method to obtain the solution. Yagi &Sawaki [103] priced the callable CBs with the utilization of the game options definedby Kifer [67]. On the other hand, there are also a few references on pricing puttableCBs in the literature. For instance, Nyborg [88] presented the boundary condition ofputtable CBs, while Lvov et al. [80] obtained the numerical solution by using MonteCarlo simulations.

In this chapter, two types of CBs mentioned above are combined together toform a new type of CBs, called callable-puttable CBs, which should be consideredon behalf of both the issuer and the holder. An integral equation formulation ispresented to price callable-puttable CBs under the Black-Scholes model with themethod of incomplete Fourier transform [26] and the Green’s function [39]. One mayargue that it is more practical to adopt stochastic interest rate models [14,29,100] forpricing CBs, as CBs are usually designed for a long time period, during which theinterest rate itself may be subject to changes. However, we still assume a constantinterest rate in this study, since the pricing exercise is already very complicated


even under this simple model, resulting from the tangled presence of callability,puttability, as well as conversion, which have led to possible co-existence of twomoving boundaries at the same time, depending on the values of the call price, theput price and the conversion ratio.

If a callable-puttable CB needs to be priced at a time sufficiently far away from theexpiry, only the moving boundary associated with the puttability needs to be dealtwith. For this situation, the partial differential equation (PDE) system governingthe price of a callable-puttable CB is presented. When the pricing time is closer toexpiry beyond a critical value, it is then possible to have two distinct cases. Whilethe two moving boundaries associated with conversion and puttability co-exist inone case, they may both disappear in another with callability remaining to be theonly issue that needs to be dealt with. The former case can be solved through one ofthe PDE systems presented in [109], while the PDE system for the latter case can bebuilt without the presence of any free boundaries. Furthermore, there exists anothercritical value, beyond which the callable-puttable CB can be treated as the vanillacounterpart, solving which requires the utilization of the PDE system presentedin [106]. In summary, the pricing problem for our issue should be designed with threedifferent scenarios, and in each case, there are three or two PDE systems governingthe price of a callable-puttable CB.

This chapter is organized as follows. In Section 2, the pricing problem is dividedinto three cases, and the PDE systems governing the price of a callable-puttable CBare established for each case, and also the form of integral equation is derived. InSection 3, we compared our results with the known benchmark. Numerical examplesare presented in Section 4, followed by some concluding remarks given in the lastsection.

4.2 Models and resultsIn this section, the PDE systems are established to price callable-puttable CBs underthe Black-Scholes model, and the integral equation formulations are obtained bysolving these systems. As mentioned above, the pricing problem should be dividedinto three scenarios according to the relationship between the values of the callprice and that of the put price. In particular, due to the fact that two additionalrights are actually added into the vanilla CB, there will be two critical moments inthe callable-puttable CB, corresponding to the time instances when the callabilitydisappears and the same instance when the puttability disappears. The differentorder of the arrival of these two moments makes the pricing problem for the bondquite different, and three scenarios distinguished by which moment arrives earlierare thus considered.


Before we discuss the difference between three scenarios of the callable-puttableCB, the two similarities among these three should be pointed out first. One is thatboth of the puttability and callability are possible at the beginning of the bond,otherwise there are no financial incentive to exercise both of the two features duringthe lifetime of the bond. The reason is that maximum and minimum values of thebond are a decreasing and an increasing function of the time, respectively, whichmakes it impossible for the bond value to reach either the value of the call price orthe value of the put price if initially the maximal and minimal bond value is lowerand higher than the value of the call price and the value of the put price, respectively.Another one is that it is not an optimal choice to call or put the bond when the timeis sufficiently close to expiry, since during this time period the maximal bond valueis smaller than the value of the call price, K, and the minimal bond value is largerthan the value of the put price, M. Therefore, the PDE systems corresponding tothese two time intervals are the same for three scenarios. On the other hand, oneshould also be noted that the two critical moments can separate the time zone intotwo or three parts, and this means that the difference between these three cases isonly the middle part. In the following, these three cases are discussed one by one.

In Case 1, when the value of the call price is sufficiently small, the callabilitydisappears later since a small value of the call price makes it harder for the maximalprice of the bond to drop down below the value of the call price compared with thecase that the minimal value of the bond hits the value of the put price. Consideringthe property of a callable CB, the moment when the callability disappears is also thetime when the value of the optimal conversion boundary gets to the value of the callprice divided by the value of the conversion ratio, K

n, where n is the conversion ratio.

Thus, for this case, the first parta actually consists of the time to expiry period whenthe value of the optimal put boundary is equal to zero and the value of the optimalconversion boundary is less than a certain value, which implies that the PDE systemfor this part is actually as same as that for the vanilla CB. The second part of thiscase represents the time to expiry period when callability is available while there isno sense to exercise puttability, which clearly shows that the value of the optimalput boundary is equal to zero and the PDE system for this part is the same as thatfor the callable CB. At last, the third part of Case 1 is the real callable-puttable CBpart, in which both the callability and the puttability are possible and the value ofthe optimal put boundary is no longer zero.

In Case 2, when the value of the call price is sufficiently large, the moment whenthe minimal value of the bond hits the value of the put price comes later. Similar

aUnder the classical treatment of the financial mathematics, we consider the pricing problemwith the increase of the time to expiry directly. Therefore, the first part in this chapter means thetime is sufficiently close to the expiry.


to Case 1, the first part still models the time to expiry period when the callabilityand the puttability are not active, and the third part denotes the situation when thecallability and the puttability both exist. However, being different from Case 1, thecallability, which is possible in the second part of Case 1, is no longer meaningful,while both the puttability and conversion come into effect in the current situation.

Case 3 is actually a special case that the two moments arrive at the same time,and thus there are only two parts with the first one being equivalent to the vanillaCB and the last one being the callable-puttable one.

Having been aware of the similarity and difference among these three cases, thevaluation of callable-puttable conversion bonds under the three cases will be dis-cussed in next three subsections, respectively.

4.2.1 Case 1

In this subsection, the pricing problem of callable-puttable CBs for Case 1 will bediscussed, in which the moment that the optimal conversion price reaches K

nlater

than the moment that the minimum value of the bond gets to the value of the putprice. Firstly, let St be an underlying asset price and we assume that its dynamicsfollows stochastic different equation (the same assumption will be used in the nexttwo cases):

dSt = (r−D0)Stdt +σStdWt , (4.1)

where Wt is a Brownian motion, σ is the volatility of the underlying asset, r is therisk-free interest rate, and D0 is the continuous dividend rate.

Let V1(S,τ) be the value of the callable-puttable CB for Case 1, with the time toexpiry, τ = T − t. Then, when the time to expiry is small enough, the value of theoptimal put boundary is always equal to zero since it is not optimal for the holderto sell the bond back to the issuer, and at the same time, the value of the optimalconversion boundary does not reach the value K

n, which means the issuer will not

choose to call back the bond. During this time interval, the callable-puttable CBcan be treated as a vanilla one. Therefore, V1(S,τ) in Part 1 should satisfy thefollowing PDE system (c.f. [106]):

−∂V1

∂τ+

12

σ2S2 ∂ 2V1

∂S2 +(r−D0)S∂V1

∂S− rV1 = 0,

V1(S,0) = minK,maxnS,Z,V1(Sc(τ),τ) = nSc(τ),∂V1

∂S(Sc(τ),τ) = n,

V1(0,τ) = Ze−rτ ,

(4.2)


with the domain of S and τ being [0,Sc(τ)] and [0,τK], respectively. Here, Z is theface value of the bond, Sc(τ) is the value of the optimal conversion boundary and τK

is the moment that the value of the optimal conversion boundary reaches Kn, which

is the maximum value of the optimal conversion boundary for a callable-puttableCB.

With the PDE system for Part 1 of Case 1 being established, we can proceedto Part 2. According to the property of the callable CB, the value of the optimalconversion boundary should be constant and be the same as the value that the issuercalls the bond back divided to the value of the conversion ratio. Also, the value ofthe optimal put boundary is equal to zero in this interval, since the minimal value ofthe bond is not bounded below by the value of the put price during this time zone.So, the PDE system can be built as follows:

−∂V1

∂τ+

12

σ2S2 ∂ 2V1

∂S2 +(r−D0)S∂V1

∂S− rV1 = 0,

V1(S,τ+K ) =V1(S,τ−K ),

V1(Kn,τ) = K,

V1(0,τ) = Ze−rτ ,

(4.3)

with the domain of S and τ being [0,Kn] and [τK,τM], respectively, where τM is the

moment that the minimal value of the bond hits the value of the put price, i.e.the value of the vanilla CB at S = 0 is bounded below by M, since the value ofthe bond at any certain time is an increasing function with the value of underlyingasset. Therefore, τM determined by the following equation: Ze−rτM = M, i.e. τM =

−1r

logMZ. Another fact that should be noted is that the value of two boundaries in

this PDE system are all constants.On the other hand, for the third time interval of Case 1, the value of the optimal

conversion boundary is still a constant due to the existence of callability, while thevalue of the optimal put boundary will change with time to expiry since it is incentiveto exercise puttability. Thus, the PDE system for Part 3 of Case 1 can be set up as:

−∂V1

∂τ+

12

σ2S2 ∂ 2V1

∂S2 +(r−D0)S∂V1

∂S− rV1 = 0,

V1(S,τ+M) =V1(S,τ−M),

V1(Kn,τ) = K,

V1(Sp(τ),τ) = M,∂V1

∂S(Sp(τ),τ) = 0,

(4.4)

with the domain of S and τ being [Sp(τ),Kn] and [τM,T ], respectively. Here, Sp(τ)


is the value of the optimal put boundary. By now, the PDE systems for pricingcallable-puttable CBs of Case 1 have been established, and the integral equationformulation will be obtained to determine the bond value, starting from Part 1.

The integral equation representation for Part 1 of Case 1

For this part, only one free boundary should be considered when we derive theintegral equation formula. According to [26], the incomplete Fourier transform canbe used to solve this problem, before applying which a classical transform needs tobe applied to the PDE system (4.2) first, so that a dimensionless PDE system canbe obtained. Let

x = ln(S), v1(x,τ) =V1(S,τ),

and then the target PDE system can be simplified to

−∂v1

∂τ+

12

σ2 ∂ 2v1

∂x2 +(r−D0 −12

σ2)∂v1

∂x− rv1 = 0,

v1(x,0) = minK,maxnex,Z,v1(ln(Sc(τ)),τ) = nSc(τ),∂v1


v1(−∞,τ) = Ze−rτ ,

(4.5)

with the domain of x and τ being [−∞, ln(Sc(τ))] and [0,τK], respectively. It needs tobe pointed out that the so-called incomplete Fourier transform can not be appliedto System (4.5) directly, since the boundary condition at infinity is non-zero. Thus,another transform:

U1(x,τ) = v1(x,τ)−Ze−rτ ,

is used, and then the PDE system (4.5) can be rewritten as:

−∂U1

∂τ+

12

σ2 ∂ 2U1

∂x2 +(r−D0 −12

σ2)∂U1

∂x− rU1 = 0,

U1(x,0) = minK −Z,maxnex −Z,0,U1(ln(Sc(τ)),τ) = nSc(τ)−Ze−rτ ,∂U1


U1(−∞,τ) = 0.

(4.6)

Considering the definition of the domain, the incomplete Fourier transform forthis part can be defined as follows:

FU1(x,τ)=∫ ln(Sc(τ))

−∞U1(x,τ) · eiωxdx , U1(ω,τ). (4.7)


Applying the incomplete Fourier transform (4.7) to System (4.6) yields the followingordinary differential equation (ODE) system (the details are presented in AppendixB.1)

∂U1

∂τ(ω,τ)+B(ω)U1(ω,τ) = f (ω,τ),

U1(ω,0) =∫ ln(Sc(0))

−∞minK −Z,maxnex −Z,0 · eiωxdx,

(4.8)

where

B(ω) =12

σ2ω2 +(r−D0 −12

σ2)iω + r, (4.9)

f (ω,τ) = (nSc(τ)−Ze−rτ)eiω ln(Sc(τ))

·[S′c(τ)

Sc(τ)+(r−D0 −

12

σ2)− 12

σ2iω]+12

σ2nSc(τ)eiω ln(Sc(τ)). (4.10)

Clearly, System (4.8) is a non-homogeneous first-order linear ODE system withan initial condition, using the general solution for which can lead to the integralequation formulation of the bond value in the Fourier space

U1(ω,τ) = U1(ω,0) · e−B(ω)τ +∫ τ

0f (ω,ξ ) · e−B(ω)(τ−ξ )dξ . (4.11)

But, it is better to derive the formulation in the original space than leaving it inthe Fourier space, since numerically inverting Fourier transform consumes a lot oftime. Therefore, the incomplete Fourier inversion transform is defined as

U1(x,τ) = F−1U1(ω,τ)= 12π

∫ ∞

−∞U1(ω,τ)e−iωxdω, (4.12)

needs to be applied to Equation (4.11).It should be noted that the definition of the incomplete Fourier inversion trans-

form is as same as the standard counterpart. In fact, according to [26], althoughthe definition of the incomplete Fourier transform can be quite different from thestandard one, the inversion transforms for these two are the same except the domainof the definition. Therefore, we will use the same definition for Fourier inversiontransform of different incomplete Fourier transforms in the following.

By applying Fourier inversion transform (4.12) on Equation (4.11), we obtain

U1(x,τ) =∫ ln(Sc(0))

−∞minK −Z,maxneu −Z,0 e−rτ

√2πσ2

· e−[(r−D0−

12 σ2)τ+x−u]2

2σ2τ du

+∫ τ

0

e−r(τ−ξ )√2πσ2(τ −ξ )

e− [(r−D0−

12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )


·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

σ2nSc(ξ )dξ . (4.13)

Then, the derivation of which is put in Appendix B.2, the integral representationin the original space can be expressed with the initial parameters as

V1(S,τ) =∫ ln(Sc(0))


√2πσ2

· e−[(r−D0−

12 σ2)τ+ln(S)−u]2

2σ2τ du

+∫ τ

0

e−r(τ−ξ )√2πσ2(τ −ξ )

e− [(r−D0−

12 σ2)(τ−ξ )+ln(S)−ln(Sc(ξ ))]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

σ2nSc(ξ )dξ +Ze−rτ .

(4.14)

By now, we have presented the integral representation for Part 1 of Case 1. How-ever, it should be noted that the integral equation formulation obtained by theincomplete Fourier transform is not perfect to price a bond [26]. Therefore, a betterone can be derived as

V1(S,τ) = D0S∫ τ

0ne−D0(τ−ξ )N (

(r−D0 +12σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)

σ√

τ −ξ)dξ

+ nSe−D0τN ((r−D0 +

12σ2)τ + ln(S)− ln(Sc(0))

σ√

τ)

− Ze−rτN ((r−D0 − 1

2σ2)τ + ln(S)− ln(Sc(0))σ√

τ)+Ze−rτ , (4.15)

after some complex computations presented in Appendix B.3. Clearly, the integralrepresentation just derived for pricing a callable-puttable CB of Part 1 for Case 1is in form of a unknown function, Sc(τ), and the method to obtain the value ofthe unknown function is displayed at the end of this subsection, after the forms ofintegral equation to PDE systems of Part 2 and Part 3 are obtained.


It is interesting to note that there is no free boundary in the PDE system (4.3), andthus its analytical solution can be directly derived using the Green’s function

V1(S,τ) =1√

2πσ2(τ − τK)e−

[(r−D0−12 σ2)2+2rσ2](τ−τK )

2σ2

·∫ ln(K

n )

−∞[exp(− [− ln(S)+ y]2

2σ2(τ − τK))− exp(−

[− ln(S)+2ln(Kn )− y]2

2σ2(τ − τK))]

·exp(r−D0 − 1

2σ2

σ2 (− ln(S)+ y)) · (V1(ey,τK)−Ze−rτK)dy


+∫ τ−τK

0

− ln(S)+ ln(Kn )√

2πσ2(τ − τK −ξ )3· exp(−

[− ln(S)+ ln(Kn )]

2

2σ2(τ − τK −ξ )) · (K −Ze−r(τK+ξ ))

·exp(r−D0 − 1

2σ2

σ2 (ln(Kn)− ln(S))+

(r−D0 − 12σ2)2 +2rσ2

2σ2 (ξ + τK − τ))dξ

+ Ze−rτ , (4.16)

the complicated solution process of which is illustrated in Appendix B.4. With inte-gral representations of bond values for the first two parts in Case 1 being obtained,the left task is to work out the third one, details of which are shown below.

The Integral representation for Part 3 of Case 1

The PDE system actually contains one free boundary, and thus the incompleteFourier transform can again be used to obtain the integral equation representation.To transform the PDE system to a dimensionless one, a classical transform is appliedfirst:

x = ln(S), v1(x,τ) =V1(S,τ),

which leads to

−∂v1

∂τ+

12

σ2 ∂ 2v1

∂x2 +(r−D0 −12

σ2)∂v1

∂x− rv1 = 0,

v1(x,τ+M) = v1(x,τ−M),

v1(ln(Kn),τ) = K,

v1(ln(Sp(τ)),τ) = M,∂v1

∂x(ln(Sp(τ)),τ) = 0,

(4.17)

with the domain of x and τ being [ln(Sp(τ)), ln(Kn)] and [τM,T ], respectively. It

should be emphasized that the domain of x for this part is different from thatfor Part 1 of Case 1, implying that another definition of the incomplete Fouriertransform will be introduced:

Fv1(x,τ)=∫ ln(K

n )

ln(Sp(τ))v1(x,τ)eiωxdx , v1(ω,τ). (4.18)

When we try to apply the incomplete Fourier transform just defined (4.18) toSystem (4.17) directly, there exists an obstacle that the first-order derivative of theCB price at S =

Kn

is unknown. Therefore, we introduce a time-dependent function

∂V1

∂S(Kn,τ) = A(τ), (4.19)


which is determined simultaneously with the unknown free boundaries later. Thus,System (4.17) should be rewritten with one more boundary condition

−∂v1

∂τ+

12

σ2 ∂ 2v1

∂x2 +(r−D0 −12

σ2)∂v1

∂x− rv1 = 0,

v1(x,τ+M) = v1(x,τ−M),

v1(ln(Kn),τ) = K,

∂v1

∂x(ln(

Kn),τ) =

Kn

A(τ),

v1(ln(Sp(τ)),τ) = M,∂v1

∂x(ln(Sp(τ)),τ) = 0.

(4.20)

In this case, formally applying the incomplete Fourier transform to the above PDEsystem, gives the following ODE system

∂ v1

∂τ(ω,τ)+B(ω)v1(ω,τ) = f1(ω,τ)− f2(ω,τ),

v1(ω,τM) =∫ ln(K

n )

ln(Sp(τM))v1(x,τM) · eiωxdx,

(4.21)

where

f1(ω,τ) = Keiω ln(Kn ) · [1

2σ2 A(τ)

n− 1

2σ2iω +(r−D0 −

12

σ2)], (4.22)

f2(ω,τ) = Meiω ln(Sp(τ)) · [S′p(τ)

Sp(τ)− 1

2σ2iω +(r−D0 −

12

σ2)], (4.23)

and B(ω) is as the same definition as before. We refer interested readers to AppendixB.5 for derivation details.

One can easily find that this is again a non-homogeneous first-order linear ODEsystem, and the solution of it can be derived as

v1(ω,τ) = e−B(ω)(τ−τM) · v1(ω,τM)

+∫ τ−τM

0f1(ω,ξ + τM) · e−B(ω)(τ−τM−ξ )dξ

−∫ τ−τM

0f2(ω,ξ + τM) · e−B(ω)(τ−τM−ξ )dξ , (4.24)

which is the solution in the Fourier space. Upon applying the Fourier inversiontransform (the procedures are in Appendix B.6), the integral representation in theoriginal space with the original parameters can be obtained

V1(S,τ) =∫ ln(K

n )


e−r(τ−τM)√2πσ2(τ − τM)

e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du


+∫ τ−τM

0

Ke−r(τ−τM−ξ )√2πσ2(τ − τM −ξ )

e− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(S)−ln(K

n )]2


·12

σ2 A(τM +ξ )n

+12[r−D0 −

12

σ2 +ln(K

n )− ln(S)τ − τM −ξ

]dξ

−∫ τ−τM

0

Me−r(τ−τM−ξ )√2πσ2(τ − τM −ξ )

e− [(r−D0−



·S′p(τM +ξ )Sp(τM +ξ )

+12(r−D0 −

12


τ − τM −ξ)dξ .(4.25)

As mentioned above, the integral representation obtained by the incompleteFourier transform may have some problems, such as the accuracy problem in theboundary and the higher requirement for the smoothness of the free boundary func-tions. As a result, it is not perfect for us to use this formula to price a bond, andan alternative one can be derived after some complex computations

V1(S,τ) =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0

Ke−r(τ−τM−ξ )√

2πe− [(r−D0−


n )]2

2σ2(τ−τM−ξ ) · σA(τM +ξ )2n√

τ − τM −ξdξ

−∫ τ−τM


·N ((r−D0 − 1



−∫ τ−τM

0rKe−r(τ−τM−ξ )

·N (−(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Kn )

σ√


−Ke−r(τ−τM)N (−(r−D0 − 1

2σ2)(τ − τM)+ ln(S)− ln(Kn )

σ√

τ − τM)

−Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM). (4.26)

Again, the details are left in Appendix B.7.By now, the integral equation representations for pricing the callable-puttable

CB of Case 1 have been derived, with three unknown functions, Sc(τ), Sp(τ) andA(τ). Fortunately, we can derive three integral equations for them using the freeboundaries conditions

nSc(τ) = nSc(τ)e−D0τN ((r−D0 +

12σ2)τ + ln(Sc(τ))− ln(Sc(0))

σ√

τ)


− Ze−rτN ((r−D0 − 1


τ)

+ D0Sc(τ)∫ τ


(r−D0 +12σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(Sc(τ))

σ√

τ −ξ)dξ

+ Ze−rτ , (4.27)

K =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(K

n )−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2πe− [(r−D0−

12 σ2)(τ−τM−ξ )]2

2σ2(τ−τM−ξ ) · σA(τM +ξ )2n√

τ − τM −ξdξ

−∫ τ−τM


·N ((r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(Kn )− ln(Sp(τM +ξ ))

σ√


−∫ τ−τM

0rKe−r(τ−τM−ξ )N (−

(r−D0 − 12σ2)(τ − τM −ξ )

σ√


− Ke−r(τ−τM)N (−(r−D0 − 1

2σ2)(τ − τM)

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1

2σ2)(τ − τM)+ ln(Kn )− ln(Sp(τM))

σ√

τ − τM), (4.28)

M =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(Sp(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2πe− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(Sp(τ))−ln(K

n )]2

2σ2(τ−τM−ξ ) · σA(τM +ξ )2n√

τ − τM −ξdξ

−∫ τ−τM


·N ((r−D0 − 1



−∫ τ−τM


·N (−(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(Sp(τ))− ln(Kn )

σ√


− Ke−r(τ−τM)N (−(r−D0 − 1

2σ2)(τ − τM)+ ln(Sp(τ))− ln(Kn )

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM). (4.29)

Therefore, using these three integral equations, the value of three unknown functionscan be obtained, after which the price of the bond can be presented through theintegral equation formulation directly.

With the valuation problem of a callable-puttable CB for Case 1 being successfully


solved, it is time to consider Case 2, which will be discussed in the next subsection.

4.2.2 Case 2

In this subsection, the PDE systems for the second case of callable-puttable CBsare established. For this case, we assume that the moment when the value of theoptimal conversion boundary reaches K

nappears earlier than the moment when the

minimum value of the bond gets to the value of the put price. Similar to Subsection4.2.1, the PDE systems also be set up with respect to the time to expiry. When thetime is sufficiently close to the expiry, it is not optimal for the holder to sell the bondback to the issuer, so the value of the optimal put boundary is always equal to zeroin this interval until the time to expiry reaches a certain value, τM. Furthermore,since it is close to expiry, the value of the optimal conversion boundary is unable toreach K

n, and thus the boundary condition for the optimal convertible boundary is

same to the vanilla CB. As a result, the PDE system for this part is as same as thatfor the vanilla CB.

If the value of the callable-puttable CB for Case 2 is assumed V2(S,τ), then thePDE system for the first part of Case 2 is

−∂V2

∂τ+

12

σ2S2 ∂ 2V2

∂S2 +(r−D0)S∂V2

∂S− rV2 = 0,


∂S(Sc(τ),τ) = n,

V2(0,τ) = Ze−rτ ,

(4.30)

with the domain of S and τ being [0,Sc(τ)] and [0,τM], respectively.For Part 2 of Case 2, the terminal time of this part will be the moment that the

value of the optimal conversion boundary reaches Kn. Thus, during this time zone,

the issuer is not willing to call the bond back either, which leads to the optimalconversion boundary condition for this part being the same as that for Part 1, whilethe value of the optimal put boundary is longer equal to zero. This can give rise to


the PDE system for this time zone

−∂V2

∂τ+

12

σ2S2 ∂ 2V2

∂S2 +(r−D0)S∂V2

∂S− rV2 = 0,

V2(S,τ+M) =V2(S,τ−M),

V2(Sc(τ),τ) = nSc(τ),∂V2

∂S(Sc(τ),τ) = n,

V2(Sp(τ),τ) = M,∂V2

∂S(Sp(τ),τ) = 0,

(4.31)

with the domain of S and τ being [Sp(τ),Sc(τ)] and [τM,τK], respectively.Now, the PDE system for the third time interval of Case 2 is set up. In fact, it

should be noted that the situation of two free boundaries in this time interval isas the same as that in Part 3 of Case 1. Therefore, the PDE system can be builtdirectly by making use of the corresponding counterpart in Case 1

−∂V2

∂τ+

12

σ2S2 ∂ 2V2

∂S2 +(r−D0)S∂V2

∂S− rV2 = 0,

V2(S,τ+K ) =V2(S,τ−K ),

V2(Kn,τ) = K,

V2(Sp(τ),τ) = M,∂V2

∂S(Sp(τ),τ) = 0,

(4.32)

with the domain of S and τ being [Sp(τ),Kn] and [τK,T ], respectively.

Obviously, the PDE systems for pricing callable-puttable CBs of Case 2 have beenset up as (4.30), (4.31) and (4.32). It is interesting to observe that the PDE systemfor Part 1 of this case is almost as same as that of Case 1, and the same phenomenonhappens for Part 3. As a result, the integral equation representations in Case 1 forthese two parts can be applied to this case directly.


Since there is no different between this part and Part 1 of Case 1, the integralequation representation can be written as

V2(S,τ) = nSe−D0τN ((r−D0 +

12σ2)τ + ln(S)− ln(Sc(0))

σ√

τ)

− Ze−rτN ((r−D0 − 1

2σ2)τ + ln(S)− ln(Sc(0))σ√

τ)


+ D0S∫ τ


(r−D0 +12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))

σ√

τ −ξ)dξ

+ Ze−rτ . (4.33)


With a careful observation, it is not difficult to find that this part is actually equiva-lent to the second part of the puttable CB, derived in [109]. Thus, the correspondingresults there can be directly utilized so that the integral equation representation canbe expressed




e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM


·N ((r−D0 +


σ√


−∫ τ−τM


·N ((r−D0 − 1



+ nSe−D0(τ−τM)N ((r−D0 +


σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM). (4.34)


The PDE system for this part shows that it is almost as the same as the Part 3of Case 1, except the domain of the time interval. Hence, the formulation of theintegral equation formulation can be presented as

V2(S,τ) =∫ ln(K

n )

ln(Sp(τK))V2(eu,τK)

e−r(τ−τK)√2πσ2(τ − τK)

e− [(r−D0−

12 σ2)(τ−τK )+ln(S)−u]2

2σ2(τ−τK ) du

+∫ τ−τK

0

Ke−r(τ−τK−ξ )√

2πe− [(r−D0−

12 σ2)(τ−τK−ξ )+ln(S)−ln(K

n )]2

2σ2(τ−τK−ξ ) · σA(τK +ξ )2n√

τ − τK −ξdξ

−∫ τ−τK

0rMe−r(τ−τK−ξ )

·N ((r−D0 − 1

2σ2)(τ − τK −ξ )+ ln(S)− ln(Sp(τK +ξ ))σ√

τ − τK −ξ)dξ

−∫ τ−τK

0rKe−r(τ−τK−ξ )


·N (−(r−D0 − 1

2σ2)(τ − τK −ξ )+ ln(S)− ln(Kn )

σ√


− Ke−r(τ−τK)N (−(r−D0 − 1

2σ2)(τ − τK)+ ln(S)− ln(Kn )

σ√

τ − τK)

− Me−r(τ−τK)N ((r−D0 − 1

2σ2)(τ − τK)+ ln(S)− ln(Sp(τK))

σ√

τ − τK). (4.35)

Here, we can find that all of these three integral representations are in form of theunknown functions, Sc(τ), Sp(τ) and A(τ). Thus, the following integral equationscan be used to obtain the value of three unknown functions by applying the boundaryconditions to above three formulae:


12σ2)τ + ln(Sc(τ))− ln(Sc(0))

σ√

τ)

− Ze−rτN ((r−D0 − 1


τ)

+ D0Sc(τ)∫ τ

0ne−D0(τ−ξ )

·N ((r−D0 +

12σ2)(τ −ξ )+ ln(Sc(τ))− ln(Sc(ξ ))

σ√

τ −ξ)dξ

+ Ze−rτ , (4.36)

nSc(τ) =∫ ln(Sc(τM))



e− [(r−D0−

12 σ2)(τ−τM)+ln(Sc(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0nD0Sc(τ)e−D0(τ−τM−ξ )

·N ((r−D0 +

12σ2)(τ − τM −ξ )+ ln(Sc(τ))− ln(Sc(τM +ξ ))

σ√


−∫ τ−τM


·N ((r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(Sc(τ))− ln(Sp(τM +ξ ))σ√


+ nSc(τ)e−D0(τ−τM)N ((r−D0 +

12σ2)(τ − τM)+ ln(Sc(τ))− ln(Sc(τM))

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1

2σ2)(τ − τM)+ ln(Sc(τ))− ln(Sp(τM))

σ√

τ − τM), (4.37)

M =∫ ln(Sc(τM))



e− [(r−D0−

12 σ2)(τ−τM)+ln(Sp(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0nD0Sp(τ)e−D0(τ−τM−ξ )

·N ((r−D0 +

12σ2)(τ − τM −ξ )+ ln(Sp(τ))− ln(Sc(τM +ξ ))

σ√



−∫ τ−τM


·N ((r−D0 − 1



+ nSp(τ)e−D0(τ−τM)N ((r−D0 +

12σ2)(τ − τM)+ ln(Sp(τ))− ln(Sc(τM))

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM), (4.38)

K =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τK )+ln(K

n )−u]2

2σ2(τ−τK ) du

+∫ τ−τK

0


2πe− [(r−D0−

12 σ2)(τ−τK−ξ )]2

2σ2(τ−τK−ξ ) · σA(τK +ξ )2n√

τ − τK −ξdξ

−∫ τ−τK


·N ((r−D0 − 1

2σ2)(τ − τK −ξ )+ ln(Kn )− ln(Sp(τK +ξ ))

σ√


−∫ τ−τK

0rKe−r(τ−τK−ξ )N (−

(r−D0 − 12σ2)(τ − τK −ξ )

σ√


− Ke−r(τ−τK)N (−(r−D0 − 1

2σ2)(τ − τK)

σ√

τ − τK)

− Me−r(τ−τK)N ((r−D0 − 1

2σ2)(τ − τK)+ ln(Kn )− ln(Sp(τK))

σ√

τ − τK), (4.39)

M =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τK )+ln(Sp(τ))−u]2

2σ2(τ−τK ) du

+∫ τ−τK

0


2πe− [(r−D0−

12 σ2)(τ−τK−ξ )+ln(Sp(τ))−ln(K

n )]2

2σ2(τ−τK−ξ ) · σA(τK +ξ )2n

√τ − τK −ξ

dξ

−∫ τ−τK


·N ((r−D0 − 1

2σ2)(τ − τK −ξ )+ ln(Sp(τ))− ln(Sp(τK +ξ ))σ√


−∫ τ−τK

0rKe−r(τ−τK−ξ )

·N (−(r−D0 − 1

2σ2)(τ − τK −ξ )+ ln(Sp(τ))− ln(Kn )

σ√


− Ke−r(τ−τK)N (−(r−D0 − 1

2σ2)(τ − τK)+ ln(Sp(τ))− ln(Kn )

σ√

τ − τK)

− Me−r(τ−τK)N ((r−D0 − 1

2σ2)(τ − τK)+ ln(Sp(τ))− ln(Sp(τK))

σ√

τ − τK). (4.40)

Hence, these five integral equations can be utilized to derive the value of the


unknown functions, which are contained in the formulation of the pricing for thecallable-puttable CB, and the bond values can be subsequently obtained, once thevalue of these unknown functions are obtained. One may get confused about thecurrent results as there are only three unknown functions, whereas five equationsgoverning these functions are derived. But, it is actually reasonable since we havethree parts, and we are trying to find the value of the unknown functions on theseparts separately. In particular, there is only one unknown term in Part 1, and weuse Equation (4.36) to solve it, while Equation (4.37) - (4.38) and (4.39) - (4.40)are used to derive the two unknown terms of Part 2 and Part 3, respectively.

Being clear about the PDE systems to value the callable-puttable CB for Case 1and Case 2 and the corresponding solutions, we can now proceed to the remainingcase, which is a special one, in the next subsection.

4.2.3 Case 3

In this subsection, the PDE systems are built for pricing the callable-puttable CBof Case 3. This is in fact a special case, and is actually the easiest one among thesethree, since the moment that the value of the optimal conversion boundary reachesKn

and the moment that the minimum value of the bond hits the value of the putprice arrive at the same time, i.e. τK = τM. Therefore, there are only two timeintervals, one of which is near the expiration, being the same as Part 1 of Case 1and Case 2, where the value of the optimal put boundary is always equal to zero.Let the value of the callable-puttable CB for this case be V3(S,τ), and the PDEsystem for this part can be written as:

−∂V3

∂τ+

12

σ2S2 ∂ 2V3

∂S2 +(r−D0)S∂V3

∂S− rV3 = 0,


∂S(Sc(τ),τ) = n,

V3(0,τ) = Ze−rτ ,

(4.41)

with the domain of S and τ being [0,Sc(τ)] and [0,τM], respectively.The other part is the one that the value of the optimal conversion boundary is a

constant while the value of the optimal put boundary is not equal to zero, which isthe same as the third part of the last two cases. Therefore, the PDE system for this


time zone is built as follows:

−∂V3

∂τ+

12

σ2S2 ∂ 2V3

∂S2 +(r−D0)S∂V3

∂S− rV3 = 0,

V3(S,τ+M) =V3(S,τ−M),

V3(Kn,τ) = K,

V3(Sp(τ),τ) = M,∂V3

∂S(Sp(τ),τ) = 0,

(4.42)

with the domain of S and τ being [Sp(τ),Kn] and [τM,T ], respectively.

With the PDE systems being built up, the integral equation representations canbe given below.


By comparing the PDE system of Part 1 among these three case, we can write theintegral equation formulation

V3(S,τ) = nSe−D0τN ((r−D0 +

12σ2)τ + ln(S)− ln(Sc(0))

σ√

τ)

− Ze−rτN ((r−D0 − 1

2σ2)τ + ln(S)− ln(Sc(0))σ√

τ)

+ D0S∫ τ


(r−D0 +12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))

σ√

τ −ξ)dξ

+ Ze−rτ . (4.43)


As mentioned above, the PDE system for this part is as the same as Part 3 of Case1 and Case 2. Following the results above, the integral equation formulation for thispart is

V3(S,τ) =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2πe− [(r−D0−


n )]2

2σ2(τ−τM−ξ ) · σA(τM +ξ )2n√

τ − τM −ξdξ

−∫ τ−τM


·N ((r−D0 − 1



−∫ τ−τM



·N (−(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Kn )

σ√


− Ke−r(τ−τM)N (−(r−D0 − 1

2σ2)(τ − τM)+ ln(S)− ln(Kn )

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM). (4.44)

Combining Equations (4.43) and (4.44), we have successfully derived the integralequation representations for the third case of the callable-puttable CB in form ofthree unknown functions, Sc(τ), Sp(τ) and A(τ). Thus, the boundary conditions canagain be used to determine the value of these unknown functions


12σ2)τ + ln(Sc(τ))− ln(Sc(0))

σ√

τ)

− Ze−rτN ((r−D0 − 1


τ)

+ D0Sc(τ)∫ τ


(r−D0 +12σ2)(τ −ξ )+ ln(Sc(τ))− ln(Sc(ξ ))

σ√

τ −ξ)dξ

+ Ze−rτ , (4.45)

K =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(K

n )−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2πe− [(r−D0−

12 σ2)(τ−τM−ξ )]2

2σ2(τ−τM−ξ ) · σA(τM +ξ )2n√

τ − τM −ξdξ

−∫ τ−τM


·N ((r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(Kn )− ln(Sp(τM +ξ ))

σ√


−∫ τ−τM

0rKe−r(τ−τM−ξ ) ·N (−

(r−D0 − 12σ2)(τ − τM −ξ )

σ√


− Ke−r(τ−τM)N (−(r−D0 − 1

2σ2)(τ − τM)

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1

2σ2)(τ − τM)+ ln(Kn )− ln(Sp(τM))

σ√

τ − τM), (4.46)

M =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(Sp(τ))−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2πe− [(r−D0−

12 σ2)(τ−τM−ξ )+ln(Sp(τ))−ln(K

n )]2

2σ2(τ−τM−ξ ) · σA(τM +ξ )2n√

τ − τM −ξdξ

−∫ τ−τM



·N ((r−D0 − 1



−∫ τ−τM


·N (−(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(Sp(τ))− ln(Kn )

σ√


− Ke−r(τ−τM)N (−(r−D0 − 1

2σ2)(τ − τM)+ ln(Sp(τ))− ln(Kn )

σ√

τ − τM)

− Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM). (4.47)

Afterwards, the integral equation representations for pricing the callable-puttableCB can be utilized to get the bond values by direct substitution of the derivedfunction values.

By now, the integral equation representations for pricing the callable-puttable CBhave been presented. However, it should be noted that, for all these three cases,the integral equations are all nonlinear, and thus a numerical method needs to beutilized to obtain the numerical solution, the details of which are provided in thenext two sections.

4.3 The numerical implementationIn this section, the numerical scheme adopted to solve the integral equations shownin the previous section will be introduced. It should be noted that although differentintegral equations corresponding to different scenarios need to be solved, the methodused for numerical implementation of each one is very similar. Therefore, we willonly introduce the scheme for solving Part 2 of Case 3 as an example; the otherscan be similarly derived.

The main procedure involved in finding numerical solutions through our approachis to solve the coupled Equation (4.47) and Equation (4.47) to obtain the value of twounknown functions, the optimal put boundary, Sp(τ), and the unknown function,A(τ), in Equation (4.44). Once these two functions are determined, the bond pricecan be calculated directly using Equation (4.44). In the following, the process indetermining the two unknown functions is illustrated.

Firstly, the time interval [τM,T ] is separated into several uniform time steps:

[s1,s2], [s2,s3] · · · [sL−1,sL], where s1 = τM and sL = T,

and the discretized unknown functions Sp(si) and A(si) are denoted by S(i)p and A(i),respectively, for i = 1,2, ...,L. And thus we obtain a set of non-linear algebraic


equations for S(i)p and A(i) with i = 1,2, ...,L, as

K =∫ ln(K

n )

ln(S(1)p )V3(eu,τM)

e−r(si−s1)√2πσ2(si − s1)

e− [(r−D0−

12 σ2)(si−s1)+ln(K

n )−u]2

2σ2(si−s1) du

+ Σi−1k=1

Ke−r(si−sk)

√2π

e− [(r−D0−

12 σ2)(si−sk)]

2

2σ2(si−sk) · σA(k)

2n√

si − sk(sk+1 − sk)

− Σi−1k=1rMe−r(si−sk) ·N (

(r−D0 − 12σ2)(si − sk)+ ln(K

n )− ln(S(k)p )

σ√


+ rMe−r(si−s1) ·N ((r−D0 − 1

2σ2)(si − s1)+ ln(Kn )− ln(S(1)p )

σ√

si − s1) · (s2 − s1)

2

− rMs2 − s1

2− rK

s2 − s1

4

− Σi−1k=1rKe−r(si−sk) ·N (−

(r−D0 − 12σ2)(si − sk)

σ√


+ rKe−r(si−s1) ·N (−(r−D0 − 1

2σ2)(si − s1)

σ√

si − s1) · (s2 − s1)

2

− Ke−r(si−s1)N (−(r−D0 − 1

2σ2)(si − s1)

σ√

si − s1)

− Me−r(si−s1)N ((r−D0 − 1

2σ2)(si − s1)+ ln(Kn )− ln(S(1)p )

σ√

si − s1), (4.48)

and

M =∫ ln(K

n )

ln(S(1)p )V3(eu,τM)

e−r(si−s1)√2πσ2(si − s1)

e− [(r−D0−

12 σ2)(si−s1)+ln(Sp(τ))−u]2

2σ2(si−s1) du

+ Σi−1k=1

Ke−r(si−sk)

√2π

e− [(r−D0−

12 σ2)(si−sk)+ln(S(i)p )−ln(K

n )]2

2σ2(si−sk) · σA(k)

2n√

si − sk(sk+1 − sk)

− Σi−1k=1rMe−r(si−sk) ·N (

(r−D0 − 12σ2)(si − sk)+ ln(S(i)p )− ln(S(k)p )

σ√


+ rMe−r(si−s1) ·N ((r−D0 − 1

2σ2)(si − s1)+ ln(S(i)p )− ln(S(1)p )

σ√

si − s1) · (s2 − s1)

2

− rMsi − si−1

4

− Σi−1k=1rKe−r(si−sk) ·N (−

(r−D0 − 12σ2)(si − sk)+ ln(S(i)p )− ln(K

n )

σ√


+ rKe−r(si−s1) ·N (−(r−D0 − 1

2σ2)(si − s1)+ ln(S(i)p )− ln(Kn )

σ√

si − s1)(s2 − s1)

− Ke−r(si−s1)N (−(r−D0 − 1

2σ2)(si − s1)+ ln(S(i)p )− ln(Kn )

σ√

si − s1)


− Me−r(si−s1)N ((r−D0 − 1

2σ2)(si − s1)+ ln(S(i)p )− ln(S(1)p )

σ√

si − s1). (4.49)

It should be pointed out that the value of the optimal put boundary at i = 1is known, because S(1)p can be obtained from the solution for the time period τ ∈[0,τM], and A(1) can be figured out from its definition, i.e. A(1) = n. Thus, we cancalculate S(i)p and A(i) for i = 2,3, ...,L, simultaneously, with a MATLAB built-in rootfinding function (lsqnonlin). Once the values of the functions Sp(τ) and A(τ) aredetermined, the value of the bond can be found directly through Equation (4.44).

Before we proceed to studying the properties of callable-puttable CBs, it is nec-essary to validate the designed numerical scheme by assessing its accuracy and effi-ciency. Unless otherwise stated, parameters listed below are also used in the nextsection.


• Conversion ratio n = 1,

• Time to expiration T = 1 (year),




• The put price M = 95,

• The call price for Case 1 K1 = 135,

• The call price for Case 2 K2 = 145.

Under these parameters, the threshold value of time to expiry, τM, is 0.5129 (year),and the value of the call price for Case 3, K3, is 140.5114.

In [109], the convergence of the binomial tree method has already been shown,and it has also pointed out that the solution of the binomial tree method with1,000 time-steps can be used as the benchmark. Thus, we omit the details here,and directly compare the accuracy and efficiency between the benchmark and ourintegral equation approach. Table 4.1 shows that all of the results obtained bythe integral equation approach with the different value of N (the number of timeintervals) match very well with the benchmark results with the maximum relativeerror being in the order of 10−2. On the other hand, the CPU time consumed by theintegral equation approach is much less than that cost by the binomial tree method.It should be remarked that the time consumed in the integral equation approach has


included the computation of computing the value of two free boundaries, while thebinomial tree method only yields the bond price within the listed time. Therefore,the benchmark test clearly demonstrates the accuracy and efficiency of our integralequation method.


Callable-puttable CB price at t = 0Case 1 Benchmark IE IE IE

S N=1,000 N=1,000 N=2,000 N=5,000100 107.6037 107.7416 107.7374 107.7351110 114.3150 114.5887 114.5845 114.5821120 121.9606 122.4874 122.4826 122.4798130 130.4558 131.3089 131.3037 131.3005

Case 2 Benchmark IE IE IES N=1,000 N=1,000 N=2,000 N=5,000

100 107.6894 107.5888 107.5806 107.5762110 114.4375 114.4180 114.4163 114.4154120 122.1723 122.1735 122.1702 122.1683130 130.7615 130.7761 130.7691 130.7648

Case 3 Benchmark IE IE IES N=1,000 N=1,000 N=2,000 N=5,000

100 107.6765 107.7163 107.7122 107.7099110 114.4205 114.4970 114.4929 114.4905120 122.1365 122.2900 122.2852 122.2824130 130.7133 130.9657 130.9603 130.9571

max. relative error - 0.0065 0.0065 0.0065Time (second) 1254.9052 25.2681 54.7545 163.7065

In the following section, the number of time intervals in solving our integral equa-tions is set to be 2000 to achieve a balance between accuracy and efficiency. Inaddition, all of our calculations in this chapter is done on a PC with the followingspecifications: Intel(R) Xeon(R), CPU E5-1640 v4 @3.60GHz 3.60 GHz, and 32.0GB of RAM.

4.4 Numerical results and discussionsIn this section, the price of the callable-puttable CB will be presented with the valueof its free boundaries, i.e., the optimal conversion boundary and the optimal putboundary. Various properties will be discussed based on the numerical experimentalr esults.

Depicted in Figure 4.1 are the values of the optimal exercise boundaries with timeto expiry for three cases. It is obvious that no matter in which case, the value of


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

0

50

100

150

200

250

300

Valu

e o

f th

e b

onudary

for

Case 1

MoKo

n=0.5

n=1.0

n=2.0

(a) Case1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

0

50

100

150

200

250

300

Valu

e o

f th

e b

onudary

for

Case 2

Mo Ko

n=0.5

n=1.0

n=2.0

(b) Case2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

0

50

100

150

200

250

300

Valu

e o

f th

e b

onudary

for

Case 3

M(

K)

o

n=0.5

n=1.0

n=2.0

(c) Case3

Figure 4.1: The value of the optimal exercise boundaries for three different valuesof the conversion ratios.


two free boundaries, including the optimal conversion boundary and the optimalput boundary, are the increasing functions with the time to expiry, τ = T − t. Also,the value of the optimal conversion boundary is constant when the time to expiry isgreater than τK , since in this situation, the call feature is switched on with the fixedboundary, K

n. In addition, the value of the optimal put boundary during the time

to expiry interval [0,τM] is always equal to zero for all the three cases, while it ariseswhen the time to expiry is greater than another critical value of the time to expiry,τM, with the put feature being meaningful. Another property that can be seen in allthe three figures is that a higher value of the conversion ratio leads to lower valuesof the free boundaries. This is because a higher value of the conversion ratio meansa larger number of the underlying assets the holder can get when they convert thebond, and thus the value of the optimal conversion boundary should be lower. Onthe other hand, the main difference in the three figures result from the magnitude ofthe two critical points, τK and τM. In particular, the value of τK is lower than thatof τM in Figure 4.1(a), while an apposite phenomenon can be observed in Figure4.1(b), with Figure 4.1(c) presenting a special case in which the value of τK is equalto that of τM.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

0

20

40

60

80

100

120

140

160

180

200

Va

lue

of

the

bo

nu

da

ry

M(

K3)

oK1o K2o

Boundary for Case 1

Boundary for Case 2

Boundary for Case 3

Figure 4.2: The optimal exercise prices for three cases

In order to compare these three scenarios clearly, we merge three sub-figures inFigure 4.1 into a single figure (the value of the conversion ratio is equal to one) toform Figure 4.2, with τK1, τK2 and τK3 being used to represent τK in Case 1, Case 2


and Case 3, respectively. Firstly, it should be noticed that the gap between τK2 andτK3 is larger than that between τK1 and τK3, with the average value of the call pricefor Case 1 and Case 2 being almost equal to the value of the call price for Case 3.This is reasonable since the value of the optimal conversion boundary climbs fastwhen the time to expiry is close to zero, while it becomes flatter and flatter withthe increase of the time to expiry. Therefore, it takes the value of the conversionboundary needs more time to increase from the value of the call price for Case 3to the value of the call price for Case 2 compared with the case when it increasesfrom the value of the call price for Case 1 to the value of the call price for Case3. If we look at these optimal conversion boundaries curves, what can be noticedfirst is that when the time to expiry is less than τK1, all three equal to each other,and when the time to expiry is close to zero, the value of the optimal conversionboundary decreases quickly to the face value divided by the value of the conversionratio. Moreover, during the time zone [τK1,τK3], the values of the optimal conversionboundary for Case 2 and Case 3 still equal to each other, increasing with the timeto expiry, while that for Case 1 remains a constant. When the time to expiry stayswithin [τK3,τK2], the value of the two optimal conversion boundaries of Case 1 andCase 3 become constants, and if we further increase the time to expiry such thatit becomes larger than τK2, the values of the optimal conversion boundary for allof three cases become constants. Overall, the value of the conversion boundary forCase 2 is obviously the highest while that of Case 1 is the lowest, since the value ofthe conversion boundaries for each case are all increasing functions with the time toexpiry before the critical values, τKi, respectively. On the other hand, if we returnto the optimal put boundaries, it is clear that they equal to zero during the timezone [0,τM], while they no longer take the value of zero when the time to expiry isgreater than τM. Although the value of three put boundaries in the time interval[τM,τK2] almost equal to each other, the value of the optimal put boundary forCase 1 and Case 2 is actually the highest and lowest one, respectively. In contract,when the time to expiry is greater than τK2, there is a surge in the optimal putboundary for Case 2, making it the highest among the three cases, which meansthat a higher value of the call price leads to a greater value of the put boundary. Itis also interesting to find that the slope of Sc(τ) near τ = 0 and that of Sp(τ) nearτ = τM are both very large, which is similar to American-style call options.

Depicted in Figure 4.3 is the bond price with the underlying asset price at differentmoments for three cases. Obviously, the price of the callable-puttable CB is alwaysan increasing function with the underlying asset price, and all of the price curvesare smoothly tangent to the payoff curve, which means the slope of the price curvesat the optimal conversion boundary (or the optimal call boundary) is equal to thevalue of the conversion ratio. Also, when the underlying asset price is very low, the


0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bond p

rice o

f C

ase 1

=0.0513

=0.3591

=0.9513

(a) Case1

0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bond p

rice o

f C

ase 2

=0.2565

=0.5616

=0.9513

(b) Case2

0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bond p

rice o

f C

ase 3

=0.2565

=0.9513

(c) Case3

Figure 4.3: The bond price for different time moments.


increase in the underlying asset price will not lead to a significant change in thebond price, and the slope of the price curve will become zero when the underlyingasset price is zero. In this situation, a greater time to expiry will result in a lowervalue of the bond, while a completely opposite phenomenon can be observed whenthe underlying asset price increases to a certain extend.

0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bo

nd

price

Case 1

Case 2

Case 3

124 125 126 127

126

128

130

Figure 4.4: The bond prices of three cases.

On the other hand, Figure 4.4 is a combination of all sub-figures in Figure 4.3,aiming at making comparison of the bond prices in three cases at the same moment,τ = T . The three prices are very similar to each other when the underlying price issmall, since the boundary conditions for the optimal put boundary of these threecases are the same. With the increase of the underlying asset price, it is interestingto find that the bond prices corresponding to Case 1 and Case 3 are still almostequal to each other, while there is a gap between these two prices and the bondprice of Case 2. This can be explained by the fact that when the time to expiry islarge enough, there exists a situation that the holder of Case 2 will still choose tosell the bond back, while the holder of Case 1 or Case 3 will keep the bond sincethe value of the optimal put boundary of Case 2 is much larger than that of Case 1and Case 3, as shown in Figure 4.2. Thus, the bond price corresponding to Case 2is still equal to the value of the put price, while that for Case 1 or Case 3 is higherthan the value of the put price. When we further increase the underlying asset price,these three prices again become close to each other. In order to demonstrate the


difference among the three cases, a zoom-in chart is embedded in this figure, whichclearly shows that the bond price for Case 1 is the highest while that for Case 2 isthe lowest.

0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bo

nd

price

PCB

CB

CPCB

91 91.5 92100

102

104

Figure 4.5: The prices of CBs, PCBs and CPCBs.

It is also interesting to compare the prices of vanilla and puttable convertiblebonds considered in Chapter 3 and those of callable-puttable convertible bonds toshow the difference between these three types. For the illustration purposes, we useCase 2 as an example, and the results are presented in Figure 4.5. One can easilyobserve from this figure that being consistent with the results in Chapter 3, the priceof a puttable convertible bond is always higher than that of the corresponding vanillaconvertible bond, since puttable convertible bonds give an additional right to theholder to sell the bond back to the issuer, when the stock price falls down to a certainlevel, potentially protecting the benefit of the holder. It is also interesting to noticethat the price of a callable-puttable convertible bond is lower than both prices ofputtable and vanillas convertible bonds when the underlying price is beyond a certainlevel. This is because when the underlying price is large enough, the possibility forthe holder to sell the bond back to the issuer becomes very low, while the call featurehas enabled the issuer to call the bond back, which means that the contract of thecallable-puttable convertible bond is more favorable to the issuer.

Figure 4.6 shows the effects of the volatility on the bond price and its optimal


0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bo

nd

price

of

Ca

se

3

=0.3

=0.4

=0.5

(a) The price of the bond for three different volatilities.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

0

20

40

60

80

100

120

140

160

180

200

Va

lue

of

the

bo

nu

da

ry f

or

Ca

se

3

Mo

=0.3

=0.4

=0.5

(b) Optimal exercise prices for three different volatilities.

Figure 4.6: Comparison by three different volatilities.


exercise boundariesb. From Figure 4.6(a), it can be noticed that the value of thecallable-puttable CB is a monotonically increasing function of volatility. It is rea-sonable since when the value of the volatility becomes larger, there is a higher risk,leading to a higher price. When we turn to Figure 4.6(b), it can be found that ahigher value of the volatility will lead to a larger value of the optimal conversionboundary while it will lead to a lower value of the optimal put boundary. The ratio-nale behind this phenomenon is that the increase in the value of the volatility willcontribute to an increase of the bond price for the same underlying price and timeto expiry, and if we increase the level of the volatility, the bond price at the con-version boundary corresponding to lower volatility is no longer equal to the value ofthe optimal conversion boundary times the value of the conversion ratio, but higherthan it. In other words, the bond holder will not be willing to convert unless theunderlying price reaches a higher level. Due to a similar reason, the bond price atthe put boundary correspond to the lower level of the volatility is higher than thevalue of the put price, and in this case the holder will not sell it back to issuer unlessthe underlying price drops to a further point.

In Figure 4.7, we show how the price of a callable-puttable CB and both itsoptimal conversion price and optimal put price change with the risk-free interestrate. Comparing the bond price as well as the values of its two free boundariesshown in this figure with those shown in Figure 4.6, one can observe that the risk-free interest rate has quite a different influence than volatility. In specific, Figure4.7(a) displays if we increase the value of the risk-free interest rate, the bond pricewill decrease, which can be easily explained since when the risk-free interest rate ishigher, investors are more willing to leave their money in a risk-free bank accountthan buying a risky bond, resulting in a lower CB price. When the two sets of freeboundaries are taken in to consideration, opposite trends are also shown in Figure4.7(b); the optimal conversion price and the optimal put price are a decreasingand an increasing function of the risk-free interest rate, respectively. The mainexplanation for this can be analogous to that for the volatility case. Taking theconversion boundary as an example. If we decrease the value of the risk free interestrate, the bond price increases for the same underlying price and time to expiry, andthus the bond price at the conversion boundary corresponding to higher interest rateis higher than the value of the conversion boundary times the value of the conversionratio, making the holder to keep the bond until the underlying price reaches a higherlevel.

bWe will only use Case 3 as example for illustration, since there is no essential difference amongthe three cases, as far as the influence of the parameters on the bond prices.


0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bo

nd

price

of

Ca

se

3

r=0.08

r=0.1

r=0.15

(a) The price of the bond for three different interest rates.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

0

20

40

60

80

100

120

140

160

180

200

Va

lue

of

the

bo

nu

da

ry f

or

Ca

se

3

M1oM2oM3o

r=0.08

r=0.1

r=0.15

(b) Optimal exercise prices for three different interest rates.

Figure 4.7: Comparison by three different interest rates..


4.5 ConclusionIn this chapter, the integral equation formulations for the valuation of the callable-puttable conversion bond on a single underlying asset with a constant dividend arederived with the incomplete Fourier transform method and the Green’s function, andthe numerical implementation of the formulations is also discussed to provide someguidance for practical application. The accuracy and efficiency of the newly derivedintegral equation formulations are demonstrated through numerical comparison withthe binomial tree method, and the quantitative impact of different parameters onthe bond price as well as it free boundaries are also studied.

Chapter 5

Pricing resettable convertible bonds withan integral equation approach

5.1 IntroductionA vanilla bond is one of the widely-used instruments, through which the issuerborrows money from the holder with a preset time period and a preset interestrate. When other clauses are added into the vanilla bond, some particular bondsarise. One of the most popular variations of vanilla bonds is the convertible bonds(CBs), which allows the holder to convert the bond into the preset number of theunderlying assets, during the lifetime of the bond or only at maturity. In contrast tovanilla CBs, we can attain other types of CBs, if additional clauses are incorporated.For example, a callable convertible bond (CCB) enables the issuer to call back thebond when the underlying price is large enough, while a puttable convertible bond(PCB) gives a right to the holder so that he/she can sell the bond back to the issuerwhen the underlying price is small enough. A resettable convertible bond (RCB),as another common CBs, contains a clause that when the underlying price dropsor increases to the preset reset price, the conversion ratio will be reseted. Due tothe existence of these additional clauses, the pricing problems of the correspondingbonds become more complicated, although they are more flexible and useful in realmarkets.

To price these financial instruments, many theoretical frameworks are proposedby the researchers. One of the most popular models is the Black-Scholes (B-S) modelproposed by Black & Scholes [9], in 1973, with the underlying price being assumedto follow a geometric Brownian motion (GBM). Under this particular model, Inger-soll [62] and Brennan & Schwartz [12] took the firm value as the underlying variableto price CBs. However, the firm value is very difficult to obtain in real markets, andthus McConnel & Schwartz [84] improved this method by using the stock price as

96

CHAPTER 5. PRICING RESETTABLE CONVERTIBLE BONDS 97

the underlying variable. Since then, various approaches have been proposed to priceCBs. While a closed-form solution was obtained for the European-style CBs, whichcan only be converted at maturity, in [88], Zhu [106] presented an analytical solutionunder the B-S model using the homotopy analysis method for the American-styleCBs, which can be converted at any time during their lifetime. Moreover, Tsiveriotis& Fernandes [99] priced the cash-only CBs by applying the finite difference methodon the coupled B-S equations, and Ohtake et al. [89] presented the definitions of thecall and reset clause. Recently, Zhu et.al. [109] derived an integral equation represen-tation to price a PCB with the utilization of the incomplete Fourier transform.

Although the B-S model is widely used in today’s financial market, it is usu-ally not adequate to model the underlying price, especially for the long-term pe-riod contracts, since the volatility and the risk free interest rate are always notconstant. Therefore, adding more random variables into the B-S model becomethe first choice, including the stochastic interest rate models and the stochasticvolatility models. Belonging to the former category, the Merton model [85], CEVmodel [27,30], Vasicek model [100], Dothan model [37], Brennan-Schwartz model [14],CIR-VR model [28], GBM model [83] and CIR model [29] are widely adopted, while theHull-White model [55] and the Heston model [52] are two of the most popular onesincluded in the latter category. Although the models with the additional randomvariables may provide better fit to the real market data, the corresponding pricingproblem would become much more complicated, and thus the numerical methodsoften have to be resorted to in finding the solution. For example, the finite differ-ence approach [105], the finite element approach [6], the finite volume approach [112],the binomial tree method [21,59] and the Monte Carlo simulation method [3,80] havealready been adopted to price CBs under these complex models.

In this chapter, a resettable convertible bond is considered, and the reset clausestudied here is that the conversion ratio will be adjusted upwards once the underlyingprice drops below the preset reset price. One may be interested in the differencebetween the vanilla CBs and the RCBs, and why we need RCBs. In the situation ofvanilla CBs, the holder will not convert the bond until the maturity to obtain theface value if the underlying price is not large enough, which causes a heavy burdenof cash flows for the issuer. However, when this particular reset clause is addedinto the vanilla CB, this bond may also be converted when the underlying price isrelatively small. Moreover, one should also notice that the value of a RCB is higherthan that of a corresponding CB since the reset clause brings benefit to the holder,which means that it is advantageous for the issuer to release the RCBs instead ofthe CBs.

In fact, pricing resettable convertible bonds has not been investigated until veryrecently, and there are only some numerical solutions derived to price RCBs [43,70],


while no partial different equation (PDE) system has been set up for the value ofRCBs. In the following, we will work under the B-S model by assuming that thevolatility of the underlying asset and the risk free interest rate are constant, andestablish a closed PDE system for the bond price. It should be pointed out thatalthough it is more appropriate to model the underlying price under the stochasticvolatility model and/or the stochastic interest rate model, we still use the B-S modelin this study. This is because it is more feasible to start with a simpler model whenintroducing a new solution approach to an already complicated problem with thefree boundary, given that very few results have been presented on how to priceRCBs.

Once the PDE system is successfully built up, a natural question is how to findits solution. It is observed that the newly established system contains an optimalconversion price, which needs to be solved with the bond price simultaneously. Todeal with this kind of problem, one of the most efficient methods is the incompleteFourier transform technique, which has been utilized to derive the value of theAmerican option and that of the PCB in [26] and [109], respectively. Therefore, theincomplete Fourier transform is adapted in our study, based on which, we obtain anintegral equation representation for the bond price, involving the unknown optimalconversion price. The optimal conversion price can then be found through numericalsolving the nonlinear equation we obtain, after which the value of the resettableconvertible bond can be derived directly.

The main contribution of this paper can be summarized from two aspects. First ofall, a closed PDE system for the pricing of RCBs under the B-S model is successfullyestablished for the first time, based on which an integral equation representationfor the prices of RCBs is derived, which is shown to be accurate and efficient fromnumerical experiments. Secondly, we clearly articulate, through a rigorous theo-retical proof of a proposition, a unique feature of RCB’s price; it is not always amonotonically increasing function of the underlying asset price, which may appearto be incomprehensible for classic convertible bonds. Such a theoretical proof is alsosupplemented by some numerical examples to further illustrate this quite amazingphenomenon.

It should be pointed out that our approach can possibly be extended for thepricing of RCBs under other models, such as stochastic volatility or interest ratemodels, and jump-diffusion models, as the essential feature of the moving boundaryis still the same. However, for any stochastic volatility or stochastic interest ratemodel, the one-dimensional free boundary curve in the B-S model will become a two-dimensional surface, and this will certainly make the target problem more complex.On the other hand, when jump-diffusion models are taken into consideration, therewill be an additional integral component in the PDE, leading to a partial integro-


differential equation. In this case, the ordinary differential equation (ODE) obtainedafter applying the Fourier transform would be different from the one derived underthe B-S model. A challenge then is to seek the analytical solution to the new ODEas well as to analytically invert the Fourier transform as what we did here.

It should also be noted that the main focus of this paper is to develop an efficientmethod for the pricing of RCBs. Of course, it is interesting to apply our newlyproposed approach to real market data. However, resettable CBs are mainly over-the-counter derivative products and thus collecting their market data is never aseasy as acquiring data of exchange-traded derivative products. Without access tothe needed data, it is impossible for us to conduct an empirical study, and thus wechoose to tackle the first part of this complicated problem (i.e., the theoretical partof the problem) first and report our methodology in this chapter.

This chapter is organized as follows: In Section 2, the PDE system for the bondprice is set up under the Black-Scholes model, and then the incomplete Fouriertransform is applied on the system to derive the integral equation representation, asshown in Section 3. Numerical schemes and several interesting results are displayedwith a set of graphs presented in Section 4. At last, some conclusion remarks aregiven in Section 5.

5.2 Model set upIn this section, the PDE system to price a resettable convertible bonds is set up.We begin by assuming the dynamic of the underlying asset price, S, satisfies thefollowing stochastic different equation:

dS = (r−D0)Sdt +σSdWt , (5.1)

where r, D0, and σ are the risk-free interest rate, the continuous dividend rate, andthe volatility of the underlying asset, respectively, and Wt is a standard Brownianmotion. Then, if the value of a resettable convertible bond is denoted by V (S, t),it should satisfy the following PDE under the Black-Scholes model, by using theFeynman-Kac formula

∂V∂ t

+12

σ2S2 ∂ 2V∂S2 +(r−D0)S

∂V∂S

− rV = 0. (5.2)

To close the PDE system, we need to give the terminal condition and boundaryconditions. With the maturity of the bond, T , the face value, Z, the initial conversionratio, n1 and the reset conversion ratio, n2, the payoff of the RCBs, or the terminal


condition can be represented as

V (S,T ) = maxn1S,Z, (5.3)

which is the same as that of the vanilla convertible bonds with the initial conversionratio, since the bond will not be reset at the maturity. It should be remarked thatthe value of the reset conversion ratio should be higher than that of the initial one,as part of the requirement according to the reset clause we set up here.

In addition, the boundary conditions at the optimal conversion boundary shouldalso be in the same form as those of the vanilla CBs, involving the initial conversionratio only, although there are two conversion ratios in our case. This can be explainedby the fact that when the underlying price is large enough, it is almost impossible forthe underlying price to drop below the reset price. In other words, the reset clausewill not be exercised in this case, and thus the RCB can be treated as the vanillaone with the initial conversion ratio. Therefore, the optimal conversion boundaryconditions can be written as

V (Sc(t), t) = n1 ·Sc(t), (5.4)∂V∂S

(Sc(t), t) = n1, (5.5)

where Sc(t) is the optimal conversion boundary. It should be pointed out that thepricing PDE system has not been closed yet, and one more boundary conditionis needed, which is the bond price at the reset price, Sr. Since the bond priceshould be a continuous function of the underlying price, we need to investigate thecase when the underlying price touches the reset price. In fact, if the underlyingprice touches the reset price from above, the bond should be reseted immediatelyand automatically, after which the RCB becomes the vanilla CB with the resetconversion ratio. One can easily deduce from this that the bond price at the resetprice should be defined as

F(t) =

n2 ·Sr, Sc2(t)≤ Sr,

V2(Sr, t), Sc2(t)> Sr,(5.6)

where V2(Sr, t) is the value of the CB with the reset conversion ratio, and Sc2(t) isits corresponding optimal conversion boundary. It should be pointed out that oncethe bond has been reset, Sc2(t) becomes the optimal exercise price of the RCB. In


summary, the closed PDE system can be formulated as follows

∂V∂ t

+12

σ2S2 ∂ 2V∂S2 +(r−D0)S

∂V∂S

− rV = 0,

V (S,T ) = maxn1S,Z,V (Sr, t) = F(t),

V (Sc(t), t) = n1 ·Sc(t),∂V∂S

(Sc(t), t) = n1.

(5.7)

It should also be noted that if the reset price equals to 0, i.e. Sr = 0, the RCBs areactually equivalent to the vanilla CBs with the initial conversion ratio. In the fol-lowing, we would like to only discuss the case when the resettable right is meaningful(otherwise the RCBs reduce to the vanilla CBs with the reset conversion ratio), i.e.Sc1(T )> Sr, where Sc1(t) is the optimal conversion boundary for the vanilla convert-ible bonds with the initial conversion ratio, in which case the domain of t and thatof S for the PDE system are [0,T ] and [Sr,Sc(t)], respectively.

5.3 Integral equation representationIn this section, the integral equation representation to price RCB is obtained byusing the incomplete Fourier transform method, which is in fact one of the standardmethods for pricing the financial derivatives. Now, we start with the following twosimple transforms

τ = T − t, x = ln(S), (5.8)

with which the initial PDE system (5.7) is transformed into

−∂V∂τ

+12

σ2 ∂ 2V∂x2 +(r−D0 −

12

σ2)∂V∂x

− rV = 0,

V (x,0) = maxn1ex,Z,V (ln(Sr),τ) = F(τ),V (ln(Sc(τ)),τ) = n1 ·Sc(τ),∂V∂x

(ln(Sc(τ)),τ) = n1 ·Sc(τ),

(5.9)

with the domain of x and τ being [ln(Sr), ln(Sc(τ))] and [0,T ], respectively.If we define the incomplete Fourier transform for our issue as

FV (x,τ)=∫ ln(Sc(τ))

ln(Sr)V (x,τ) · eiωxdx , V (ω,τ), (5.10)

and then when we apply this incomplete Fourier transform on the PDE system (5.9),


a problem immediately arises that the value of the first-order derivative, ∂V∂x

(x,τ),at x = ln(Sr) is needed in the computation, which is not available in the system.Therefore, we add one more boundary condition, ∂V

∂S(Sr, t) = G(t), into the original

PDE system (5.7) so that the PDE system (5.9) becomes

−∂V∂τ

+12

σ2 ∂ 2V∂x2 +(r−D0 −

12

σ2)∂V∂x

− rV = 0,

V (x,0) = maxn1ex,Z,V (ln(Sr),τ) = F(τ),∂V∂x

(ln(Sr),τ) = Sr ·G(τ),

V (ln(Sc(τ)),τ) = n1 ·Sc(τ),∂V∂x

(ln(Sc(τ)),τ) = n1 ·Sc(τ).

(5.11)

Applying the incomplete Fourier transform (5.10) on the new PDE system (5.11)yields the following ordinary diffident equation (ODE) system

∂V∂τ

(ω,τ)+B(ω)V (ω,τ) = F1(ω,τ)−F2(ω,τ),

V (ω,0) =∫ ln(Sc(0))

ln(Sr)maxn1ex,Z · eiωxdx,

(5.12)

where

B(ω) =12

σ2ω2 +(r−D0 −11

σ2)iω + r, (5.13)

F1(ω,τ) = [S′c(τ)Sc(τ)

− 12

σ2iω +(r−D0)] ·n1Sc(τ)eiω ln(Sc(τ)), (5.14)

F2(ω,τ) = [12

σ2G(τ)Sr −12

σ2iωF(τ)+(r−D0 −12

σ2)F(τ)] · eiω ln(Sr), (5.15)

with the derivation process being left in Appendix C.1. It should be noted thatthis is a non-homogeneous first-order linear ODE with an initial condition, thus thesolution to which can be easily derived as follows

V (ω,τ) = V (ω,0) · e−B(ω)τ +∫ τ

0[F1(ω,ξ )−F2(ω,ξ )] · e−B(ω)(τ−ξ )dξ . (5.16)

Clearly, the integral equation formulation for the RCB price in the Fourier spacehas been presented. However, it should be pointed out that it costs a lot of timeto use numerical methods to conduct the Fourier inversion transform, and it isdesired to find the solution in the original space. Fortunately, after some complexcomputation, with details left in Appendix C.2, we have successfully derived ananalytical expression of the solution in the original space by applying the Fourier


inversion transform on Equation (5.16), with its formulation presented as

V (S,τ) =e−rτ

√2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+ln(S)−u]2

2σ2τ ·maxn1eu,Zdu

+∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√2πσ2(τ −ξ )

· e−[(r−D0−

12 σ2)(τ−ξ )+ln(S)−ln(Sc(ξ ))]2

2σ2(τ−ξ )

·S′c(ξ )Sc(ξ )

+(r−D0 +

12σ2)(τ −ξ )− ln(S)+ ln(Sc(ξ ))

2(τ −ξ )dξ

−∫ τ

0

12

σ2G(ξ )Sr ·e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ ) dξ

−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− ln(S)+ ln(Sr)

2(τ −ξ )dξ . (5.17)

It should be noted that this integral equation representation contains the unknownfunctions, Sc(τ) and G(τ), and the numerical method has to be used to obtain theirvalues, since it is impossible to obtain their explicit expressions. Moreover, thisintegral equation representation also contains the first-derivative of the unknownfunction, S′c(τ), which leads to the requirement of the higher smoothness of theinterpolation function used in the numerical solution procedure. Therefore, a newrepresentation is obtained based on the formation (5.17), using the integration byparts, and it has the form of

V (S,τ) =e−rτ

√2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+ln(S)−u]2


+n1S · e−D0τ ·N ((r−D0 +

12σ2)τ + ln(S)− ln(Sc(0))√

σ2τ)

+∫ τ

0D0n1S · e−D0(τ−ξ ) ·N (

(r−D0 +12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

−∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ ) dξ

−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− ln(S)+ ln(Sr)

2(τ −ξ )dξ . (5.18)

We refer interested readers to Appendix C.3 for the technical details. By now, thefinal integral equation representation has been obtained. However, this formulationcan not be directly used since it involves the unknown functions, Sc(τ) and G(τ),


implying that two equations are needed to obtain the values of these two unknownfunctions. If we recall two of the boundary conditionsa here

V (Sr,τ) =12

F(τ), (5.19)V (Sc(τ),τ) = n1 ·Sc(τ), (5.20)

substituting these two boundary conditions into representation (5.18) can lead tothe following two integral equations

12

F(τ) =e−rτ

√2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+ln(Sr)−u]2


+n1Sr · e−D0τ ·N ((r−D0 +

12σ2)τ + ln(Sr)− ln(Sc(0))√

σ2τ)

+∫ τ

0D0n1Sr · e−D0(τ−ξ ) ·N (

(r−D0 +12σ2)(τ −ξ )+ ln(Sr)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

−∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(Sr)−ln(Sr)]2

2σ2(τ−ξ ) dξ

−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(Sr)−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− ln(Sr)+ ln(Sr)

2(τ −ξ )dξ , (5.21)

and

n1 ·Sc(τ) =e−rτ

√2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+ln(Sc(τ))−u]2


+n1Sc(τ) · e−D0τ ·N ((r−D0 +

12σ2)τ + ln(Sc(τ))− ln(Sc(0))√

σ2τ)

+∫ τ

0D0n1Sc(τ) · e−D0(τ−ξ ) ·N (

(r−D0 +12σ2)(τ −ξ )+ ln(Sc(τ))− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

−∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(Sc(τ))−ln(Sr)]2

2σ2(τ−ξ ) dξ

−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(Sc(τ))−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− ln(Sc(τ))+ ln(Sr)

2(τ −ξ )dξ . (5.22)

In summary, using Equations (5.21) and (5.22), the values of the two unknownaThe factor of 1/2 contained in one of the boundary conditions is a result of applying the

incomplete Fourier transform when deriving the integral equation representation.


functions, contained in the integral equation representation (5.18), can be obtained,and then the value of the RCBs can be computed directly. In the next section, thenumerical scheme will be designed to obtain the value of the optimal conversionprice, based on which, the bond price can be derived. It should be pointed out thatnumerically deriving the optimal conversion boundary requires the knowledge of theoptimal conversion price at expiry, and thus its value should be determined first,which is provided in the following proposition.

Proposition 5.3.1 The optimal conversion price at expiry is Sc(τ)|τ=0 = Z/n1.

The details of the proof are left in Appendix C.4.Before we present numerical examples in the next section, it needs to be pointed

out that because of the additional reset clause, the bond price is not always amonotonically increasing function of the underlying price, which is impossible forother types of convertible bonds. To address this unique property, we use a particularexample presented in the following proposition.

Proposition 5.3.2 (Non-monotonicity) The price of the resettable convertible bondis not a monotonically increasing function of the underlying price when the time toexpiry is sufficiently small, when Sc(τ)|τ=0 = Z/n1 = Sr.

The detailed proof is left in Appendix C.5. It should be remarked that the propertyshown by Proposition 5.3.2 also holds when Sc(τ)|τ=0 −Sr is not very large.

5.4 Numerical schemes and the resultsIn this section, we will provided the numerical scheme to obtain the value of theunknown functions, Sc and G, numerically. After that, some results will be displayedto illustrate the properties of the RCBs. Before we present the scheme, the timeinterval, [0,T ], should be discretized uniformly as: 0= τ1 < τ2 < · · ·< τN < τN+1 = T ,with τn = (n−1)∗T/N. In this case, the discretized unknown functions Sc(τn) andG(τn) are denoted as S(n)c and G(n), respectively, and at the same time, the knownfunction F(τn) is denoted as F(n). Therefore, the numerical scheme is constructedby two sets of the coupled non-linear algebraic equations:

12

F(n) =e−rτn√2πσ2τn

∫ ln(S(1)c )

ln(Sr)e− [(r−D0−

12 σ2)τn+ln(Sr)−u]2

2σ2τn ·maxn1eu,Zdu

+n1Sr · e−D0τn ·N ((r−D0 +

12σ2)τn + ln(Sr)− ln(S(1)c )√

σ2τn)

+Σi=n−1i=1 D0n1Sr · e−D0(τn−τi) ·N (

(r−D0 +12σ2)(τn − τi)+ ln(Sr)− ln(S(i)c )√

σ2(τn − τi)) · T

N


−12

D0n1Sr · e−D0τn ·N ((r−D0 +

12σ2)τn√

σ2τn) · T

N

−Σi=n−1i=1

12

σ2G(i)Sr ·e−r(τn−τi)√

2πσ2(τn − τi)· e

− [(r−D0−12 σ2)(τn−τi)]

2

2σ2(τn−τi) · TN

−Σi=n−1i=1 F(i) · e−r(τn−τi)√


− [(r−D0−12 σ2)(τn−τi)]

2

2σ2(τn−τi) ·r−D0 − 1

2σ2

2· T

N, (5.23)

and

n1 ·S(n)c =

e−rτn√2πσ2τn

∫ ln(S(1)c )

ln(Sr)e− [(r−D0−

12 σ2)τn+ln(S(n)c )−u]2

2σ2τn ·maxn1eu,Zdu

+n1S(n)c · e−D0τn ·N ((r−D0 +

12σ2)τn + ln(S(n)c )− ln(S(1)c )√

σ2τn)

+Σi=n−1i=1 D0n1S(n)c · e−D0(τn−τi) ·N (

(r−D0 +12σ2)(τn − τi)+ ln(S(n)c )− ln(S(i)c )√

σ2(τn − τi)) · T

N

−12

D0n1S(n)c · e−D0τn ·N ((r−D0 +

12σ2)τn + ln(S(n)c )− ln(S(1)c )√

σ2τn) · T

N

+14

D0n1S(n)c · TN

−Σi=n−1i=1

12

σ2G(i)Sr ·e−r(τn−τi)√


− [(r−D0−12 σ2)(τn−τi)+ln(S(n)c )−ln(Sr)]2

2σ2(τn−τi) · TN

−Σi=n−1i=1 F(i) · e−r(τn−τi)√


− [(r−D0−12 σ2)(τn−τi)+ln(S(n)c )−ln(Sr)]2

2σ2(τn−τi)

·(r−D0 +

12σ2)(τn − τi)− ln(S(i)c )+ ln(Sr)

2(τn − τi)· T

N. (5.24)

Since the terminal value of the optimal conversion boundary, S(1)c =Z/n1, is known,we can use Equation (5.23) to calculate the value of the unknown function, G(1),with n = 2, and then Equation (5.24) can be utilized to calculate the value of S(2)c

with n = 2. With the values of the unknown functions computed in the previoussteps, Equation (5.23) and Equation (5.24) can be used to find the value of unknownfunction, G, and the value of the optimal conversion boundary, Sc, in the followingsteps, until n = N +1.

With the numerical scheme being established, we are now able to present numer-ical solutions. Unless otherwise stated, the parameters used in our study are listedbelow:


• Reset price Sr = 100,






• Initial conversion ratio n1 = 1,

• Reset conversion ratio n2 = 1.2.

It should be noted that all of our calculations in this paper are done using Mat-lab R2017a on a PC with the following specifications: Intel(R) Core(TM), [email protected] 3.60 GHz, and 16.0 GB of RAM.

To validate our numerical scheme, the results obtained through the Monte-Carlosimulation are chosen as a benchmark, the implementation of which is through theLeast Square Monte-Carlo (LSMC) approach proposed by Longstaff & Schwartz [79].The main idea of LSMC is to assume that there is a finite number of possible exercisedates, and the holder of the bond needs to determine whether the bond should beexercised or not at each discrete exercise time. Such a decision is made by comparingthe immediate exercise value, which is the amount that the holder can obtain if thebond is exercised now, and the continuation value, which is defined as the amountthat the holder can receive if the bond is exercised at a future time. Therefore,it is vital to estimate the continuation value, which can be achieved through theleast squares regression with the cross-sectional information coming from the Monte-Carlo simulation. The comparison between the estimated continuation value andthe immediate exercise value will then determine the optimal stopping rule. Thisprocedure is conducted backward in time as we know the payoff function, and theresulted cash flow, if discounted back to the current time, would yield the bond price.It should be pointed out that we can almost apply the same procedure as illustratedabove to price RCBs using LSMC, and the only exception is that one should alwaysdetermine the conversion ratio according to the simulated underlying price in eachsimulation path before any calculation is carried out, due to the existence of “reset”feature. With 5,000 time steps and 200,000 sampling paths, we are now ready topresent the comparison results.

Table 5.1 clearly demonstrates the accuracy of our scheme, with the maximum rel-ative error for these two methods being less than 0.5%. However, one should noticethat the Monte-Carlo simulation is very time-intensive, while the integral equa-tion method is computationally efficient [109]. In particular, the average CPU timeconsumed by the Monte-Carlo simulations to calculate the RCB price is 100.0745seconds, while it only takes 0.0173 seconds to produce one price with the integral


equation approach, which demonstrates the superiority of the integral equation ap-proach, as far as the computational efficiency is concerned.

Table 5.1: MC method vs IE method

RCB price at t = 0S=120 S=130 S=140 S=150 S=160 CPU time

MC 130.2665 136.4142 143.7459 151.8093 160.6040 100.0745IE 130.8426 136.9983 144.1337 152.1121 160.8153 0.0173

Relative error 4.42∗10−3 4.28∗10−3 2.70∗10−3 1.99∗10−3 1.32∗10−3 -

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiry

80

90

100

110

120

130

140

150

160

170

180

Sc(

)

RCB

CB for n1

CB for n2

Figure 5.1: The optimal conversion price of RCB and also two vanilla CBs.

In Figure 5.1, we show the value of the optimal conversion boundary of the RCBand that of the vanilla convertible bonds with two conversion ratios. What we cansee first is that all three optimal conversion prices are the monotonically increasingfunctions of the time to expiry, with the optimal conversion price of the RCB beingthe highest one. This is reasonable since the holder of the RCB has one more benefitthat the conversion ratio can be reset to a higher value when the underlying priceis low enough, implying that the optimal conversion price of the RCB should behigher than that of the CB with initial conversion ratio. Moreover, when the timeto expiry is equal to zero, or at maturity, the optimal conversion price of the RCBis as same as that of the CB with initial conversion ratio, and both of which are


equal to Z/n1, because the holder of both contracts face the same choice betweenreceiving the face value Z and getting n1 shares of stocks.

0 20 40 60 80 100 120 140 160

Underlying price

0

20

40

60

80

100

120

140

160

180B

on

d p

rice

Sr*

Sc( )

*

V(Sr, )

*V(S

c( ), )*

RCB

CB for n1

CB for n2

Figure 5.2: The bond price of RCB and also two CBs at τ=0.05.

Figures 5.2-5.5 show the price of the RCB with respect to the underlying priceat three time moments, τ = 0.05, τ = 0.50 and τ = 0.95, respectively. One of themost noticeable phenomena is the non-monotonicity of the bond price displayed inFigure 5.2, which was discussed in theory in Proposition 5.3.2 already. Financially,this is a natural consequence of the introduction of the resettable clause; in thisparticular case forcing the bond price being bounded below, as a result of bondholders would benefit from the bond being reset to a higher conversion ratio whenthe underlying price is sufficiently small. On the other hand, our observations showthat such a non-monotonic phenomenon never occurs when it is still quite far awayfrom expiry (see Figure 5.3 and Figure 5.4). Financially, the fact that the non-monotonicity can only be observed when the time is sufficiently close to expiry, witha sufficiently large reset conversion ratio as discussed in Proposition 5.3.2, is becausethe optimal conversion price is a decreasing function of time and when the expirytime is approached, the bond price corresponding to this price, V (Sc(τ),τ), becomessmaller than the bond price corresponding to the reset, V (Sr,τ). Thus, unless thebond holder chooses not to convert, then he/she should be prepared to accept theprice non-monotonicity between Sr and Sc(τ).

In addition, it should also be pointed out that all these three figures have shown


0 20 40 60 80 100 120 140 160 180

Underlying price

0

20

40

60

80

100

120

140

160

180

Bo

nd

price

Sr*

RCB

CB for n1

CB for n2


0 20 40 60 80 100 120 140 160 180

Underlying price

0

20

40

60

80

100

120

140

160

180

Bo

nd

price

Sr*

RCB

CB for n1

CB for n2


that the value of the resettable convertible bond is lower than that of the convertiblebond with the reset conversion ratio, while is higher than that of the convertible


100 110 120 130 140 150 160 170 180

Underlying price

100

110

120

130

140

150

160

170

180

Bo

nd

price Price at =0.01

Price at =0.50

Price at =0.99

Payoff of n1

Figure 5.5: The bond price of RCB at three moments.

bond with the initial conversion ratio. This is reasonable because the price of theconvertible bond is an increasing function of the conversion ratio, according to theproperties of the convertible bond in [106], implying that the price of the bond withthe reset conversion ratio is higher than that of the bond with the initial conversionratio. With this in mind, the value of the resettable convertible bond is higher thanthat of the convertible bond with the initial conversion ratio because it is boundedbelow due to the existence of the reset clause, while it is lower than that of theconvertible bond with the reset conversion ratio because it will only become a morevaluable contract when the underlying price is small enough.

To further demonstrate the properties of the bond price, Figure 5.5 combines thethree resettable convertible bond price curves together. From this figure, one canclearly observe that all three price curves are tangent to the payoff line correspond-ing to the initial conversion ratio, instead of that with the reset conversion ratio.Moreover, the value of the optimal conversion boundary is a monotonically increas-ing function of the time to expiry, which is consistent with the property shown inFigure 5.1.

Figure 5.6 and Figure 5.7 are used to show the impact of different reset priceson the bond price and that on the optimal conversion price, respectively. When wefocus on Figure 5.6, what can be observed is that a higher value of the reset priceleads to a higher value of the resettable convertible bonds. This can be explained


0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bo

nd

price

Sr=85

Sr=100

Sr=115

Figure 5.6: The bond price of RCB.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiry

100

120

140

160

180

200

220

Sc(

)

Sr=85

Sr=100

Sr=115

Figure 5.7: The conversion boundary price of RCB.

from the property mentioned above that the price of the RCB should lay betweenthe convertible bond prices with the two conversion ratios, leading to the case that a


larger reset price yields a higher value of the RCB at the reset price. Take the RCBprices at the highest reset price, Sr = 115, as an example to further illustrate this.When the underlying price equals to the reset price, the price of the RCB with thereset price being to 115 is actually the price of the convertible bond with the resetconversion ratio, while the prices of another two RCBs should be lower than it sincethey have not been reseted yet. Moreover, it is also clear in this figure that a highervalue of the reset price leads to a higher value of the optimal conversion boundary,which is also shown in Figure 5.7. It is simply because the higher value of the resetprice implies a higher value of the resettable convertible bond, and thus the holderwill naturally hold the bond unless the underlying price increases to a higher level.

0 20 40 60 80 100 120 140 160 180 200

Underlying price

0

20

40

60

80

100

120

140

160

180

200

Bo

nd

price

n2=1.1

n2=1.2

n2=1.3

Figure 5.8: The bond price of RCB.

Depicted in Figure 5.8 and Figure 5.9 are the changes of the bond price and theoptimal conversion price with three different values of the reset conversion ratios,respectively. In specific, Figure 5.8 clearly shows that the bond price appears tobe increasing with respect to the reset conversion ratio. The rational behind thisphenomenon is that the price of the vanilla convertible bond is an increasing functionof the conversion ratio, as discussed above, and the increase in the reset conversionratio actually raises the lower bound of the RCB price. This also explains thephenomenon in Figure 5.9 that a higher value of the bond leads to a higher optimalconversion price.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiry

100

110

120

130

140

150

160

170

180

190

200

Sc(

)

n2=1.1

n2=1.2

n2=1.3

Figure 5.9: The conversion boundary price of RCB.

5.5 ConclusionIn this chapter, the pricing problem of a resettable convertible bond is considered.A PDE system for the bond price is built up, and an integral equation formulationfor the bond price is obtained with the use of the incomplete Fourier transform,involving the optimal conversion price as an unknown function to be solved. Af-ter the establishment of an appropriate numerical scheme, the value of the optimalconversion boundary is obtained, with which the bond price can be directly calcu-lated from the integral equation representation, and some numerical examples areprovided to show the properties of the resettable convertible bond price as well asthe optimal conversion price.

Chapter 6

Pricing convertible bonds under stochasticvolatility or interest rate

6.1 IntroductionA bond is an instrument that can be treated as the issuer borrowing money fromthe holders for a pre-specified period. If a clause is added to the contract so that theholders can choose to convert the bond into a predetermined number of stocks ornot, it then becomes a convertible bond (CB). While it provides a great incentives toa bond holder to invest in CBs rather than a conventional bond, this additional rightof the holder indeed makes its pricing problem much more complicated, since thebond price and the optimal conversion pricea should be determined simultaneously.

In 1973, Black & Scholes [9] proposed to model the underlying price with a geo-metric Brownian motion (GBM) for the option pricing problem, and shortly after,Ingersoll [62] and Brennan & Schwartz [12] considered the valuation problem of CBswith the firm value being taken as underlying variable following this particular Black-Scholes (B-S) model. This approach was improved by McConnel & Schwartz [84] byreplacing the firm value with the stock price as the underlying variable since thefirm value can not be directly observed in real markets. Since then, various ap-proaches have been proposed to price CBs. For example, Nyborg [88] obtained aclosed-form solution under the B-S model for a simple convertible bond, which canonly be converted at maturity, while Zhu [106] presented an analytical solution underthe same model for a convertible bond, which can be converted at any time on orbefore maturity, using the homotopy analysis method.

If the issuer or the holder of a CB is entitled with some additional rights, differentkinds of CBs will be formulated, such as callable CBs, puttable CBs, resettable CBsand so on, making the corresponding pricing problem even more complex. Thus,

aThe optimal conversion price is referred to as the critical stock price beyond which the holderwill choose to convert the bond into stocks.

115

CHAPTER 6. CONVERTIBLE BONDS UNDER STOCHASTIC VOLATILITY OR INTEREST116

numerical methods must be adopted in most cases. For example, Tsiveriotis &Fernandes [99] priced the cash-only CBs by applying the finite difference method onthe coupled B-S equations. Ohtake et al. [89] presented the definitions of the calland reset clause, depending on which resettable CBs were considered by Kimura &Shinohara [70] with the Monte Carlo method. Recently, Zhu et al. [109] derived anintegral equation formulation for pricing the puttable convertible bond.

Of course, the B-S model is usually not adequate to model the underlying price,and one of the most popular approaches is to introduce additional random variablesinto the B-S model, which can mainly be divided into two categories, i.e., stochasticinterest rate models and stochastic volatility models. Examples in the former cat-egory include the Merton model [85], CEV model [27,30], Vasicek model [100], Dothanmodel [37], Brennan-Schwartz model [14], CIR-VR model [28], GBM model [83] and CIRmodel [29], while the Heston model [52] and Hull-White model [55] are very popularamong many others included in the latter category. Unfortunately, the additionalrandom variables make the pricing problem much more complicated, and thus thenumerical methods are often resorted to in these cases. In particular, the finite dif-ference approach [105], the finite element approach [6], the finite volume approach [112],the binomial tree method [21,59] and the Monte Carlo simulation method [3,80] havealready been adopted to price the convertible bonds under these complex models.

Another popular numerical approach is the predictor-corrector scheme [16,93]. Itis a method to solve the ordinary differential equation (ODE) with two steps; aprediction step computing the value of the function at a preceding set of pointsto obtain the value of this function at a subsequent point, and then a correctionstep refining the value of the unknown function at the same subsequent point usinga suitable approach. In other words, it is a method with suitable association ofan implicit scheme and an explicit scheme. In fact, this method has already beenapplied to solve the pricing problem of the security instruments, even though thegoverning equations for these pricing problems are all partial differential equations(PDEs). A typical example is provided in [110], where Zhu & Zhang chose a suitablecombination of a prediction scheme and a correction scheme to obtain a new schemefor evaluating American options under the B-S model. On the other hand, theAlternating Direction Implicit (ADI) method is a very useful technique to solve thePDEs on the rectangular domains [32,33,58,91], especially for the parabolic ones as forthe other two cases the problem can become quite complex [96]. Fortunately, theequations governing the prices of financial derivatives under most existing models,including the B-S model and the stochastic volatility/interest-rate models, are allparabolic differential equations, and thus the ADI method is ideal to be utilized forthese pricing problems [48,60].

In this chapter, we adopt a particular predictor-corrector scheme, constructed by


the two methods mentioned above being combined together with the ADI schemeused as the correction step. It was proposed by Zhu & Chen [107] in solving thepricing problem of American puts with stochastic volatility, which was utilized byChen et al [25] for the pricing of the stock loan with stochastic interest rate. In orderto determine the price of CBs, we firstly establish two PDE systems for the price ofCBs under a stochastic volatility and a stochastic interest rate model, respectively.Since the moving boundary exists in both of the two systems, Landau transform [71]

is used to transform the free boundary problem into a fixed one, at the cost of theoriginal linear PDE becoming a nonlinear one, after which the predictor-correctorscheme is adopted for each time step to convert the nonlinear PDE into two lin-earized difference equations associated with the prediction and correction phase,respectively. For the prediction step, an explicit Euler scheme is used to predict thevalue of the optimal conversion boundary, and at the correction step, the value ofthe bond is then determined through the ADI scheme, based on which the correctionof the optimal conversion boundary is obtained. Another contribution of this paperis the proposition of the boundary conditions along the volatility and interest ratedirection, which contribute to the development of the closed PDE system for pricingCBs under stochastic volatility and interest rate models.

The chapter is organized as follows. In Section 2, the pricing problem for CBsunder a stochastic volatility model is considered, presenting the numerical schemeas well as the numerical results we obtain. In Section 3, how to price CBs undera stochastic interest rate model is illustrated. Concluding remarks are given in thelast section.

6.2 Pricing convertible bonds with stochastic volatilityIn this section, the pricing problem of convertible bonds when the volatility is madeto be another random variable is discussed. We will use the Heston model as anexample to illustrate this since the processes in solving the pricing problem underdifferent stochastic volatility models are very similar and the Heston model is oneof the most popular models.

In the following, the PDE system governing the price of the CBs is firstly setup and then how to obtain the predictor-corrector scheme with the ADI method tovalue the CBs are illustrated, after which the accuracy of the proposed method isdemonstrated through numerical experiments and the properties of the CBs withstochastic volatility are also studied.


6.2.1 The PDE system under the Heston model

To build the PDE system for pricing a CB, the dynamics of the adopted modelshould be specified first. Let St be the underlying asset price, and then its dynamicunder a risk-neutral measure is assumed to satisfy the following stochastic differentequation (SDE):

dSt = (r−D0)Stdt +√

vtStdW1, (6.1)

where r is the risk-free interest rate and D0 is the continuous dividend rate. W1 is astandard Brownian motion, and vt is the stochastic volatility, which is governed bythe following SDE:

dvt = κ(η − vt)dt +σ√

vtdW2, (6.2)

with κ denoting the rate of relaxation to this mean, η representing the long timemean of vt , and σ being the volatility of volatility. W2 is also a standard Brownianmotion, being correlated W1 with ρ ∈ [−1,1]. If the value of the bond is denoted byU(S,v, t), its governing PDE can be written as follows:

12

vS2 ∂ 2U∂S2 +σρvS

∂ 2U∂S∂v

+12

σ2v∂ 2U∂v2 +(r−D0)S

∂U∂S

+κ(η − v)∂U∂v

− rU +∂U∂ t

= 0.(6.3)

The terminal condition for Equation (6.3) is actually the payoff function of CBs

U(S,v,T ) = maxCRS,Z, (6.4)

where CR is the conversion ratio and Z is the face value of the bond. The boundaryconditions in the direction of the underlying asset price S are given as

U(0,v, t) = Ze−r(T−t), (6.5)U(S f (v, t),v, t) =CR ·S f (v, t), (6.6)∂U∂S

(S f (v, t),v, t) =CR, (6.7)

where S f is the optimal conversion price. It should be noted that all of the condi-tions mentioned above are very similar to that under the B-S model, and the maindifference between the B-S model and the stochastic volatility model is that thebond price and the optimal conversion price are both the functions of the volatilityfor the stochastic volatility model. Therefore, the boundary conditions for v areneeded to close the PDE system. In this study, the boundary conditions are chosenas

limv→0

U(S,v, t) = maxCRS,Ze−r(T−t), (6.8)


limv→∞

∂U∂v

(S,v, t) = 0. (6.9)

We would like to explain a bit on how we choose the boundary conditions in thedirection of v. On one hand, for the boundary condition at v = 0, it needs to bepointed out that this boundary condition is not necessary if the Fichera function [44]

along v = 0 satisfies κη − σ2

2 ≥ 0, while the boundary condition at v = 0 is neededto close the system when κη < σ2

2 . Moreover, the solution of SDE (6.1) when v = 0can be approximated as, S = e(r−D0)tS0, there is virtually no risk with the underlyingasset. This demonstrates that if CRS > Ze−r(T−t), there is no sense to hold the bond,and it should be exercised immediately, implying that the bond price at this situationis CRS. In contrast, if CRS ≤ Ze−r(T−t), the bond should be held until the expiry andits value should be Ze−r(T−t) instead. Therefore, both cases show that the boundarycondition at v = 0 is limv→0U(S,v, t) = maxCRS,Ze−r(T−t). On the other hand,when the volatility approaches infinity, the bond price should be independent of thevolatility change, otherwise, the bond price will reach infinity, since the bond priceis an increasing function with respect to the volatility.

In summary, the PDE system for pricing a CB under the Heston model can beestablished as

12

vS2 ∂ 2U∂S2 +σρvS

∂ 2U∂S∂v

+12

σ2v∂ 2U∂v2 +(r−D0)S

∂U∂S

+κ(η − v)∂U∂v

− rU +∂U∂ t

= 0,

U(S,v,T ) = maxCRS,Z,U(0,v, t) = Ze−r(T−t),

U(S f (v, t),v, t) =CR ·S f (v, t),∂U∂S

(S f (v, t),v, t) =CR,

limv→0

U(S,v, t) = maxCRS,Ze−r(T−t),

limv→∞

∂U∂v

(S,v, t) = 0,

(6.10)

for S ∈ [0,S f (v, t)], v ∈ [0,∞] and t ∈ [0,T ]. In the following, we are going to presentthe details on how to apply the predictor-corrector method with ADI scheme on thePDE system governing the value of CBs.

6.2.2 Discretize the PDE system

In this subsection, the PDE system is discretized with some rules. Before discretiza-tion, it should be noted that one of the boundaries of System (6.10) in the direction ofS is not fixed, which poses an obstacle in applying the predictor-corrector method.Therefore, a classical transform, Landau transform [71], i.e. x = ln(

SS f

), should be


adopted to this PDE system to solve this issue, and at the same time the valueof the optimal conversion boundary is now a part of the solution. Moreover, asimple transform, τ = T − t, is also applied to the PDE system to change the ter-minal condition problem to an initial counterpart. In addition, another transform,V (x,v,τ) = U(x,v,τ)−Ze−rτ , is also made here so that the PDE system (6.10) canbe rewritten as

LV (x,v,τ) = 0,V (x,v,0) = maxCR ·S f (v,0) · ex −Z,0,lim

x→−∞V (x,v,τ) = 0,

V (0,v,τ) =CR ·S f (v,τ)−Ze−rτ ,∂V∂x

(0,v,τ) =CR ·S f (v,τ),

limv→0

V (x,v,τ) = maxCR ·S f (0,τ) · ex −Ze−rτ ,0,

limv→∞

∂V∂v

(x,v,τ) = 0,

(6.11)

with x ∈ (−∞,0], v ∈ [0,∞) and τ ∈ [0,T ], and

L = [12

v+12

σ2v1S2

f(∂S f

∂v)2 − ρσv

S f

∂S f

∂v]

∂ 2

∂x2 +12

σ2v∂ 2

∂v2 +(ρσv− σ2vS f

∂S f

∂v)

∂ 2

∂x∂v

+[−12

v+12

σ2vS2

f(∂S f

∂v)2 − 1

2σ2vS f

∂ 2S f

∂v2 + r−D0 −κ(η − v)1S f

∂S f

∂v+

1S f

∂S f

∂τ]

∂∂τ

+κ(η − v)∂∂v

− r− ∂∂τ

. (6.12)

To further simplify the notation of L , some new notations are defined here

ξ =1S f

·∂S f

∂v, β =

1S f

·∂ 2S f

∂v2 , λ =1S f

·∂S f

∂τ, (6.13)

with which L can be represented by

L = a(v)∂ 2

∂x2 +b(v)∂ 2

∂v2 + c(v)∂ 2

∂x∂v+[d(v)+λ ]

∂∂x

+ e(v)∂∂v

− r− ∂∂τ

, (6.14)

where

a(v) =12

v+12

σ2vξ 2 −ρσvξ , (6.15)

b(v) =12

σ2v, (6.16)

c(v) = ρσv−σ2vξ , (6.17)

d(v) = −12

v+12

ξ 2σ2v− 12

σ2vβ + r−D0 −κ(η − v)ξ , (6.18)


e(v) = κ(η − v). (6.19)

Before our approach can be applied to obtain the numerical solution, System(6.11) should be discretized first. Specifically, the semi-infinite domain should befirstly truncated into a finite one as follows

(x,v,τ)|x ∈ [xmin,0],v ∈ [0,vmax],τ ∈ [0,T ], (6.20)

and the values of xmin and vmax will be chosen when we present the numerical results.Then, the finite domain will be separated into Nτ uniform grids in the direction ofτ , Nx uniform grids in the direction of x and Nv uniform grids in the direction of v.Thus, we have

∆τ =TNτ

, ∆x =−xmin

Nx, ∆v =

vmax

Nv, (6.21)

and

τn = n∆τ, with τ0 = 0 and τNτ = T, (6.22)xi = xmin + i∆x, with x0 = xmin and xNx = 0, (6.23)v j = j∆v, with v0 = 0 and vNv = vmax. (6.24)

In this case, the value of the unknown functions at a grid point, V (xi,v j,τn) andS f (v j,τn), are denoted as V (n)

i, j and S(n)f ( j), respectively, for i = 0,1, · · · ,Nx, j =

0,1, · · · ,Nv and n = 0,1, · · · ,Nτ .In order to apply our numerical method, we now classify the entire domain into

two parts, with the first one being the interior of the domain

D = (xi,v j)|i = 1, · · · ,Nx −1, j = 1, · · · ,Nv −1, (6.25)

and another one representing the boundaries. For each grid point in D , the standardcentral difference scheme and the second-order half-central difference scheme areused to approximate the first-order derivative (including the cross-derivative) andthe second-order derivative, respectively. Thus, all the derivatives belonging to D

are discretized as following:

∂V (n)i, j

∂x=

V (n)i+1, j −V (n)

i−1, j

2∆x, (6.26)

∂V (n)i, j

∂v=

V (n)i, j+1 −V (n)

i, j−1

2∆v, (6.27)

∂ 2V (n)i, j

∂x2 =V (n)

i+1, j −2V (n)i, j +V (n)

i−1, j

(∆x)2 , (6.28)


∂ 2V (n)i, j

∂v2 =V (n)

i, j+1 −2V (n)i, j +V (n)

i, j−1

(∆v)2 , (6.29)

∂ 2V (n)i, j

∂x∂v=

V (n)i+1, j+1 −V (n)

i+1, j−1 −V (n)i−1, j+1 +V (n)

i−1, j−1

4∆v∆x. (6.30)

For the boundary part, it is easy to deal with the Dirichlet boundary conditions,while it is quite difficult to approximate the Neumann boundary condition. This isbecause in general, the first-order derivative at x = 0 should be presented

∂V (n)Nx, j

∂x=

V (n)Nx+1, j −V (n)

Nx−1, j

2∆x, (6.31)

whereas it is impossible to obtain the value of V (n)Nx+1, j. Therefore, we have to use an

alternative approach, the so-called one-sided difference, instead of the central one.It is a form of extrapolation that determines the value of the unknown function onthe boundary in terms of its values at the interior grid points [96]. With the use ofthe Taylor series, we can obtain the following equations:

V (n)Nx−1 = V (n)

Nx, j −∆x∂V (n)

Nx, j

∂x+

12(∆x)2 ∂ 2V (n)

Nx, j

∂x2 +o((∆x)3), (6.32)

V (n)Nx−2, j = V (n)

Nx, j −2∆x∂V (n)

Nx, j

∂x+

12(2∆x)2 ∂ 2V (n)

Nx, j

∂x2 +o((∆x)3). (6.33)

By eliminating∂ 2V (n)

Nx, j

∂x2 , we can further obtain

∂V (n)Nx, j

∂x=

3V (n)Nx, j +V (n)

Nx−2, j −4V (n)Nx−1, j

2∆x+o((∆x)3), (6.34)

where the value of∂V (n)

Nx, j

∂xis expressed in the form of the values for V (n)

Nx−2, j, V (n)Nx−1, j,

and the unknown boundary value V (n)Nx, j approximately. In summary, the finite dif-

ference equation (FDE) system written on a grid point for System (6.11) can be


specified as

∂V (n)i, j

∂τ = a j∂ 2V (n)

i, j∂x2 +b j

∂ 2V (n)i, j

∂v2 + c j∂ 2V (n)

i, j∂x∂v +[d j +λ j]

∂V (n)i, j

∂x + e j∂V (n)

i, j∂v − rV (n)

i, j ,

V (0)i, j = maxCR ·S(0)f ( j)exi −Z,0,

V (n)0, j = 0,

V (n)Nx, j =CR ·S(n)f ( j)−Ze−rτn,

3V (n)Nx, j+V (n)

Nx−2, j−4V (n)Nx−1, j

2∆x =CR ·S(n)f ( j),

V (n)i,0 = maxCR ·S(n)f (0)exi −Ze−rτn,0,

V (n)i,Nv

= 0,

(6.35)

where

a j =12

v j +12

σ2v jξ 2j −ρσv jξ j, (6.36)

b j =12

σ2v j, (6.37)

c j = ρσv j −σ2v jξ j, (6.38)

d j =−12

v j +12

σ2v jξ j −12

σ2v jβ j + r−D0 −κ(η − v j)ξ j, (6.39)e j = κ(η − v j), (6.40)

with

ξ j =1

S f ( j)·

S f ( j+1)−S f ( j−1)2∆v

, (6.41)

β j =1

S f ( j)·

S f ( j+1)−2S f ( j)+S f ( j−1)(∆v)2 , (6.42)

λ j =1

S f ( j)·

∂S f ( j)∂τ

. (6.43)

It should be remarked here that the boundary condition in the FDE system ischanged from the Neumann one to the Dirichlet counterpart when v → ∞. This canbe explained by nothing that the initial Neumann boundary condition implies thatthe value of the bond at v → ∞ should be a constant (independent on v), and sucha constant should be equal to 0 since the holder will not choose to convert the bondas the market is too volatile and the bond price in this case should equal to Ze−rτ ,or in other words, limv→∞V (x,v,τ) = 0. Another thing should also be noted that thetime derivatives have not been discretized by now, and the explicit Euler scheme

and the implicit Euler scheme are applied to the time derivative∂V (n)

i, j

∂τand

∂S(n)f ( j)

∂τ,

respectively, in the process of applying the predictor-corrector scheme, the detailsof which are illustrated in the next two subsections.


6.2.3 Numerical scheme for the prediction step

In this subsection, the numerical scheme for predicting the value of the optimalconversion boundary will be presented. Firstly, we should recall the discretizedformulation of the boundary conditions at x = 0

V (n)Nx, j = CR ·S(n)f ( j)−Ze−rτn, (6.44)

CR ·S(n)f ( j) =3V (n)

Nx, j +V (n)Nx−2, j −4V (n)

Nx−1, j

2∆x. (6.45)

Then, we can obtain

CR ·S(n)f ( j) =3V (n)

Nx, j +V (n)Nx−2, j −4V (n)

Nx−1, j

2∆x

⇒ CR ·S(n)f ( j) =3(CR ·S(n)f ( j)−Ze−rτn)+V (n)

Nx−2, j −4V (n)Nx−1, j

2∆x⇒ 2∆x ·CR ·S(n)f ( j) = 3(CR ·S(n)f ( j)−Ze−rτn)+V (n)

Nx−2, j −4V (n)Nx−1, j

⇒ (3−2∆x) ·CR ·S(n)f ( j) = 3Ze−rτn +4V (n)Nx−1, j −V (n)

Nx−2, j

⇒ S(n)f ( j) =3Ze−rτn +4V (n)

Nx−1, j −V (n)Nx−2, j

(3−2∆x) ·CR. (6.46)

If we assume the values of the bond and its optimal conversion boundary at thenth time step are known, the optimal conversion boundary at the (n+ 1)th timestep can be determined by using the formulation above

S(n+1)f ( j) =

3Ze−rτn+1 +4V (n+1)Nx−1, j −V (n+1)

Nx−2, j

(3−2∆x) ·CR. (6.47)

With this newly obtained expression, if the explicit Euler scheme and the implicit

Euler scheme are applied to the time derivative∂V (n)

i, j

∂τand

∂S(n)f ( j)

∂τ, respectively, we

can obtain

V (n+1)i, j = V (n)

i, j +∆τa j∂ 2V (n)

i, j

∂x2 +b j∂ 2V (n)

i, j

∂v2 + c j∂ 2V (n)

i, j

∂x∂v+[d j +

S(n+1)f ( j)−S(n)f ( j)

∆τS(n)f ( j)]∂V (n)

i, j

∂x

+ e j∂V (n)

i, j

∂v− r∆τV (n)

i, j , i = Nx −2,Nx −1. (6.48)

After some complex computation, the predicted value of the optimal conversion


boundary at (n+1)th time step, S f(n+1)

( j), can be presented

S f(n+1)

( j) =3Ze−rτn+1 +F [4V (n)

Nx−1, j −V (n)Nx−2, j]

(3−2∆x)CR −δx(4V (n)

Nx−1, j−V (n)Nx−2, j])

S(n)f ( j)

, (6.49)

withF = I+∆τ[a jδxx +b jδvv + c jδxv +(d j −

1∆τ

)δx + e jδv]− r∆τ. (6.50)

Here, all the derivatives are replaced by the corresponding δ∗. Therefore, the pre-dicted value of the corresponding bond price, V (n+1)

Nx, j , can be calculated as

V (n+1)Nx, j =CR · S f

(n+1)( j)−Ze−rτn+1. (6.51)

In summary, the value of the optimal conversion boundary and the bond price onthis boundary can be predicted by Equation (6.50) and Equation (6.51), respectively,with which the algorithm of ADI method is used to obtain all the values at the(n+1)th time step, V (n+1), in the next subsection. Afterwards, we can correct thevalues of S(n+1)

f and V (n+1)Nx

by using the newly obtained V (n+1)Nx−2 and V (n+1)

Nx−1 .

6.2.4 Numerical scheme for the correction step

In this subsection, ADI method is utilized for the correction step. As mentionedabove, this method is a very useful technique to solve PDEs on rectangular domains,especially for the parabolic ones. Moreover, what we choose in this study is theDouglas-Rachford (D-R) method, whose accuracy is of first-order in time and second-order in space. Now, the FDE should be rewritten, so that the ADI method can beapplied

(I−ϕA1)(I−ϕA2)V (n+1) = [I+A0 +(1−ϕ)A1 +A2]V (n)− (I−ϕA1)ϕA2V (n), (6.52)

where ϕ ∈ [0,1]. The procedures to obtain all operators, A0, A1 and A2, are left inAppendix D.1. For the D-R method, there are two steps that need to be conductedbefore we can obtain the final scheme. Firstly, we should calculate the intermediatevalue, Y , with the following equation

(I−ϕA1)Y = [I+A0 +(1−ϕ)A1 +A2]V (n), (6.53)

where we fix the v direction. Then, the above equation can be simplified as

AYj = Pj +Bx j, (6.54)


with the details of A, Yj, Pj and Bx j being presented in Appendix D.2. It shouldbe noted that matrix A is a tridiagonal, and thus the Thomas algorithm [96] can beapplied to improve the speed and the accuracy of our method. It should also bepointed out that the value Y is actually obtained by the loop of j and each Yj is avector with (Nx+1)×1. Once the value of Y is known, we can then compute V (n+1)

from the following equation

(I−ϕA2)V (n+1)+ϕA2V (n) = Y, (6.55)

with the x direction being fixed. This equation can be further represented as

CV (n+1)i = Qi +Bvi, (6.56)

and the details of C, V (n+1)i , Qi and Bvi are left in Appendix D.3. Similarly to

Equation (6.54), Matrix C is also a tridiagonal, and the Thomas algorithm is utilizedhere again. For this equation, the loop of i is used to obtain the value of V (n+1),leading to our desired result, since it can be easily proved that solving Equation(6.54) and Equation (6.56) means solving the initial Equation (6.52). Clearly, withthese two steps illustrated in this subsection, the algorithm to derive the correctorscheme has already been designed.

In order to numerically implement our scheme in the next subsection, we needto make it clear how to calculate the boundary value of the intermediate value Y ,which is presented as

Y0 = 0, (6.57)YNx = (I−ϕA2)V

(n+1)Nx

+ϕA2V (n)Nx

= (I−ϕA2)(CR · S f(n+1)−Ze−rτn+1)+ϕA2V (n)

Nx, (6.58)

with the use of the predicted value of S f(n+1). Clearly, this has completed the

establishment of the predictor-corrector scheme with the ADI method for pricingCBs if we combine the predictor scheme presented in the previous subsection andthe corrector scheme shown in this subsection, and we are now ready to conductnumerical experiments to study the properties of CBs under the stochastic volatilitymodel, the details of which are shown in the next subsection.

6.2.5 Numerical examples

In this subsection, the accuracy of our method is tested first, and then the numericalresults are provided to illustrate several properties of the convertible bond under astochastic volatility model. Unless otherwise stated, the parameters used are listed


below:


• Conversion ratio CR = 1,




• Reversion rate κ = 1.5,

• Reversion level η = 0.16,

• Volatility of the volatility σ = 0.4,

• Correlation factor ρ = 0.1.

Before we present the numerical results, it is necessary to determine the domainthat we are going to operate on. Although x can take any value being less than 0,we need to truncate the semi-infinite domain into a finite one so as to implementour numerical scheme. As mentioned in [102] that it suffices to set the minimum of x

to be − ln5, the domain of our model is assumed as

(x,v,τ)|x ∈ [− ln5,0]× [0,1]× [0,1]. (6.59)

Here, setting vmax = 1 is sufficient and reasonable since the value of the volatilityis usually very small. After the domain is uniformly discretized, with the step sizein the direction of x, v and t being 101, 201 and 5001, respectively, the numericalresults are presented in the following. It should also be pointed out that all of ourcalculations in this paper are done using Matlab R2017a on a PC with the followingspecifications: Intel(R) Core(TM), i7-4790 [email protected] 3.60 GHz, and 16.0 GBof RAM.

To validate our numerical scheme, a degenerate case is considered as the bench-mark, where the volatility is a fixed value instead of being stochastic, and the valuesof the optimal conversion boundary calculated with our method and those derivedthrough the integral equation approach [109] are displayed in both Table 6.1 and Fig-ure 6.1. In particular, Table 6.1 shows the prices of the optimal conversion boundaryat the current time, t = 0, and one can easily observe that with the increase in thenumber of grid points, our results converge to the benchmark, as the relative errorbetween the two prices are decreasing. If we turn to Figure 6.1, it is clear thatboth values match very well with each other, demonstrating the accuracy of our


Table 6.1: Convergence test

Volatility value (Nx,Nv,Nt) ADI IE Relative error(25,50,1250) 12.8669 5.97×10−3

v = 0.1 (50,100,2500) 12.8110 12.7905 1.61×10−3

(100,200,5000) 12.7969 5.00×10−4

(25,50,1250) 15.7500 6.17×10−3

v = 0.2 (50,100,2500) 15.6796 15.6533 1.67×10−3

(100,200,5000) 15.6618 5.38×10−4

(25,50,1250) 21.2244 8.19×10−3

v = 0.4 (50,100,2500) 21.1017 21.0519 2.36×10−3

(100,200,5000) 21.0709 9.00×10−4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

10

10.5

11

11.5

12

12.5

13

Va

lue

of

the

bo

un

da

ry

Our method

Intergal Equation method

Figure 6.1: The comparison of the optimal conversion boundary obtained by ourmethod and that from the integral equation method [109], at v = 0.1.

method for this case. Of course, we still need to check whether our approach worksfor the general case when the stochastic volatility is incorporated. Thus, the valuesof CBs obtained from our method are also compared with those generated throughthe Monte Carlo method, the results of which are presented in Figure 6.2. It can beeasily noticed that both prices are point-wisely close to each other, which certainlyreflects that our method is reliable. On the other hand, it should be point out thatthe average CPU time consumed by the Monte Carlo method, with 100 time stepsand 500,000 simple paths, is 10.3828 seconds, while it only takes 7.9×10−7 secondsto produce one price with the method introduced by this paper. Such a great differ-


0 2 4 6 8 10 12 14 16 18 20

Underlying price

0

2

4

6

8

10

12

14

16

18

20

Bo

nd

price

Our method

MC method

Payoff

Figure 6.2: The comparison of the bond prices obtained by our method and thosefrom the Monte Carlo method, at t = 0 and v = 0.4.

ence certainly indicates that the proposed predictor-corrector scheme is much moreefficient.

With the confidence in our approach, we are now studying the properties of theoptimal conversion boundary as well as the bond prices when the stochastic volatil-ity is considered. Depicted in Figure 6.3 are the values of the optimal conversionboundary with respect to the volatility and the time to expiry. In order to showclearly the effects of the volatility and the time to expiry on the optimal conversionprice, two figures, Figure 6.4 and Figure 6.5, are displayed by fixing one direction.An interesting phenomenon that can be observed in Figure 6.4 is that the optimalconversion boundary is not a monotonic increasing function of the time to expiry; itincreases with the time to expiry when the time to expiry is small, and it will show adownward trend once the time to expiry is large enough. This is consistent with thetheoretical result [104] that the value of the perpetual optimal conversion boundaryequals to zero. From the financial point of view, it can be understood from theextreme case that when the time to expiry approaches infinity, the current value ofthe face value is almost zero, implying that it is meaningless to continue to holdthe CB and the investor should convert it into stocks immediately. On the otherhand, when we turn to Figure 6.5, it should be noted that the optimal conversionboundary is always a monotonic increasing function of the volatility, no matter the


Figure 6.3: The optimal conversion price.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

10

11

12

13

14

15

16

17

Va

lue

of

the

bo

un

da

ry

v=0.2

v=0.5

v=0.8


value of the time to expiry, which is reasonable since a higher volatility means ahigher risk, leading to a higher premium of the CB.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

v

11

12

13

14

15

16

17

18

Va

lue

of

the

bo

un

da

ry

=0.2

=0.5

=0.8


8

1

10

12

20

Bo

nd

price

14

15

v

16

0.5

S

18

10

50 0

Figure 6.6: The bond price at τ = 1.

What is shown in Figure 6.6 is the change of the bond price with respect tothe volatility and the underlying price, when the CB has not been converted into


0 2 4 6 8 10 12 14 16 18 20

S

0

2

4

6

8

10

12

14

16

18

20

Va

lue

of

the

bo

un

da

ry

v=0.2

v=0.5

v=0.8


stocks. Clearly, no matter what the value of the underlying asset is, the bond price isa monotonic increasing function of the volatility as a larger volatility always impliesa higher risk. Moreover, a higher value of the underlying asset leads to the largerslop of the bond price with respect to the volatility. In other words, the bond priceis also an increasing function of the underlying asset, when the volatility is fixed,which is also clearly presented in Figure 6.7. This is financially meaningful sincewhen the underlying asset price increases, there will be a higher probability forthe holder to convert the bond, leading to the higher value of the bond. Anotherphenomenon that can be noticed here is that increasing the value of the volatilityis actually increasing the value of the optimal conversion boundary, which confirmsthe result presented in Figure 6.3.

6.3 Pricing convertible bonds with stochastic interest rateIn this section, we study the pricing problem of the convertible bond with a stochas-tic interest rate model (CIR model). Given the fact that the PDE system in thissection is very similar to that in the last section and the same predictor-correctorscheme will also be utilized here, the details on some tedious but very similar com-putational processes are thus omitted.


6.3.1 The PDE system and its numerical scheme

We now begin by assuming that the stochastic interest rate satisfies the followingSDE

dr = κ(η − r)dt +ξ√

rdW 3t , (6.60)

where κ , η and ξ are the mean reversion speed, the long term mean and the volatilityof the interest rate, respectively, while W3 is another standard Brownian motion. Inaddition, the dynamics of the underlying asset price is assumed as

dSt = (r−D0)Stdt +σStdW 1t , (6.61)

which is the same as Equation (6.1) except that the constant volatility of the under-lying asset is denoted as σ . The correlation between W1 and W3 is also representedby ρ , which can vary within [−1,1]. In this case, if the bond price is denoted asU(S,r, t), its governing PDE system can be set up

12

σ2S2 ∂ 2U∂S2 +σρξ

√rS

∂ 2U∂S∂ r

+12

ξ 2r∂ 2U∂ r2 +(r−D0)S

∂U∂S

+κ(η − r)∂U∂ r

− rU +∂U∂ t

= 0,

U(S,r,T ) = maxCR ·S,Z,U(0,r, t) = E[Ze−

∫ Tt r(s)ds|rt ], Z ·F(r, t),

U(Sc(r, t),r, t) =CR ·Sc(r, t),∂U∂S

(Sc(r, t),r, t) =CR,

limr→0

U(S,r, t) =UBS(S, t),

limr→∞

U(S,r, t) = 0,

(6.62)

where Sc(r, t) and UBS(S, t) are the optimal conversion boundary and the convertiblebond price under the Black-Scholes model with r = 0, respectively, and F(r, t) satisfiesthe following PDE system

∂F∂ t

(r, t)+κ(η − r)∂F∂ r

(r, t)+12

ξ 2r∂ 2F∂ r2 (r, t)− rF(r, t) = 0,

F(r,T ) = 1.(6.63)

The solution to this PDE system can be found as

F(r, t) = eA(t)−B(t)r, (6.64)


where

A(t) =−κη 4(m−κ)(m+κ)

ln[2m+(m+κ)(em(T−t)−1)

2m+

2κ −m

(T − t)],(6.65)

B(t) =2(e

√m(T−t)−1)

2m+(κ +m)(em(T−t)−1), (6.66)

with m =√

κ +2ξ 2, the derivation of which can be found in [50]. Before we proceedfurther, it is necessary for us to explain the boundary conditions we gave in thesystem in the direction of r. On one hand, for the boundary condition at r = 0, itneeds to be pointed out that this boundary condition is not necessary if the Ficherafunction [44] along r = 0 satisfies κη − ξ 2

2 ≥ 0, while a boundary condition at r = 0is needed to close the system when κη < ξ 2

2 . If a boundary condition is needed, weadopt the bond price under the Black-Scholes model with r = 0 as an approximation,which is based on an assumption that when O(κη

∂U∂ r

) is much smaller than theorder of the other terms in (6.62) when r = 0, the resulting equation degeneratesto the Black-Scholes equation. On the other hand, when the value of the risk-freeinterest rate goes to infinity, the best way to achieve the best return for an investoris leave the money in a risk-free bank account, which implies that no one wouldinvest in a convertible bond, resulting in the bond value being equal to zero.

Now, applying the following transform

V (S,r, t) =U(S,r, t)−E[Ze−∫ T

t r(s)ds|rt ] =U(S,r, t)−Z ·F(r, t), (6.67)

to the above PDE system yields

12

σ2S2 ∂ 2V∂S2 +σρξ

√rS

∂ 2V∂S∂ r

+12

ξ 2r∂ 2V∂ r2 +(r−D0)S

∂V∂S

+κ(η − r)∂V∂ r

− rV +∂V∂ t

= 0,

V (S,r,T ) = maxCR ·S−Z,0,V (0,r, t) = 0,V (Sc(r, t),r, t) =CR ·Sc(r, t)−Z ·F(r, t),∂V∂S

(Sc(r, t),r, t) =CR,

limr→0

V (S,r, t) =UBS(S, t)−Z ·F(0, t),

limr→∞

V (S,r, t) = 0.

(6.68)

Then, in order to transform the target PDE system to a dimensionless one and alsotransform the free boundary problem into a fixed boundary one to facilitate the


numerical computation, we make the following transformation

τ = T − t, x = ln(SSc), (6.69)

so that we have

L [V ] = 0,V (x,r,0) = maxCR ·Sc(r,0) · ex −Z,0,lim

x→−∞V (x,r,τ) = 0,

V (0,r,τ) =CR ·Sc(r,τ)−Z ·F(r,τ),∂V∂x

(0,r,τ) =CR ·Sc(r,τ),

limr→0

V (x,r,τ) =UBS(Sc(0,τ) · ex,τ)−Z ·F(0,τ),

limr→∞

V (x,r,τ) = 0,

(6.70)

where

L = a(r)∂ 2

∂x2 +b(r)∂ 2

∂ r2 + c(r)∂ 2

∂x∂ r+[d(r)+λ ]

∂∂x

+ e(r)∂∂ r

− r− ∂∂τ

, (6.71)

with

a(r) =12

σ2 +12

ξ 2rζ 2 −σρξ√

rζ , (6.72)

b(r) =12

ξ 2r, (6.73)

c(r) = σρξ√

r−ξ 2rζ , (6.74)

d(r) = −12

σ2 +12

ξ 2rζ 2 − 12

ξ 2rβ + r−D0 −κ(η − r)ζ , (6.75)e(r) = κ(η − r). (6.76)

Here, we denote

ζ =1Sc

· ∂Sc

∂ r, β =

1Sc

· ∂ 2Sc

∂ r2 , λ =1Sc

· ∂Sc

∂τ. (6.77)

Since this system is very similar to the last one, the FDE system is directly provided


below without presenting all the details

∂V (n)i, j

∂τ= a j

∂ 2V (n)i, j

∂x2 +b j∂ 2V (n)

i, j

∂ r2 + c j∂ 2V (n)

i, j

∂x∂ r+[d j +λ j]

∂V (n)i, j

∂x+ e j

∂V (n)i, j

∂ r− r jV

(n)i, j ,

V (0)i, j = maxCR ·S(0)c ( j)exi −Z,0,

V (n)0, j = 0,

V (n)Nx, j =CR ·S(n)c ( j)−Z ·F(n)( j),

3V (n)Nx, j+V (n)

Nx−2, j−4V (n)Nx−1, j

2∆x =CR ·S(n)c ( j),

V (n)i,0 =UBS(S

(n)c (0) · exi,τn)−Z ·F(n)(0),

V (n)i,Nr

= 0,

(6.78)

with the divided finite domain being (xi,r j,τn)|xi = xmin+ i∆x, f or i= 0, · · · ,Nx; r j =

j∆r, f or j = 0, · · · ,Nr; τn = n∆τ, f or n = 0, · · · ,Nτ. Here, we have ∆x = −xmin

Nx,

∆r =rmax

Nrand ∆τ =

TNτ

, and the parameters are

a j =12

σ2 +12

ξ 2r jζ 2j −σρξ√r jζ j, (6.79)

b j =12

ξ 2r j, (6.80)

c j = σρξ√r j −ξ 2r jζ j, (6.81)

d j = −12

σ2 +12

ξ 2r jζ 2j −

12

ξ 2r jβ + r j −D0 −κ(η − r j)ζ j, (6.82)e j = κ(η − r j), (6.83)

with

ζ j =1

Sc( j)· Sc( j+1)−Sc( j−1)

2∆r, (6.84)

β j =1

Sc( j)· Sc( j+1)−2Sc( j)+Sc( j−1)

(∆r)2 , (6.85)

λ j =1

Sc( j)· ∂Sc( j)

∂τ. (6.86)

In order to numerically solve the FDE system, we are now ready to set up thepredictor-corrector scheme again with two steps. We will first briefly discuss howthe predictor scheme can be established. By using the boundary conditions at x = 0

CR ·S(n)c ( j) =3V (n)

Nx, j +V (n)Nx−2, j −4V (n)

Nx−1, j

2∆x, (6.87)

V (n)Nx, j = CR ·S(n)c ( j)−Z ·F(n)( j), (6.88)


we can further obtain

CR ·S(n)c ( j) =3V (n)

Nx, j +V (n)Nx−2, j −4V (n)

Nx−1, j

2∆x

⇒ CR ·S(n)c ( j) =3(CR ·S(n)c ( j)−Z ·F(n)( j))+V (n)

Nx−2, j −4V (n)Nx−1, j

2∆x⇒ 2∆x ·CR ·S(n)c ( j) = 3(CR ·S(n)c ( j)−Z ·F(n)( j))+V (n)

Nx−2, j −4V (n)Nx−1, j

⇒ (3−2∆x) ·CR ·S(n)c ( j) = 3Z ·F(n)( j)−V (n)Nx−2, j +4V (n)

Nx−1, j

⇒ S(n)c ( j) =3Z ·F(n)( j)−V (n)

Nx−2, j +4V (n)Nx−1, j

(3−2∆x) ·CR. (6.89)

Therefore, if we assume the value of the bond and that of the optimal conversionboundary at nth time step are known, then the optimal conversion price at (n+1)thtime step can be expressed

S(n+1)c ( j) =

3Z ·F(n+1)( j)+4V (n+1)Nx−1, j −V (n+1)

Nx−2, j

(3−2∆x) ·CR. (6.90)

With the explicit Euler scheme and implicit Euler scheme being applied to the

time derivative∂V (n)

i, j

∂τand ∂S(n)c ( j)

∂τ, respectively, the following formulation can be

obtained

V (n+1)i, j = V (n)

i, j +∆τa j∂ 2V (n)

i, j

∂x2 +b j∂ 2V (n)

i, j

∂ r2 + c j∂ 2V (n)

i, j

∂x∂ r+[d j +

S(n+1)c ( j)−S(n)c ( j)

∆τ ·S(n)c ( j)]∂V (n)

i, j

∂x

+ e j∂V (n)

i, j

∂ r− r j∆τV (n)

i, j , i = Nx −2,Nx −1. (6.91)

As a result, the representation of the predicted optimal conversion price at (n+1)thtime step, Sc

(n+1)( j), should be

Sc(n+1)

( j) =3Z ·F(n+1)( j)+F [4V (n)

Nx−1, j −V (n)Nx−2, j]

(3−2∆x)CR −δx(4V (n)

Nx−1, j−V (n)Nx−2, j)

S(n)c ( j)

, (6.92)

where

F = I+∆τ[a jδxx +b jδrr + c jδxr +(d j −1

∆τ)δx + e jδr]− r j∆τ. (6.93)

Here, all the derivatives are again replaced by the corresponding δ∗. Hence, the


predicted convertible bond price at x = 0, V (n+1)Nx, j , is

V (n+1)Nx, j =CR · Sc

(n+1)( j)−Z ·F(n+1)( j). (6.94)

By now, the numerical scheme for the prediction step, where the optimal conver-sion price is predicted via Equation (6.93), and the bond price at x = 0 is predictedthrough Equation (6.94), have been obtained, and the next step is to correct thevalues of S(n+1)

c and V (n+1)Nx

by using the obtained V (n+1)Nx−2 and V (n+1)

Nx−1 at the predictionstep using the ADI technique.

We are now again presenting the numerical scheme for the correction step directly,while omitting the details for the derivation. In order to apply the ADI method, theFDE also needs to be reformulated as

(I−ϕA1)(I−ϕA2)V (n+1) = [I+A0 +(1−ϕ)A1 +A2]V (n)− (I−ϕA1)ϕA2V (n), (6.95)

where the definitions of all operators, A0, A1 and A2, are left in Appendix D.4. Forthe adopted D-R method, two steps should be taken into consideration. Firstly,we should fix the r direction, and a intermediate value, Y , satisfies the followingequation

(I−ϕA1)Y = [I+A0 +(1−ϕ)A1 +A2]V (n), (6.96)

should be determined. Simplifying this equation leads to

AYj = Pj +Bx j, (6.97)

with the definitions of A, Yj, Pj and Bx j left in Appendix D.5, and solving the setof equations here will give the value of Y . Once the value of Y is known, the valueof V (n+1) can be obtained from the following equation

(I−ϕA2)V (n+1)+ϕA2V (n) = Y, (6.98)

with the x direction being fixed. To deal with the above equation, it is again trans-formed into a set of equations

CV (n+1)i = Qi +Bri, (6.99)

with the details of C, V (n+1)i , Qi and Bri left in Appendix D.6. Of course, it is also

easy to show that solving Equation (6.97) and Equation (6.99) means solving theinitial Equation (6.95). In order to numerically implement our scheme, we need tomake it clear how to calculate the boundary value of the intermediate value Y , which


is presented as

Y0 = 0, (6.100)YNx = (I−ϕA2)V

(n+1)Nx

+ϕA2V (n)Nx

= (I−ϕA2)(CR · Sc(n+1)−Z ·F(n+1))+ϕA2V (n)

Nx, (6.101)

with the use of the predicted value of Sc(n+1). With the numerical scheme being

established for the PDE system governing the bond price under the stochastic in-terest rate model, we are now ready to conduct numerical experiments, the detailsof which are presented in the following subsection.

6.3.2 Numerical examples

In this subsection, the numerical examples are presented with the same values ofthe corresponding parameters used in the last section. The only exception is thatthe constant volatility, σ , is assumed to be 0.4 in this section. It should be pointedout that in this section, we will do not check the accuracy of our method, sincethe algorithm of the stochastic interest rate model is almost same as that of thestochastic volatility model, which has been confirmed as accurate in the last section.


Figure 6.8 displays that the value of the optimal conversion boundary with respect


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time to expiration

8

8.5

9

9.5

10

10.5

11

11.5

12

Va

lue

of

the

bo

un

da

ry

r=0.2

r=0.5

r=0.8


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r

7.5

8

8.5

9

9.5

10

10.5

11

11.5

12

12.5

Va

lue

of

the

bo

un

da

ry

=0.2

=0.5

=0.8


to the time to expiry and the interest rate. To clearly demonstrate the properties ofthe optimal conversion boundary, two figures are presented by fixing one direction.


Firstly, a similar phenomenon as shown in the case of stochastic volatility can beobserved in Figure 6.9 that the optimal convertible price is not an increasing functionof the time to expiry; the value of the optimal conversion boundary increases withthe time to expiry initially, before it decreases. The main explanation for this is alsothe fact that the optimal conversion boundary of a perpetual CB is zero, as discussedin the case of stochastic volatility. When we look at Figure 6.10, we can find thatthe value of the optimal conversion boundary is actually a decreasing function ofthe interest rate, no matter what the lifetime of the bond is. This is also financiallyreasonable since the higher the interest rate, the lower the present value of the facevalue will be, implying that the holder will choose to convert the bond at a smallerunderlying price.

154

1

6

0.8 10

8

S

0.6

r

Bo

nd

price

10

50.4

12

0.2

14

00


In Figure 6.11, the effects of the interest rate and the underlying price on the bondprice with a certain time to expiry are demonstrated, and we again only considerthe case when the CB has not been converted. When the interest rate is takeninto consideration, it is not difficult to find that the price of the CB is a monotonicdecreasing function with respect to the interest rate, which is reasonable since ahigher value of the interest rate means that it is more incentive for the holder toleave their money in a risk-free environment than buying a risky bond, leading toa lower bond value. When we turn to the underlying price, the effect of which isdisplayed in Figure 6.12, it can be observed that the price of the convertible bond


0 2 4 6 8 10 12 14 16 18 20

S

0

2

4

6

8

10

12

14

16

18

20

Va

lue

of

the

bo

un

da

ry

r=0.2

r=0.5

r=0.8


is actually an increasing function of the underlying price, which can be understoodwith a similar reason as provided for the case of stochastic volatility.

6.4 ConclusionIn this paper, the pricing problems of convertible bonds with a stochastic volatilitymodel and a stochastic interest rate model are considered, respectively. An efficientnumerical scheme, the predictor-corrector scheme, is established for these two cases.Being able to provide the entire optimal conversion boundary as part of the solutionprocedure, this new approach requires no embedded iterations at all. Finally, nu-merical experiments are also carried out to show the reliability of our approach, anddifferent properties of the convertible bond price as well as the optimal convertibleboundary are investigated.

Chapter 7

Summary and Conclusion

In this thesis, we consider the pricing of various types of American-style convertiblebonds under different models, and both semi-analytical and numerical approachesare applied. In particular, integral equation approaches are applied to evaluateputtable convertible bonds, callable-puttable convertible bonds and resettable con-vertible bonds under the B-S model. We have also discussed the pricing problemof vanilla convertible bonds when stochastic volatility or stochastic interest rate isincorporated into the B-S model, utilizing a predictor-corrector scheme.

Firstly, two integral equation formulations for the puttable convertible bond pricesunder the B-S model are presented in Chapter 3. The bond prices as well as optimalconversion and put boundaries can be numerically derived by solving the obtainedintegral equations. This approach is shown to be superior to the binomial treemethod, as far as the accuracy and efficiency is concerned. Numerical results alsoconfirm that the price of a puttable convertible bond is higher than that of thecorresponding vanilla one, which is a result of the holder being entitled to the rightto sell the bond back to the issuer, potentially protecting the benefit of the holder.

Chapter 4 moves a step further to price callable-puttable convertible bonds, whichcombine the call and put feature together, under the B-S model. Being different fromvanilla convertible bonds, three different cases, depending on the parameters of thetarget contract, are discussed, and the corresponding PDE systems for these casesare established. By using the incomplete Fourier transform method and Green’sfunction, the integral equation formulations for the target bond prices correspondingto the three cases are all presented, and the details of their numerical implementationare further discussed. The newly derived formulations are shown to be very accurate,while they are much more efficient than the binomial tree method.

In Chapter 5, the reset feature is embedded into vanilla convertible bonds to for-mulate resettable convertible bonds. To value these contracts under the B-S model,a closed PDE system is established, and applying the incomplete Fourier transformleads to an integral equation formulation for the bond price. This new formulation

143

CHAPTER 7. SUMMARY AND CONCLUSION 144

involves an unknown optimal conversion boundary, which is solved after an appro-priate numerical scheme is designed. It is also shown through a rigorous theoreticalproof that the price of a resettable convertible bond is not always a monotonically in-creasing function of the underlying asset price, and this quite amazing phenomenonis further illustrated through some numerical examples.

Lastly, stochastic volatility and interest rate are respectively introduced into theB-S model for the valuation of vanilla convertible bonds in Chapter 6. We establisha new predictor-corrector scheme, which involves no embedded iterations, so thatthe entire optimal conversion boundary can be simultaneously determined, togetherwith the bond prices. The reliability of this approach is tested through the com-parison with different existing approaches, and it is also shown that our approach isfar more efficient than the Monte-Carlo method. Interestingly, it is demonstratedthrough numerical experiments that the optimal conversion boundary is not alwaysan increasing function of the time to expiry; rather it will show a downward trendwith respect to the time to expiry when the time to expiry reaches a critical value.

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Appendix A

Appendix for Chapter 3

A.1 Appendix A.1In this appendix, we will present the detail of the computation process from System(3.7) to Equation (3.12). Under the definition of (3.8), we can transform System(3.7) to System (3.9), and here are several important processes. Since the incompleteFourier transform operator is linear, we can obtain the following equation:

−F∂U∂τ

+ 12

σ2F∂ 2U∂x2 +(r−D0 −

12)F∂U

∂x− rFU= 0. (A.1)

Now, we compute every term of Equation (A.1):

F∂U∂τ

=∫ ln(Sc(τ))

−∞

∂U∂τ

(x,τ)eiωxdx

=∂

∂τ[∫ ln(Sc(τ))

−∞U(x,τ)eiωxdx]− S′c(τ)

Sc(τ)U(ln(Sc(τ)),τ)eiω ln(Sc(τ))

=∂U∂τ

(ω,τ)− S′c(τ)Sc(τ)

(nSc(τ)−Ze−rτ)eiω ln(Sc(τ)), (A.2)

F∂U∂x

=∫ ln(Sc(τ))

−∞

∂U∂x

(x,τ)eiωxdx

=∫ ln(Sc(τ))

−∞eiωxdU(x,τ)

= eiωxU(x,τ)|ln(Sc(τ))−∞ − iω

∫ ln(Sc(τ))

−∞eiωxU(x,τ)dx

= eiω ln(Sc(τ))[nSc(τ)−Ze−rτ ]− iωU(ω,τ), (A.3)

F∂ 2U∂x2 =

∫ ln(Sc(τ))

−∞

∂ 2U∂x2 (x,τ)eiωxdx

=∫ ln(Sc(τ))

−∞eiωxd

∂U∂x

(x,ω)

= eiωx ∂U∂x

(x,τ)|ln(Sc(τ))−∞ − iω

∫ ln(Sc(τ))

−∞

∂U∂x

(x,τ)eiωxdx

154

APPENDIX A. APPENDIX FOR CHAPTER 3 155

= nSc(τ)eiω ln(Sc(τ))− iω[eiω ln(Sc(τ))(nSc(τ)−Ze−rτ)− iωU(ω,τ)]

= (1− iω)nSc(τ)eiω ln(Sc(τ))+ iωZe−rτeiω ln(Sc(τ))−ω2U(ω,τ). (A.4)

Apply these three results to Equation (A.1), we can obtain the following equation:

∂U∂τ

(ω,τ)+ [12

σ2ω2 +(r−D0 −12

σ2)iω + r]U(ω,τ)

= (nSc(τ)−Ze−rτ)eiω ln(Sc(τ))[S′c(τ)Sc(τ)

+(r−D0 −12

σ2)− 12

σ2iω]

+12

σ2eiω ln(Sc(τ))nSc(τ). (A.5)

Therefore, System (3.9) can be derived directly. Using the technique of the solutionof ODE system, we can write the solution of System (3.9)

U(w,τ) = U(ω,0)e−B(ω)τ +∫ τ

0f (ω,ξ )e−B(ω)(τ−ξ )dξ . (A.6)

Now, the integral equation formulation in the Fourier space has been presented.To obtain the integral equation formulation in the original space, the Fourier Inver-sion transform should be applied to Equation (A.6). And then we obtain

U(x,τ) =1

2π

∫ ∞

−∞e−iωxU(ω,τ)dω

=1

2π

∫ ∞

−∞e−iωxU(ω,0)e−B(ω)τdω

+1

2π

∫ ∞

−∞e−iωx

∫ τ

0f (ω,ξ )e−B(ω)(τ−ξ )dξ dω

, I1 + I2. (A.7)

Now, we compute I1 and I2 respectively, and I1 first.

I1 =1

2π

∫ ∞

−∞e−iωxU(ω,0)e−B(ω)τdω

=1

2π

∫ ∞

−∞e−iωxU(ω,0) · e−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r]τdω

, 12π

∫ ∞

−∞e−iωxU(ω,0) ·G(ω,τ)dω, (A.8)

where G(ω,τ) = e−[ 12 σ2ω2+(r−D0− 1

2 σ2)iω+r]τ .In order to use the Convolution theorem [11] to obtain the value of I1, we should

obtain the Fourier Inversion transform of G(ω,τ) first. Define

g(x,τ) , F−1G(ω,τ)= 12π

∫ ∞

−∞e−iωxG(ω,τ)dω


=1

2π

∫ ∞

−∞e−iωxe−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r]τdω

=e−rτ

2π

∫ ∞

−∞e−

12 σ2τω2−[(r−D0− 1

2 σ2)τ+x]iωdω

=e−rτ

2πe−

[(r−D0−12 σ2)τ+x]2

2σ2τ

∫ ∞

−∞e−

12 σ2τ[ω+i

(r−D0−12 )τ+x

σ2τ]2dω

=e−rτ√

2πσe−

[(r−D0−12 σ2)τ+x]2

2σ2τ . (A.9)

Therefore, I1 can be expressed

I1 =1

2π

∫ ∞

−∞e−iωxU(ω,0) ·G(ω,τ)dω

= U(x,0)∗g(x,τ)

=∫ ln(Sc(0))

−∞maxneu −Z,0 e−rτ

√2πτσ

e−[(r−D0−

12 σ2)τ+x−u]2

2σ2τ du. (A.10)

Then, we compute I2

I2 =1

2π

∫ ∞

−∞e−iωx

∫ τ

0f (ω,ξ )e−B(ω)(τ−ξ )dξ dω

=∫ τ

0

12π

∫ ∞

−∞e−iωx f (ω,ξ )e−B(ω)(τ−ξ )dωdξ

=∫ τ

0

12π

∫ ∞

−∞e−iωx−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r](τ−ξ )eiω ln(Sc(ξ ))

·(nSc(ξ )−Ze−rξ )[S′c(ξ )Sc(ξ )

+(r−D0 −12

σ2)− 12

σ2iω]+12

σ2nSc(ξ )dωdξ

,∫ τ

0

12π

∫ ∞

−∞e−iωx−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r](τ−ξ )eiω ln(Sc(ξ )) · f1(ξ )− f2(ξ )ωdωdξ ,

(A.11)

where

f1(ξ ) = (nSc(ξ )−Ze−rξ )[S′c(ξ )Sc(ξ )

+(r−D0 −12

σ2)]+12

σ2nSc(ξ ),

f2(ξ ) =12

σ2i(nSc(ξ )−Ze−rξ ).

Now, we compute I2:

I2 =∫ τ

0

e−r(τ−ξ )

2π

∫ ∞

−∞e−

12 σ2(τ−ξ )ω2−[(r−D0− 1

2 σ2)(τ−ξ )+x−ln(Sc(ξ ))]iω · f1(ξ )− f2(ξ )ωdωdξ

=∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )σ

· e−[(r−D0−

12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

nσ2Sc(ξ )dξ , (A.12)


where the following equation is used∫ ∞

−∞e−pω2−qωωndω = (−1)n

√πp

∂ n

∂qn eq24p .

Combining I1 and I2, U(x,τ) can be expressed as

U(x,τ) =∫ ln(Sc(0))

−∞maxneu −Z,0 e−rτ

√2πτσ

e−[(r−D0−

12 σ2)τ+x−u]2

2σ2τ du

+∫ τ

0


· e−[(r−D0−

12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

nσ2Sc(ξ )dξ , (A.13)

which gives Equation (3.12).

A.2 Appendix A.2In this appendix, we will present the detail of the computation process from System(3.5) to Equation (3.18). First, we apply the incomplete Fourier transform (3.14) toSystem (3.5). Since incomplete Fourier transform operator is linear, we can obtainthe following equation:

−F∂v2

∂τ+ 1

2σ2F∂ 2v2

∂x2 +(r−D0 −12

σ2)F∂v2

∂x− rFv2= 0. (A.14)

Now, we compute every term of Equation (A.14):

F∂v2

∂τ(x,τ) =

∫ ln(Sc(τ))

ln(Sp(τ))

∂v2

∂τ(x,τ)eiωxdx

=∂


ln(Sp(τ))v2(x,τ)eiωxdx]− S′c(τ)

Sc(τ)nSc(τ)eiω ln(Sc(τ))+

S′p(τ)Sp(τ)

Meiω ln(Sp(τ))

=∂ v2

∂τ(ω,τ)− S′c(τ)

Sc(τ)nSc(τ)eiω ln(Sc(τ))+

S′p(τ)Sp(τ)

Meiω ln(Sp(τ)), (A.15)

F∂v2

∂x(x,τ) =

∫ ln(Sc(τ))

ln(Sp(τ))

∂v2

∂x(x,τ)eiωxdx

=∫ ln(Sc(τ))

ln(Sp(τ))eiωxdv2(x,τ)

=∂v2

∂x(x,τ)eiωx|ln(Sc(τ))

ln(Sp(τ))− iω∫ ln(Sc(τ))

ln(Sp(τ))v2(x,τ)eiωxdx

= eiω ln(Sc(τ))nSc(τ)− eiω ln(Sp(τ))M− iω v2(ω,τ), (A.16)

F∂ 2v2

∂x2 (x,τ) =∫ ln(Sc(τ))

ln(Sp(τ))

∂ 2v2

∂x2 (x,τ)eiωxdx


=∫ ln(Sc(τ))

ln(Sp(τ))eiωxd

∂v2

∂x(x,τ)

= eiωx ∂v2

∂x(x,τ)|ln(Sc(τ))

ln(Sp(τ))− iω∫ ln(Sc(τ))

ln(Sp(τ))eiωx ∂v2

∂x(x,τ)dx

= eiω ln(Sc(τ))nSc(τ)− iω[eiω ln(Sc(τ))nSc(τ)− eiω ln(Sp(τ))M− iω v2(ω,τ)]

= eiω ln(Sc(τ))[nSc(τ)− iωnSc(τ)+ iωM]−ω2v2(ω,τ). (A.17)

Substituting these three expressions into Equation (A.14), the following equationcan be obtained:

∂ v2

∂τ(ω,τ)+ [

12

σ2ω2 +(r−D0 −12

σ2)iω + r]v2(ω,τ)

= nSc(τ)eiω ln(Sc(τ))[S′c(τ)Sc(τ)

− 12

σ2iω + r−D0]−Meiω ln(Sp(τ))[S′p(τ)Sp(τ)

− 12

σ2iω + r−D0 −12

σ2].

(A.18)

Therefore, the ODE System (3.5) is derived directly. Using the technique of thesolution of the ODE system, we obtain

v2(ω,τ) = v1(ω,τM)e−B(ω)(τ−τM)+∫ τ−τM

0f (ω,τM +ξ )e−B(ω)(τ−τM−ξ )dξ ,

(A.19)

which gives the integral equation formulation in the Fourier space. We apply theFourier inversion transform to the last equation to obtain the integral equationformulation v2(x,τ) in the original space,

v2(x,τ) =1

2π

∫ ∞

−∞v2(ω,τ)e−iωxdω

=1

2π

∫ ∞

−∞e−iωxv1(ω,τM)e−B(ω)(τ−τM)dω

+1

2π

∫ ∞

−∞e−iωx

∫ τ−τM

0f (ω,τM +ξ )e−B(ω)(τ−τM−ξ )dξ dω

, I1 + I2. (A.20)

We compute I1 first.

I1 =1

2π

∫ ∞

−∞e−iωxv1(ω,τM)e−B(ω)(τ−τM)dω

=1

2π

∫ ∞

−∞e−iωxv1(ω,τM)e−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r](τ−τM)dω

, 12π

∫ ∞

−∞e−iωxv1(ω,τM) ·G(ω,τ)dω, (A.21)


where G(ω,τ) = e−[ 12 σ2ω2+(r−D0− 1

2 σ2)iω+r](τ−τM). To use the Convolution theoremhere, we need to obtain the inverse Fourier transform of G(ω,τ). Define

g(x,τ) , G(ω,τ)

= F−1G(ω,τ)

=1

2π

∫ ∞

−∞e−iωxe−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r](τ−τM)dω

=e−r(τ−τM)

2π· e

− [(r−D0−12 σ2)(τ−τM)+x]2

2σ2(τ−τM)

∫ ∞

−∞e− 1

2 σ2(τ−τM)[ω+i(r−D0−

12 σ2)(τ−τM)+x

σ2(τ−τM)]2

dω

=e−r(τ−τM)√

2π(τ − τM)σ· e

− [(r−D0−12 σ2)(τ−τM)+x]2

2σ2(τ−τM) . (A.22)

Then

I1 =1

2π

∫ ∞

−∞e−iωxv1(ω,τM) ·G(ω,τ)dω

= v1(x,τ)∗g(x,τ)

=∫ ln(Sc(τM))

ln(Sp(τM))v1(u,τM)

e−r(τ−τM)√2π(τ − τM)σ

e− [(r−D0−

12 σ2)(τ−τM)+x−u]2

2σ2(τ−τM) du. (A.23)

Now, we compute I2

I2 =1

2π

∫ ∞

−∞e−iωx

∫ τ−τM

0f (ω,τM +ξ )e−B(ω)(τ−τM−ξ )dξ dω

=∫ τ−τM

0

12π

∫ ∞

−∞e−iωx f (ω,τM +ξ )e−B(ω)(τ−τM−ξ )dωdξ

=∫ τ−τM

0

12π

∫ ∞

−∞e−iωx−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r](τ−τM−ξ )

·nSc(τM +ξ )eiω ln(Sc(τM+ξ ))[S′c(τM +ξ )Sc(τM +ξ )

− 12

σ2iω + r−D0]

−Meiω ln(Sp(τM+ξ ))[S′p(τM +ξ )Sp(τM +ξ )

− 12

σ2iω + r−D0 −12

σ2]dωdξ

,∫ τ−τM

0


2π∫ ∞

−∞e−

12 σ2(τ−τM−ξ )ω2−[(r−D0− 1

2 σ2)(τ−τM−ξ )+x−ln(Sc(τM+ξ ))]iω · f1(ξ )− f2(ξ )ωdωdξ

−∫ τ−τM

0


2π∫ ∞

−∞e−

12 σ2(τ−τM−ξ )ω2−[(r−D0− 1

2 σ2)(τ−τM−ξ )+x−ln(Sp(τM+ξ ))]iω · f3(ξ )− f4(ξ )ωdωdξ ,

(A.24)


where

f1(ξ ) =S′c(τM +ξ )Sc(τM +ξ )

+ r−D0, f2(ξ ) =12

σ2i,

f3(ξ ) =S′p(τM +ξ )Sp(τM +ξ )

+ r−D0 −12

σ2, f4(ξ ) =12

σ2i.

So, I2 can be computed as

I2 =∫ τ−τM

0


2π∫ ∞

−∞e−

12 σ2(τ−τM−ξ )ω2−[(r−D0− 1

2 σ2)(τ−τM−ξ )+x−ln(Sc(τM+ξ ))]iω · f1(ξ )− f2(ξ )ωdωdξ

−∫ τ−τM

0


2π∫ ∞

−∞e−

12 σ2(τ−τM−ξ )ω2−[(r−D0− 1

2 σ2)(τ−τM−ξ )+x−ln(Sp(τM+ξ ))]iω · f3(ξ )− f4(ξ )ωdωdξ

=∫ τ−τM

0

nSc(τM +ξ )e−r(τ−τM−ξ )√2π(τ − τM −ξ )σ

· e− [(r−D0−

12 σ2)(τ−τM−ξ )+x−ln(Sc(τM+ξ ))]2


·S′c(τM +ξ )Sc(τM +ξ )

+12[r−D0 +

12

σ2 +ln(Sc(τM +ξ ))− x


−∫ τ−τM

0

Me−r(τ−τM−ξ )√2π(τ − τM −ξ )σ

· e− [(r−D0−

12 σ2)(τ−τM−ξ )+x−ln(Sp(τM+ξ ))]2



+12[r−D0 −

12

σ2 +ln(Sp(τM +ξ ))− x

τ − τM −ξ]dξ ,

(A.25)

where the follow equation is used:∫ ∞


√πp

∂ n

∂qn eq24p .

Combining I1 and I2, the expression of v2(x,τ) is displayed as

v2(x,τ) =∫ ln(Sc(τM))

ln(Sp(τM))V1(u,τM)

e−r(τ−τM)√2π(τ − τM)σ

e− [(r−D0−

12 σ2)(τ−τM)+x−u]2

2σ2(τ−τM) du

+∫ τ−τM

0

nSc(τM +ξ )e−r(τ−τM−ξ )√2π(τ − τM −ξ )σ

· e− [(r−D0−

12 σ2)(τ−τM−ξ )+x−ln(Sc(τM+ξ ))]2


·S′c(τM +ξ )Sc(τM +ξ )

+12[r−D0 +

12

σ2 +ln(Sc(τM +ξ ))− x


−∫ τ−τM

0

Me−r(τ−τM−ξ )√2π(τ − τM −ξ )σ

· e− [(r−D0−

12 σ2)(τ−τM−ξ )+x−ln(Sp(τM+ξ ))]2




+12[r−D0 −

12


τ − τM −ξ]dξ . (A.26)

A.3 Appendix A.3Now, we present the detail of computing Equation (3.22) from Equation (3.13).First, we rewrite Equation (3.13) as follows

V1(S,τ) =∫ τ

0


e− [(r−D0−

12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(S)]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12


(A.27)

It should be noted that the first term of Equation (3.13) is missing. Actually, it isalways zero, since Sc(0) =

Zn. Now, we define

h(S,ξ ) ,[(r−D0 − 1

2σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)]2

2σ2(τ −ξ )

=1

2(τ −ξ )[(r−D0 − 1

2σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)σ

]2

=1

2(τ −ξ )[(r−D0 − 1

2σ2)τ + ln(S)σ

−(r−D0 − 1

2σ2)ξ + ln(Sc(ξ ))σ

]2

, 12(τ −ξ )

[y−P(ξ )]2, (A.28)

where y =(r−D0 − 1

2σ2)τ + ln(S)σ

and P(ξ ) =(r−D0 − 1


. It can

be found that P′(ξ ) =1σ[S′c(ξ )Sc(ξ )

+ r−D0−12

σ2]. Therefore, we apply it to Equation

(A.27) and obtain

V1(S,τ) =∫ τ

0


e− [(r−D0−

12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(S)]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

nσ2Sc(ξ )dξ +Ze−rτ

=∫ τ

0


e−[y−P(ξ )]2

2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+ r−D0 −

12

σ2 − 12(r−D0 −

12

σ2 − ln(Sc(ξ ))− ln(S)τ −ξ

)]

+12

nσ2Sc(ξ )dξ +Ze−rτ


=∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[P′(ξ )− 12σ

(r−D0 −12


)]+12

nσSc(ξ )dξ +Ze−rτ

=∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[P′(ξ )− 12(τ −ξ )

((r−D0 − 1

2σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)σ

)]

+12


=∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · (nSc(ξ )−Ze−rξ )[P′(ξ )− y−P(ξ )2(τ −ξ )

]

+12


=∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·nSc(ξ )[σ2+P′(ξ )− y−P(ξ )

2(τ −ξ )]dξ

−∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·Ze−rξ [P′(ξ )− y−P(ξ )2(τ −ξ )

]dξ +Ze−rτ

, R1(S,τ)−ZR2(S,τ)+Ze−rτ , (A.29)

where

R1(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·nSc(ξ )[σ2+P′(ξ )− y−P(ξ )

2(τ −ξ )]dξ ,

R2(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · e−rξ [P′(ξ )− y−P(ξ )2(τ −ξ )

]dξ .

Now, we compute R1(S,τ) and R2(S,τ), respectively.

R1(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·nSc(ξ )[σ2+P′(ξ )− y−P(ξ )

2(τ −ξ )]dξ

=∫ τ

0

e−r(τ−ξ )√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ ) e(y−P(ξ ))σ+

σ2(τ−ξ )2 ·nSc(ξ )[

σ2 +P′(ξ )− y−P(ξ )

2(τ−ξ )√τ −ξ

]dξ

=∫ τ

0

e−r(τ−ξ )+(y−P(ξ ))σ+σ2(τ−ξ )

2√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )σ(τ −ξ )+2P′(ξ )(τ −ξ )− y+P(ξ )

2(τ −ξ )√

τ −ξdξ

=∫ τ

0

e−r(τ−ξ )+(r−D0− 12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(S)+σ2(τ−ξ )

2√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )P(ξ )− y−σ(τ −ξ )+2P′(ξ )(τ −ξ )−2σ(τ −ξ )

2(τ −ξ )√

τ −ξdξ


=∫ τ

0

e−D0(τ−ξ )−ln(Sc(ξ ))+ln(S)√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )[P(ξ )− y−σ(τ −ξ )]+ [P′(ξ )−σ ] ·2(τ −ξ )

2(√

τ −ξ )2√

τ −ξdξ

=∫ τ

0

SSc(ξ )

· e−D0(τ−ξ )√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )1

2√

τ−ξ[P(ξ )− y−σ(τ −ξ )]+(P′(ξ )+σ)

√τ −ξ

(√

τ −ξ )2dξ

=∫ τ

0

nSe−D0(τ−ξ )√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ ) · ∂

∂ξ[P(ξ )− y−σ(τ −ξ )√

τ −ξ]dξ

=∫ τ

0−nSe−D0(τ−ξ ) ∂

∂ξN (

y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ . (A.30)

R2(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · e−rξ [P′(ξ )− y−P(ξ )2(τ −ξ )

]dξ

=∫ τ

0

e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · [2P′(ξ )(τ −ξ )− y+P(ξ )

2(τ −ξ )√

τ −ξ]dξ

=∫ τ

0

e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · [P

′(ξ ) ·2(τ −ξ )+(P(ξ )− y)

2(τ −ξ )√

τ −ξ]dξ

=∫ τ

0

e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · [

P′(ξ )√

τ −ξ + P(ξ )−y2√

τ−ξ

(√

τ −ξ )2]dξ

=∫ τ

0− e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · ∂

∂ξ(y−P(ξ )√

τ −ξ)dξ

=∫ τ

0−e−rτ ∂

∂ξN (

y−P(ξ )√τ −ξ

)dξ . (A.31)

Thus, we substitute the expression of R1(S,τ) and R2(S,τ) into Equation (A.29).Equation (3.22) can be obtained

V1(S,τ) = R1(S,τ)−ZR2(S,τ)+Ze−rτ

=∫ τ

0−nSe−D0(τ−ξ ) ∂

∂ξN (

y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ

+Z∫ τ

0e−rτ ∂

∂ξN (


)dξ +Ze−rτ

=∫ τ

0−nSe−D0(τ−ξ )dN (

y−P(ξ )+σ(τ −ξ )√τ −ξ

)

+Z∫ τ

0e−rτdN (


)+Ze−rτ

= −nSe−D0(τ−ξ )N (y−P(ξ )+σ(τ −ξ )√

τ −ξ)|ξ=τ


+nSe−D0(τ−ξ )N (y−P(ξ )+σ(τ −ξ )√

τ −ξ)|ξ=0

+∫ τ

0nD0Se−D0(τ−ξ )N (

y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ

+Ze−rτN (y−P(ξ )√

τ −ξ)|ξ=τ −Ze−rτN (


)|ξ=0 +Ze−rτ

= −nSN (ln(S)− ln(Sc(ξ ))

σ√

τ −ξ)|ξ=τ

+nSe−D0τN ((r−D0 +

12σ2)τ − lnSc(0)+ ln(S)

σ√

τ)

+∫ τ


y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ

+Ze−rτN (ln(S)− ln(Sc(ξ ))

σ√

τ −ξ)|ξ=τ

−Ze−rτN ((r−D0 − 1

2σ2)τ − ln(Sc(0))+ ln(S)σ√

τ)+Ze−rτ

= nSe−D0τN ((r−D0 +

12σ2)τ − lnSc(0)+ ln(S)

σ√

τ)

−Ze−rτN ((r−D0 − 1

2σ2)τ − ln(Sc(0))+ ln(S)σ√

τ)

+∫ τ

0nD0Se−D0(τ−ξ )

·N ((r−D0 +

12σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)

σ√

τ −ξ)dξ

−1S=Sc(τ)(nS−Ze−rτ)+Ze−rτ , (A.32)

where

1S=Sc(τ) =

12 S = Sc(τ),0 S < Sc(τ).

(A.33)

A.4 Appendix A.4Now, we give the detail of computing Equation (3.23) from Equation (3.18), usingthe same method mentioned in Appendix A.3. First, we rewrite Equation (3.18) asfollows



e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−




·S′c(τM +ξ )

Sc(τM +ξ )+

12[r−D0 +

12



−∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−



·S′p(τM +ξ )

Sp(τM +ξ )+

12[r−D0 −

12



,∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+R1(S,τ)−R2(S,τ), (A.34)

where

R1(S,τ) =∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−



·S′c(τM +ξ )

Sc(τM +ξ )+

12[r−D0 +

12


τ − τM −ξ]dξ ,

R2(S,τ) =∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−



·S′p(τM +ξ )

Sp(τM +ξ )+

12[r−D0 −

12


τ − τM −ξ]dξ .

Now, we compute R1(S,τ) first. Define

h1(S,ξ ) ,[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Sc(τM +ξ ))]2

2σ2(τ − τM −ξ )

=1

2(τ − τM −ξ )[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Sc(τM +ξ ))σ

]2

=1

2(τ − τM −ξ )[(r−D0 − 1

2σ2)(τ − τM)+ ln(S)σ

−(r−D0 − 1

2σ2)ξ + ln(Sc(τM +ξ ))σ

]2

, 12(τ − τM −ξ )

[y1 −P1(ξ )]2, (A.35)

where y1 =(r−D0 − 1


and P1(ξ )=(r−D0 − 1

2σ2)ξ + ln(Sc(τM +ξ ))σ

.

It should be noted that P′1(ξ ) =

S′c(τM+ξ )Sc(τM+ξ ) + r−D0 − 1

2σ2

σ. Therefore, we can obtain

R1(S,ξ ) =∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−



·S′c(τM +ξ )

Sc(τM +ξ )+

12[r−D0 +

12




=∫ τ−τM

0


σ√

2π(τ − τM −ξ )e−

[y1−P1(ξ )]2

2(τ−τM−ξ )

·S′c(τM +ξ )

Sc(τM +ξ )+ r−D0 +

12

σ2 − 12[r−D0 +

12

σ2 − ln(Sc(τM +ξ ))− ln(S)τ − τM −ξ

]dξ

=∫ τ−τM

0


σ√

2π(τ − τM −ξ )e−

[y1−P1(ξ )]2

2(τ−τM−ξ )

·S′c(τM +ξ )

Sc(τM +ξ )+ r−D0 −

12

σ2 − 12[r−D0 −

12

σ2 − ln(Sc(τM +ξ ))− ln(S)τ − τM −ξ

]+σ2

2dξ

=∫ τ−τM

0

nSc(τM +ξ )e−r(τ−τM−ξ )√2π(τ − τM −ξ )

e−[y1−P1(ξ )]

2

2(τ−τM−ξ ) · [P′1(ξ )−

y1 −P1(ξ )2(τ − τM −ξ )

+σ2]dξ

=∫ τ−τM

0

nSc(τM +ξ )e−r(τ−τM−ξ )√

2πe−

[y1−P1(ξ )+σ(τ−τM−ξ )]22(τ−τM−ξ ) · eσ(y1−P1(ξ ))+

σ2(τ−τM−ξ )2

·2P′1(ξ )(τ − τM −ξ )− y1 +P1(ξ )+σ(τ − τM −ξ )

2(τ − τM −ξ )√

τ − τM −ξdξ

=∫ τ−τM

0−nSc(τM +ξ )e−r(τ−τM−ξ )+σ(y1−P1(ξ ))+

σ2(τ−τM−ξ )2

√2π

e−[y1−P1(ξ )+σ(τ−τM−ξ )]2

2(τ−τM−ξ )

· [y1 −P1(ξ )+σ(τ − τM −ξ )]− [P′1(ξ )+σ ] ·2(τ − τM −ξ )

2(τ − τM −ξ )√

τ − τM −ξdξ

=∫ τ−τM

0−nSc(τM +ξ )e−r(τ−τM−ξ )+(r−D0− 1

2 σ2)(τ−τM−ξ )+ln(S)−ln(Sc(τM+ξ ))+ σ2(τ−τM−ξ )2

√2π

·e−[y1−P1(ξ )+σ(τ−τM−ξ )]2

2(τ−τM−ξ ) · [y1 −P1(ξ )+σ(τ − τM −ξ )]− [P′1(ξ )+σ ] ·2(τ − τM −ξ )

2(τ − τM −ξ )√

τ − τM −ξdξ

=∫ τ−τM

0− S

Sc(τM +ξ )· nSc(τM +ξ )e−D0(τ−τM−ξ )

√2π

· e−[y1−P1(ξ )+σ(τ−τM−ξ )]2

2(τ−τM−ξ )

·1

2√

τ−τM−ξ[y1 −P1(ξ )+σ(τ − τM −ξ )]− [P′

1(ξ )+σ ] ·√

τ − τM −ξ

(√

τ − τM −ξ )2dξ

=∫ τ−τM

0−nSe−D0(τ−τM−ξ )

√2π

· e−[y1−P1(ξ )+σ(τ−τM−ξ )]2

2(τ−τM−ξ ) · ∂∂ξ

(y1 −P1(ξ )+σ(τ − τM −ξ )√


=∫ τ−τM

0−nSe−D0(τ−τM−ξ ) · ∂

∂ξN (

y1 −P1(ξ )+σ(τ − τM −ξ )√τ − τM −ξ

)dξ . (A.36)

Now, we compute R2(S,τ). Define

h2(S,ξ ) ,[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Sp(τM +ξ ))]2

2σ2(τ − τM −ξ )

=1

2(τ − τM −ξ )[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Sp(τM +ξ ))σ

]2

=1

2(τ − τM −ξ )[(r−D0 − 1


−(r−D0 − 1

2σ2)ξ + ln(Sp(τM +ξ ))σ

]2

, 12(τ − τM −ξ )

[y2 −P2(ξ )]2, (A.37)


where y2 =(r−D0 − 1


and P2(ξ )=(r−D0 − 1


.

It should be noted that P′2(ξ ) =

S′p(τM+ξ )Sp(τM+ξ ) + r−D0 − 1

2σ2

σ. Therefore, we can obtain

R2(S,ξ ) =∫ τ−τM

0


σ√

2π(τ − τM −ξ )e− [(r−D0−



·S′p(τM +ξ )

Sp(τM +ξ )+

12[r−D0 −

12



=∫ τ−τM

0


σ√

2π(τ − τM −ξ )e−

[y2−P2(ξ )]2

2(τ−τM−ξ )

·S′p(τM +ξ )

Sp(τM +ξ )+ r−D0 −

12

σ2 − 12[r−D0 −

12

σ2 −ln(Sp(τM +ξ ))− ln(S)


=∫ τ−τM

0

Me−r(τ−τM−ξ )√2π(τ − τM −ξ )

e−[y2−P2(ξ )]

2

2(τ−τM−ξ ) · [P′2(ξ )−

y2 −P2(ξ )2(τ − τM −ξ )

]dξ

=∫ τ−τM

0

Me−r(τ−τM−ξ )√

2πe−

[y2−P2(ξ )]2

2(τ−τM−ξ ) · 2P′

2(ξ )(τ − τM −ξ )− y2 +P2(ξ )2(τ − τM −ξ )

√τ − τM −ξ

dξ

=∫ τ−τM

0−Me−r(τ−τM−ξ )

√2π

e−[y2−P2(ξ )]

2

2(τ−τM−ξ ) · [y2 −P2(ξ )] 1

2√

τ−τM−ξ−P′

2(ξ )√

τ − τM −ξ√τ − τM −ξ

2 dξ

=∫ τ−τM

0−Me−r(τ−τM−ξ )

√2π

e−[y2−P2(ξ )]

2

2(τ−τM−ξ ) · ∂∂ξ

(y2 −P2(ξ )√τ − τM −ξ

)dξ

=∫ τ−τM

0−Me−r(τ−τM−ξ ) · ∂

∂ξN (

y2 −P2(ξ )√τ − τM −ξ

)dξ . (A.38)

Now, we substitute the expression of R1(S,τ) and R2(S,τ) into V2(S,τ), we obtain



e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+R1(S,τ)−R2(S,τ)

=∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

−∫ τ−τM

0nSe−D0(τ−τM−ξ ) · ∂

∂ξN (

y1 −P1(ξ )+σ(τ − τM −ξ )√τ − τM −ξ

)dξ

+∫ τ−τM

0Me−r(τ−τM−ξ ) · ∂

∂ξN (

y2 −P2(ξ )√τ − τM −ξ

)dξ

=∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

−∫ τ−τM

0nSe−D0(τ−τM−ξ ) ·dN (

y1 −P1(ξ )+σ(τ − τM −ξ )√τ − τM −ξ

)


+∫ τ−τM

0Me−r(τ−τM−ξ ) ·dN (

y2 −P2(ξ )√τ − τM −ξ

)

=∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

−nSe−D0(τ−τM−ξ ) ·N (y1 −P1(ξ )+σ(τ − τM −ξ )√

τ − τM −ξ)|ξ=τ−τM

+nSe−D0(τ−τM−ξ ) ·N (y1 −P1(ξ )+σ(τ − τM −ξ )√

τ − τM −ξ)|ξ=0

+∫ τ−τM

0nD0Se−D0(τ−τM−ξ )N (

y1 −P1(ξ )+σ(τ − τM −ξ )√τ − τM −ξ

)dξ

+Me−r(τ−τM−ξ ) ·N (y2 −P2(ξ )√τ − τM −ξ

)|ξ=τ−τM

−Me−r(τ−τM−ξ ) ·N (y2 −P2(ξ )√τ − τM −ξ

)|ξ=0

−∫ τ−τM

0rMe−r(τ−τM−ξ ) ·N (

y2 −P2(ξ )√τ − τM −ξ

)dξ

=∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

−nSN (ln(S)− ln(Sc(τM +ξ ))

σ√


+nSe−D0(τ−τM)N ((r−D0 +


σ√

τ − τM)

+∫ τ−τM


N ((r−D0 +


σ√


+MN (ln(S)− ln(Sp(τM +ξ ))

σ√


−Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM)

−∫ τ−τM


N ((r−D0 − 1



=∫ ln(Sc(τM))


e−r(τ−τM)

σ√

2π(τ − τM)e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+nSe−D0(τ−τM)N ((r−D0 +


σ√

τ − τM)

−nS1|S=Sc(τ)+∫ τ−τM



·N ((r−D0 +


σ√


−Me−r(τ−τM)N ((r−D0 − 1


σ√

τ − τM)

+M ·1S=Sp(τ)−∫ τ−τM


·N ((r−D0 − 1


τ − τM −ξ)dξ ,(A.39)

where

1S=Sc(τ) =

12 S = Sc(τ),0 S < Sc(τ),

(A.40)

1S=Sp(τ) =

12 S = Sp(τ),1 S > Sp(τ).

(A.41)

Appendix B


B.1 Appendix B.1In this appendix, we present the details of applying the incomplete Fourier transformto the PDE system (4.6). Firstly, it should be noted that the incomplete Fouriertransform operate is a linear transform. Thus, when we apply it to the PDE, thefollowing equation can be obtained:

−F∂U1

∂τ+ 1

2σ2F∂ 2U1

∂x2 +(r−D0 −12

σ2)F∂U1

∂x− rFU1= 0. (B.1)

Now, we compute every term of the Equation (B.1):

F∂U1

∂τ =

∫ ln(Sc(τ))

−∞

∂U1

∂τ(x,τ) · eiωxdx

=∂


−∞U1(x,τ) · eiωxdx]− S′c(τ)

Sc(τ)U1(ln(Sc(τ)),τ)eiω ln(Sc(τ))

=∂U1

∂τ(ω,τ)− S′c(τ)

Sc(τ)(nSc(τ)−Ze−rτ)eiω ln(Sc(τ)), (B.2)

F∂U1

∂x(x,τ) =

∫ ln(Sc(τ))

−∞

∂U1

∂x(x,τ) · eiωxdx

=∫ ln(Sc(τ))

−∞eiωxdU1(x,τ)

= U1(x,τ)eiωx|ln(Sc(τ))−∞ − iω

∫ ln(Sc(τ))

−∞U1(x,τ) · eiωxdx

= (nSc(τ)−Ze−rτ)eiω ln(Sc(τ))− iωU1(ω,τ), (B.3)

F∂ 2U1

∂x2 (x,τ) =∫ ln(Sc(τ))

−∞

∂ 2U1

∂x2 (x,τ) · eiωxdx

=∫ ln(Sc(τ))

−∞eiωxd

∂U1

∂x(x,τ)

=∂U1

∂x(x,τ)eiωx|ln(Sc(τ))

−∞ − iω∫ ln(Sc(τ))

−∞

∂U1


170

APPENDIX B. APPENDIX FOR CHAPTER 4 171

= nSc(τ)eiω ln(Sc(τ))− iω[(nSc(τ)−Ze−rτ)eiω ln(Sc(τ))− iωU1(ω,τ)]

= nSc(τ)eiω ln(Sc(τ))− iω(nSc(τ)−Ze−rτ)eiω ln(Sc(τ))−ω2U1(ω,τ)

= (1− iω)nSc(τ)eiω ln(Sc(τ))+ iωZe−rτeiω ln(Sc(τ))−ω2U1(ω,τ).

(B.4)

Substituting these three results into Equation (B.1), the following equation canbe obtained:

∂U1

∂τ(ω,τ)+ [

12

σ2ω2 +(r−D0 −12

σ2)iω + r]U1(ω,τ)

= (nSc(τ)−Ze−rτ)eiω ln(Sc(τ))[S′c(τ)Sc(τ)

+(r−D0 −12

σ2)− 12

σ2iω]+12

σ2eiω ln(Sc(τ))nSc(τ).

(B.5)

Then, System (4.8) is obtained.

B.2 Appendix B.2In this appendix, the Fourier inversion transform is considered. Firstly, we rewriteEquation (4.11) as follows

U1(ω,τ) = U1(ω,0) · e−B(ω)τ +∫ τ

0f (ω,ξ ) · e−B(ω)(τ−ξ )dξ . (B.6)

Then, the Fourier inversion transform should be applied to this equation, and thefollowing formulation can be obtained

U1(x,τ) =1

2π

∫ ∞

−∞[U1(ω,0) · e−B(ω)τ +

∫ τ

0f (ω,ξ ) · e−B(ω)(τ−ξ )dξ ] · e−iωxdω

=1

2π

∫ ∞

−∞U1(ω,0) · e−B(ω)τ · e−iωxdω +

12π

∫ ∞

−∞

∫ τ

0f (ω,ξ ) · e−B(ω)(τ−ξ ) · e−iωxdξ dω

, I1 + I2, (B.7)

where I1 =1

2π

∫ ∞

−∞U1(ω,0)·e−B(ω)τ ·e−iωxdω and I2 =

12π

∫ ∞

−∞

∫ τ

0f (ω,ξ )·e−B(ω)(τ−ξ ) ·

e−iωxdξ dω .Now, we calculate I1 and I2, respectively, and I1 first

I1 =1

2π

∫ ∞

−∞U1(ω,0) · e−B(ω)τ · e−iωxdω

=1

2π

∫ ∞

−∞U1(ω,0) · e−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r]τ · e−iωxdω

, 12π

∫ ∞

−∞U1(ω,0) · e−iωx ·G(ω,τ)dω, (B.8)


where G(ω,τ) = e−[ 12 σ2ω2+(r−D0− 1

2 σ2)iω+r]τ . To use the Convolution theorem, weshould apply the Fourier inversion transform to G(ω,τ) first.

g(x,τ) , F−1G(ω,τ)= 12π

∫ ∞

−∞e−iωx ·G(ω,τ)dω

=1

2π

∫ ∞

−∞e−iωx · e−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r]τdω

=e−rτ

2π

∫ ∞

−∞e−iωx− 1

2 σ2ω2τ−(r−D0− 12 σ2)iωτdω

=e−rτ

2π· e−

[(r−D0−12 σ2)τ+x]2

2σ2τ

∫ ∞

−∞e−

12 σ2τ[ω+i

(r−D0−12 σ2)τ+x

σ2τ]2dω

=e−rτ

√2πσ2

· e−[(r−D0−

12 σ2)τ+x]2

2σ2τ . (B.9)

Thus, I1 can be rewritten

I1 =1

2π

∫ ∞

−∞U1(ω,0) · e−iωx ·G(ω,τ)dω

= U1(x,0)∗g(x,τ)

=∫ ln(Sc(0))


√2πσ2

· e−[(r−D0−

12 σ2)τ+x−u]2

2σ2τ du. (B.10)

Then, we compute I2

I2 =1

2π

∫ ∞

−∞

∫ τ

0f (ω,ξ ) · e−B(ω)(τ−ξ ) · e−iωxdξ dω

=∫ τ

0

12π

∫ ∞

−∞f (ω,ξ ) · e−B(ω)(τ−ξ ) · e−iωxdωdξ

=∫ τ

0

12π

∫ ∞

−∞(nSc(ξ )−Ze−rξ ) · eiω ln(Sc(ξ )) · [S

′c(ξ )

Sc(ξ )+ r−D0 −

12

σ2 − 12

σ2iω]

+12

σ2nSc(ξ )eiω ln(Sc(ξ )) · e−B(ω)(τ−ξ ) · e−iωxdωdξ

=∫ τ

0

12π

∫ ∞

−∞e−iωx−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r](τ−ξ )+iω ln(Sc(ξ )) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+ r−D0 −

12

σ2 − 12

σ2iω]+12

σ2nSc(ξ )dωdξ

,∫ τ

0

12π

∫ ∞

−∞e−iωx−[ 1

2 σ2ω2+(r−D0− 12 σ2)iω+r](τ−ξ )+iω ln(Sc(ξ ))

· f1(ξ )− f2(ξ )ωdωdξ , (B.11)

where

f1(ξ ) = (nSc(ξ )−Ze−rξ ) · [S′c(ξ )

Sc(ξ )+ r−D0 −

12

σ2]+12

σ2nSc(ξ ), (B.12)

f2(ξ ) =12

σ2(nSc(ξ )−Ze−rξ )i. (B.13)


With the following equation used∫ ∞


√πp

∂ n

∂qn eq24p . (B.14)

We can obtain

I2 =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )σ2

· e−[(r−D0−

12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rτ)

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

nσ2Sc(ξ )dξ . (B.15)

Combining I1 and I2, U1 can be presented

U1(x,τ) =∫ ln(Sc(0))


√2πσ2

· e−[(r−D0−

12 σ2)τ+x−u]2

2σ2τ du

+∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )σ2

· e−[(r−D0−

12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

nσ2Sc(ξ )dξ . (B.16)

B.3 Appendix B.3Now, we present the details of computing Equation (4.15) from Equation (4.14).First, we rewrite Equation (4.14) as follows

V1(S,τ) =∫ τ

0


e− [(r−D0−

12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(S)]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]

12


(B.17)

It should be noted that the first term of Equation (4.14) is missing. Actually, it isalways zero, due to Sc(0) =

Zn. Now, we define

h(S,ξ ) ,[(r−D0 − 1

2σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)]2

2σ2(τ −ξ )

=1

2(τ −ξ )[(r−D0 − 1

2σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)σ

]2

=1

2(τ −ξ )[(r−D0 − 1

2σ2)τ + ln(S)σ

−(r−D0 − 1


]2

, 12(τ −ξ )

[y−P(ξ )]2, (B.18)



2σ2)τ + ln(S)σ

and P(ξ ) =(r−D0 − 1


. It can

be found that P′(ξ ) =1σ[S′c(ξ )Sc(ξ )

+ r−D0−12

σ2]. Therefore, we apply it to Equation

(B.17) and obtain

V1(S,τ) = Ze−rτ +∫ τ

0


e− [(r−D0−

12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(S)]2

2σ2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+

12(r−D0 −

12


τ −ξ)]+

12

nσ2Sc(ξ )dξ

= Ze−rτ +∫ τ

0


e−[y−P(ξ )]2

2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[S′c(ξ )

Sc(ξ )+ r−D0 −

12

σ2 − 12(r−D0 −

12


)]+12

nσ2Sc(ξ )dξ

= Ze−rτ +∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[P′(ξ )− 12σ

(r−D0 −12


)]+12

nσSc(ξ )dξ

= Ze−rτ +∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · (nSc(ξ )−Ze−rξ )

·[P′(ξ )− 12(τ −ξ )

((r−D0 − 1

2σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)σ

)]+12

nσSc(ξ )dξ

=∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ )

·(nSc(ξ )−Ze−rξ ) · [P′(ξ )− y−P(ξ )2(τ −ξ )

]+12


=∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·nSc(ξ )[σ2+P′(ξ )− y−P(ξ )

2(τ −ξ )]dξ

−∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·Ze−rξ [P′(ξ )− y−P(ξ )2(τ −ξ )

]dξ +Ze−rτ

, R1(S,τ)−ZR2(S,τ)+Ze−rτ , (B.19)

where

R1(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·nSc(ξ )[σ2+P′(ξ )− y−P(ξ )

2(τ −ξ )]dξ ,(B.20)

R2(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · e−rξ [P′(ξ )− y−P(ξ )2(τ −ξ )

]dξ . (B.21)


Now, we compute R1(S,τ) and R2(S,τ), respectively.

R1(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) ·nSc(ξ )[σ2+P′(ξ )− y−P(ξ )

2(τ −ξ )]dξ

=∫ τ

0

e−r(τ−ξ )√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ ) e(y−P(ξ ))σ+

σ2(τ−ξ )2 ·nSc(ξ )[

σ2 +P′(ξ )− y−P(ξ )

2(τ−ξ )√τ −ξ

]dξ

=∫ τ

0

e−r(τ−ξ )+(y−P(ξ ))σ+σ2(τ−ξ )

2√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )σ(τ −ξ )+2P′(ξ )(τ −ξ )− y+P(ξ )

2(τ −ξ )√

τ −ξdξ

=∫ τ

0

e−r(τ−ξ )+(r−D0− 12 σ2)(τ−ξ )−ln(Sc(ξ ))+ln(S)+σ2(τ−ξ )

2√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )P(ξ )− y−σ(τ −ξ )+2P′(ξ )(τ −ξ )−2σ(τ −ξ )

2(τ −ξ )√

τ −ξdξ

=∫ τ

0

e−D0(τ−ξ )−ln(Sc(ξ ))+ln(S)√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )[P(ξ )− y−σ(τ −ξ )]+ [P′(ξ )−σ ] ·2(τ −ξ )

2(√

τ −ξ )2√

τ −ξdξ

=∫ τ

0

SSc(ξ )

· e−D0(τ−ξ )√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ )

·nSc(ξ )1

2√

τ−ξ[P(ξ )− y−σ(τ −ξ )]+(P′(ξ )+σ)

√τ −ξ

(√

τ −ξ )2dξ

=∫ τ

0

nSe−D0(τ−ξ )√

2πe−

[y−P(ξ )+σ(τ−ξ )]22(τ−ξ ) · ∂

∂ξ[P(ξ )− y−σ(τ −ξ )√

τ −ξ]dξ

=∫ τ

0−nSe−D0(τ−ξ ) ∂

∂ξN (

y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ , (B.22)

R2(S,τ) =∫ τ

0

e−r(τ−ξ )√2π(τ −ξ )

e−[y−P(ξ )]2

2(τ−ξ ) · e−rξ [P′(ξ )− y−P(ξ )2(τ −ξ )

]dξ

=∫ τ

0

e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · [2P′(ξ )(τ −ξ )− y+P(ξ )

2(τ −ξ )√

τ −ξ]dξ

=∫ τ

0

e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · [P

′(ξ ) ·2(τ −ξ )+(P(ξ )− y)

2(τ −ξ )√

τ −ξ]dξ

=∫ τ

0

e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · [

P′(ξ )√

τ −ξ + P(ξ )−y2√

τ−ξ

(√

τ −ξ )2]dξ

=∫ τ

0− e−rτ√

2πe−

[y−P(ξ )]22(τ−ξ ) · ∂

∂ξ(y−P(ξ )√

τ −ξ)dξ

=∫ τ

0−e−rτ ∂

∂ξN (


)dξ . (B.23)


Thus, we substitute the expression of R1(S,τ) and R2(S,τ) into Equation (B.19).Then, Equation (4.15) can be obtained

V1(S,τ) = R1(S,τ)−ZR2(S,τ)+Ze−rτ

=∫ τ

0−nSe−D0(τ−ξ ) ∂

∂ξN (

y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ

+Z∫ τ

0e−rτ ∂

∂ξN (


)dξ +Ze−rτ

=∫ τ

0−nSe−D0(τ−ξ )dN (

y−P(ξ )+σ(τ −ξ )√τ −ξ

)+Z∫ τ

0e−rτdN (


)+Ze−rτ

= −nSe−D0(τ−ξ )N (y−P(ξ )+σ(τ −ξ )√

τ −ξ)|ξ=τ

+nSe−D0(τ−ξ )N (y−P(ξ )+σ(τ −ξ )√

τ −ξ)|ξ=0

+∫ τ


y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ

+Ze−rτN (y−P(ξ )√

τ −ξ)|ξ=τ −Ze−rτN (


)|ξ=0 +Ze−rτ

= −nSN (ln(S)− ln(Sc(ξ ))

σ√

τ −ξ)|ξ=τ +Ze−rτN (

ln(S)− ln(Sc(ξ ))σ√

τ −ξ)|ξ=τ


12σ2)τ − lnSc(0)+ ln(S)

σ√

τ)

+∫ τ


y−P(ξ )+σ(τ −ξ )√τ −ξ

)dξ

−Ze−rτN ((r−D0 − 1

2σ2)τ − ln(Sc(0))+ ln(S)σ√

τ)+Ze−rτ

= −1S=Sc(τ)(nS−Ze−rτ)


12σ2)τ − lnSc(0)+ ln(S)

σ√

τ)

+∫ τ

0nD0Se−D0(τ−ξ ) ·N (

(r−D0 +12σ2)(τ −ξ )− ln(Sc(ξ ))+ ln(S)

σ√

τ −ξ)dξ

−Ze−rτN ((r−D0 − 1

2σ2)τ − ln(Sc(0))+ ln(S)σ√

τ)+Ze−rτ , (B.24)

where

1S=Sc(τ) =

12 S = Sc(τ),0 S < Sc(τ).

(B.25)


B.4 Appendix B.4In this appendix, Green’s function is used to obtain Equation (4.16) from System(4.3). Firstly, we rewrite the system here

−∂V1

∂τ+

12

σ2S2 ∂ 2V1

∂S2 +(r−D0)S∂V1

∂S− rV1 = 0,

V1(S,τ+K ) =V1(S,τ−K ),

V1(Kn,τ) = K,

V1(0,τ) = Ze−rτ .

(B.26)

A classical transform should be applied to this system to obtain a dimensionlessPDE system, mentioned before

x = log(S), v1(x,τ) =V1(S,τ),

then, we obtain

−∂v1

∂τ+

12

σ2 ∂ 2v1

∂x2 +(r−D0 −12

σ2)∂v1

∂x− rv1 = 0,

v1(x,τ+K ) = v1(x,τ−K ),

v1(ln(Kn),τ) = K,

v1(−∞,τ) = Ze−rτ .

(B.27)

To apply the Green’s function method to the PDE system, we should make surethat the boundary condition at negative infinity is equal to zero. Thus, a simpletransform can reach to it

U(x,τ) = v1(x,τ)−Ze−rτ , (B.28)

and then

−∂U∂τ

+12

σ2 ∂ 2U∂x2 +(r−D0 −

12

σ2)∂U∂x

− rU = 0,

U(x,τ+K ) =U(x,τ−K ),

U(ln(Kn),τ) = K −Ze−rτ ,

U(−∞,τ) = 0.

(B.29)

Next step, this PDE system is transformed to a normal one using the transform

W (x,τ) = er−D0−

12 σ2

σ2 x+(r−D0−

12 σ2)2+2rσ2

2σ2 τ ·U(x,τ), (B.30)


then

∂W∂τ

=12

σ2 ∂ 2W∂x2 ,

W (x,τ+K ) =W (x,τ−K ),

W (ln(Kn),τ) = (K −Ze−rτ) · e

r−D0−12 σ2

σ2 ln(Kn )+

(r−D0−12 σ2)2+2rσ2

2σ2 τ,

W (−∞,τ) = 0.

(B.31)

By now, the Green’s function can be applied to the PDE system, and then thesolution can be obtained

W (x,τ) =1√

2πσ2(τ − τK)

∫ ∞

0[exp(−

[−x+ ln(Kn )− y]2

2σ2(τ − τK))

−exp(−[−x+ ln(K

n )+ y]2

2σ2(τ − τK))] ·W (ln(

Kn)− y,τK)dy

+∫ τ−τK

0

−x+ ln(Kn )√

2σ2π(τ − τK −ξ )3· exp(−

[−x+ ln(Kn )]

2

2σ2(τ − τK −ξ )) · (K −Ze−r(ξ+τK))

·exp(r−D0 − 1

2σ2

σ2 ln(Kn)+

(r−D0 − 12σ2)2 +2rσ2

2σ2 (ξ + τK))dξ .

Then, we rewrite the form of integral equation with the original parameters

V1(S,τ) =1√

2πσ2(τ − τK)e−

[(r−D0−12 σ2)2+2rσ2](τ−τK )

2σ2

·∫ ln(K

n )

−∞[exp(− [− ln(S)+ y]2

2σ2(τ − τK))− exp(−

[− ln(S)+2ln(Kn )− y]2

2σ2(τ − τK))]

·exp(r−D0 − 1

2σ2

σ2 (− ln(S)+ y)) · (V1(ey,τK)−Ze−rτK)dy

+∫ τ−τK

0

− ln(S)+ ln(Kn )√

2πσ2(τ − τK −ξ )3· exp(−

[− ln(S)+ ln(Kn )]

2

2σ2(τ − τK −ξ )) · (K −Ze−r(τK+ξ ))

·exp(r−D0 − 1

2σ2

σ2 (ln(Kn)− ln(S))+

(r−D0 − 12σ2)2 +2rσ2

2σ2 (ξ + τK − τ))dξ

+ Ze−rτ , (B.32)

it is Equation (4.16).

B.5 Appendix B.5In this appendix, the details of applying the incomplete Fourier transform (4.18) tothe PDE System (4.20) are displayed. Firstly, it should be noted that the incompleteFourier transform operator is linear, and thus we can obtain the following equation:


−F∂v1

∂τ+ 1

2σ2F∂ 2v1

∂x2 +(r−D0 −12

σ2)F∂v1

∂x− rFv1= 0. (B.33)

Now, we compute every term of Equation (B.33)

F∂v1

∂τ(x,τ) =

∫ ln(Kn )

ln(Sp(τ))

∂v1

∂τ(x,τ) · eiωxdx

=∂

∂τ[∫ ln(K

n )

ln(Sp(τ))v1(x,τ) · eiωxdx]+

S′p(τ)Sp(τ)

v1(ln(Sp(τ)),τ)eiω ln(Sp(τ))

=∂ v1

∂τ(ω,τ)+M

S′p(τ)Sp(τ)

eiω ln(Sp(τ)), (B.34)

F∂v1

∂x(x,τ) =

∫ ln(Kn )

ln(Sp(τ))

∂v1


=∫ ln(K

n )

ln(Sp(τ))eiωxdv1(x,τ)

= v1(x,τ)eiωx|ln(Kn )

ln(Sp(τ))− iω∫ ln(K

n )

ln(Sp(τ))v1(x,τ)eiωxdx

= Keiω ln(Kn )−Meiω ln(Sp(τ))− iω v1(ω,τ), (B.35)

F∂ 2v1

∂x2 (x,τ) =∫ ln(K

n )

ln(Sp(τ))

∂ 2v1

∂x2 (x,τ) · eiωxdx

=∫ ln(K

n )

ln(Sp(τ))eiωxd

∂v1

∂x(x,τ)

=∂v1

∂x(x,τ)eiωx|ln(

Kn )

ln(Sp(τ))− iω∫ ln(K

n )

ln(Sp(τ))

∂v1


=Kn

A(τ)eiω ln(Kn )− iω[Keiω ln(K

n )−Meiω ln(Sp(τ))− iω v1(ω,τ)]

=Kn

A(τ)eiω ln(Kn )− iωKeiω ln(K

n )+ iωMeiω ln(Sp(τ))−ω2v1(ω,τ).

(B.36)

Substituting these three formulations into Equation (B.33), we can obtain

∂v1

∂τ(ω,τ)+ [

12

σ2ω2 +(r−D0 −12

σ2)iω + r]v1(ω,τ)

= Keiω ln(Kn ) · [1

2σ2 A(τ)

n− 1

2σ2iω +(r−D0 −

12

σ2)]

−Meiω ln(Sp(τ)) · [S′p(τ)Sp(τ)

− 12

σ2iω +(r−D0 −12

σ2)]. (B.37)

Then, the ODE System (4.21) is obtained.


B.6 Appendix B.6In this appendix, the Fourier inversion transform is applied to Equation (4.24), andEquation (4.25) will be obtained. Firstly, we rewrite Equation (4.24) here

v1(ω,τ) = e−B(ω)(τ−τM) · v1(ω,τM)+∫ τ−τM

0f1(ω,ξ + τM) · e−B(ω)(τ−τM−ξ )dξ

−∫ τ−τM

0f2(ω,ξ + τM) · e−B(ω)(τ−τM−ξ )dξ , (B.38)

where

B(ω) =12

σ2ω2 +(r−D0 −12

σ2)iω + r, (B.39)

f1(ω,τ) = Keiω ln(Kn ) · [1

2σ2 A(τ)

n− 1

2σ2iω +(r−D0 −

12

σ2)], (B.40)

f2(ω,τ) = Meiω ln(Sp(τ)) · [S′p(τ)

Sp(τ)− 1

2σ2iω +(r−D0 −

12

σ2)]. (B.41)

Then, the integral equation formation of v1(x,τ) can be presented

v1(x,τ) =1

2π

∫ ∞

−∞v1(ω,τ)e−iωxdω

=1

2π

∫ ∞

−∞e−B(ω)(τ−τM) · v1(ω,τM) · e−iωxdω

+1

2π

∫ ∞

−∞

∫ τ−τM

0f1(ω,ξ + τM) · e−B(ω)(τ−τM−ξ ) · e−iωxdξ dω

− 12π

∫ ∞

−∞

∫ τ−τM


, I1 + I2 − I3, (B.42)

where I1, I2 and I3 are the integral equation formulations, respectively. As follows,we compute these three one by one with I1 first. It should be noted that the formulaof I1 in this appendix is very similar to the I1 in the Appendix B. Therefore, we willonly give the solution of it with no process

I1 =∫ ln(K

n )

ln(Sp(τM))v1(u,τM)

e−r(τ−τM)√2π(τ − τM)

· e− [(r−D0−

12 σ2)(τ−τM)+x−u]2

2σ2(τ−τM) du. (B.43)

Now, we compute I2

I2 =1

2π

∫ ∞

−∞

∫ τ−τM


=1

2π

∫ τ−τM

0

∫ ∞

−∞f1(ω,ξ + τM) · e−B(ω)(τ−τM−ξ ) · e−iωxdωdξ


=∫ τ−τM

0

12π

∫ ∞

−∞Keiω ln(K

n ) · e−[ 12 σ2ω2+(r−D0− 1

2 σ2)iω+r](τ−τM−ξ ) · e−iωx

·[12

σ2 A(τ)n

− 12

σ2iω +(r−D0 −12

σ2)]dωdξ

=∫ τ−τM

0

Ke−r(τ−τM−ξ )

2π

∫ ∞

−∞eiω ln(K

n )−[ 12 σ2ω2+(r−D0− 1

2 σ2)iω](τ−τM−ξ )−iωx

·[12

σ2 A(τ)n

− 12

σ2iω +(r−D0 −12

σ2)]dωdξ

=∫ τ−τM

0

Ke−r(τ−τM−ξ )

2π

∫ ∞

−∞e−

12 σ2(τ−τM−ξ )ω2−[(r−D0− 1

2 σ2)(τ−τM−ξ )+x−ln(Kn )]iω

·[g1(ξ )−g2(ξ )ω]ωdξ , (B.44)

where g1(ξ ) =12

σ2 A(τ)n

+(r−D0 −12

σ2) and g2(ξ ) =12

σ2i. Thus, with the using

of the conclusion∫ ∞


√πp

∂ n

∂qn eq24p , we obtain

I2 =∫ τ−τM

0

Ke−r(τ−τM−ξ )√2π(τ − τM −ξ )σ2

· e− [(r−D0−

12 σ2)(τ−τM−ξ )+x−ln(K

n )]2


·12

σ2 A(τM +ξ )n

+12[r−D0 −

12

σ2 +ln(K

n )− xτ − τM −ξ

]dξ . (B.45)

With the same method, we can obtain

I3 =∫ τ−τM

0

Me−r(τ−τM−ξ )√2π(τ − τM −ξ )σ2

· e− [(r−D0−

12 σ2)(τ−τM−ξ )+x−ln(Sp(τM+ξ ))]2



+12[r−D0 −

12


τ − τM −ξ]dξ . (B.46)

Combining these three formulae, and rewriting the integral equation formulation inthe original parameters, then Equation (4.25) can be obtained.

B.7 Appendix B.7In this appendix, the details of how to obtain Equation (4.26) form Equation (4.25)can be presented. Firstly, we rewrite Equation (4.25)

V1(S,τ) =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0

Ke−r(τ−τM−ξ )√2πσ2(τ − τM −ξ )

e− [(r−D0−


n )]2


·12

σ2 A(τM +ξ )n

+12[r−D0 −

12

σ2 +ln(K


]dξ


−∫ τ−τM

0

Me−r(τ−τM−ξ )√2πσ2(τ − τM −ξ )

e− [(r−D0−




+12(r−D0 −

12


τ − τM −ξ)dξ .

(B.47)

It should be noted that there are only two items should be done the further com-putations, I2 and I3, where we use the same marks as in Appendix B.6. Thus, do I2

first

I2 =∫ τ−τM

0

Ke−r(τ−τM−ξ )√2π(τ − τM −ξ )σ2

· e− [(r−D0−


n )]2


·12

σ2 A(τM +ξ )n

+12[r−D0 −

12

σ2 +ln(K


]dξ , (B.48)

and we define

h2(S,ξ ) ,[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Kn )]

2

2σ2(τ − τM −ξ )

=1

2(τ − τM −ξ )[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Kn )

σ]2

=1

2(τ − τM −ξ )[(r−D0 − 1


−(r−D0 − 1

2σ2)ξ + ln(Kn )

σ]2

, 12(τ − τM −ξ )

[y−P(ξ )]2, (B.49)



and P(ξ ) =(r−D0 − 1

2σ2)ξ + ln(Kn )

σ. It is

interesting to find that P′(ξ ) =r−D0 − 1

2σ2

σ. Then, we apply it to the I2 and obtain

I2 =∫ τ−τM

0

Ke−r(τ−τM−ξ )√2π(τ − τM −ξ )

· e−[y−P(ξ )]2

2(τ−τM−ξ )

·12

σA(τM +ξ )

n+

12σ

[r−D0 −12

σ2 +ln(K


]dξ

=∫ τ−τM

0


· e−[y−P(ξ )]2

2(τ−τM−ξ )

·12

σA(τM +ξ )

n+P′(ξ )− 1

2σ(r−D0 −

12

σ2)+ln(K

n )− ln(S)2σ(τ − τM −ξ )

dξ

=∫ τ−τM

0


· e−[y−P(ξ )]2

2(τ−τM−ξ )


·12

σA(τM +ξ )

n+P′(ξ )− 1

2σ(r−D0 −

12

σ2)+ln(K

n )− ln(S)2σ(τ − τM −ξ )

dξ

=∫ τ−τM

0


· e−[y−P(ξ )]2

2(τ−τM−ξ )

·12

σA(τM +ξ )

n+P′(ξ )−

(r−D0 − 12σ2)(τ − τM −ξ )+ ln(S)− ln(K

n )

2σ(τ − τM −ξ )dξ

=∫ τ−τM

0


· e−[y−P(ξ )]2

2(τ−τM−ξ )

·12

σA(τM +ξ )

n+P′(ξ )− y−P(ξ )

2(τ − τM −ξ )dξ

=∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ )

·12

σA(τM +ξ )

n√

τ − τM −ξ+

2P′(ξ )(τ − τM −ξ )− y+P(ξ )2(τ − τM −ξ )

√τ − τM −ξ

dξ

=∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ )

·12

σA(τM +ξ )

n√

τ − τM −ξ+

2P′(ξ )√

τ − τM −ξ − (y−P(ξ )) 1√τ−τM−ξ


=∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ ) · 1

2σ

A(τM +ξ )n√

τ − τM −ξdξ

+∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ ) ·

2P′(ξ )√

τ − τM −ξ − (y−P(ξ )) 1√τ−τM−ξ


=∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ ) · σA(τM +ξ )

2n√

τ − τM −ξdξ

−∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ ) · ∂

∂ξ y−P(ξ )√

τ − τM −ξdξ

=∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ ) · σA(τM +ξ )

2n√

τ − τM −ξdξ

−∫ τ−τM

0Ke−r(τ−τM−ξ ) · ∂

∂ξN y−P(ξ )√

τ − τM −ξdξ . (B.50)

Then is the I3

I3 =∫ τ−τM

0

Me−r(τ−τM−ξ )√2π(τ − τM −ξ )σ2

· e− [(r−D0−




+12[r−D0 −

12


τ − τM −ξ]dξ . (B.51)


As the same method as before, we define

h3(S,ξ ) ,[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Sp(τM +ξ ))]2

2σ2(τ − τM −ξ )

=1

2(τ − τM −ξ )[(r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Sp(τM +ξ ))σ

]2

=1

2(τ − τM −ξ )[(r−D0 − 1


−(r−D0 − 1


]2

, 12(τ − τM −ξ )

[x− J(ξ )]2, (B.52)

where x=(r−D0 − 1


and J(ξ )=(r−D0 − 1


.

It is interesting to notice that J′(ξ ) =1σ[S′p(τM +ξ )Sp(τM +ξ )

+r−D0−12

σ2]. Then, we apply

it to the I3 and obtain

I3 =∫ τ−τM

0


· e−[x−J(ξ )]2

2(τ−τM−ξ )

·J′(ξ )− 12σ

(r−D0 −12

σ2)+ln(Sp(τM +ξ ))− ln(S)

2σ(τ − τM −ξ )dξ

=∫ τ−τM

0


· e−[x−J(ξ )]2

2(τ−τM−ξ )

·J′(ξ )−(r−D0 − 1

2σ2)(τ − τM −ξ )− ln(Sp(τM +ξ ))+ ln(S)2σ(τ − τM −ξ )

dξ

=∫ τ−τM

0


· e−[x−J(ξ )]2

2(τ−τM−ξ ) · J′(ξ )− x− J(ξ )2(τ − τM −ξ )

dξ

=∫ τ−τM

0


2π· e−

[x−J(ξ )]22(τ−τM−ξ )2J′(ξ )(τ − τM −ξ )− x+ J(ξ )

2(τ − τM −ξ )√

τ − τM −ξdξ

=∫ τ−τM

0


2π· e−

[x−J(ξ )]22(τ−τM−ξ ) ·

2J′(ξ )√

τ − τM −ξ − (x− J(ξ )) 1√τ−τM−ξ


= −∫ τ−τM

0


2π· e−

[x−J(ξ )]22(τ−τM−ξ )

∂∂ξ

x− J(ξ )√τ − τM −ξ

dξ

= −∫ τ−τM

0Me−r(τ−τM−ξ ) · ∂

∂ξN x− J(ξ )√

τ − τM −ξdξ . (B.53)

Then, we substitute the expression of I2 and I3 into Equation (B.47), and obtain

V1(S,τ) =∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2π· e−

[y−P(ξ )]22(τ−τM−ξ ) · σA(τM +ξ )

2n√

τ − τM −ξdξ


−∫ τ−τM

0Ke−r(τ−τM−ξ ) · ∂

∂ξN y−P(ξ )√

τ − τM −ξdξ

+∫ τ−τM

0Me−r(τ−τM−ξ ) · ∂

∂ξN x− J(ξ )√

τ − τM −ξdξ

=∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2π· e

− [(r−D0−12 σ2)(τ−τM−ξ )+ln(S)−ln(K

n )]2

2σ2(τ−τM−ξ ) · σA(τM +ξ )2n

√τ − τM −ξ

dξ

−∫ τ−τM

0Ke−r(τ−τM−ξ ) · ∂

∂ξN

(r−D0 − 12σ2)(τ − τM −ξ )+ ln(S)− ln(K

n )

σ√

τ − τM −ξdξ

+∫ τ−τM

0Me−r(τ−τM−ξ )

· ∂∂ξ

N (r−D0 − 1


τ − τM −ξdξ

=∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2π· e

− [(r−D0−12 σ2)(τ−τM−ξ )+ln(S)−ln(K

n )]2


√τ − τM −ξ

dξ

−Ke−r(τ−τM−ξ ) ·N (r−D0 − 1

2σ2)(τ − τM −ξ )+ ln(S)− ln(Kn )

σ√

τ − τM −ξ|τ−τM

0

+∫ τ−τM

0rKe−r(τ−τM−ξ ) ·N

(r−D0 − 12σ2)(τ − τM −ξ )+ ln(S)− ln(K

n )

σ√

τ − τM −ξdξ

+Me−r(τ−τM−ξ ) ·N (r−D0 − 1


τ − τM −ξ|τ−τM

0

−∫ τ−τM

0rMe−r(τ−τM−ξ ) ·N

(r−D0 − 12σ2)(τ − τM −ξ )+ ln(S)− ln(Sp(τM +ξ ))

σ√

τ − τM −ξdξ

=∫ ln(K

n )



e− [(r−D0−

12 σ2)(τ−τM)+ln(S)−u]2

2σ2(τ−τM) du

+∫ τ−τM

0


2π· e

− [(r−D0−12 σ2)(τ−τM−ξ )+ln(S)−ln(K

n )]2


√τ − τM −ξ

dξ

+Ke−r(τ−τM) ·N (r−D0 − 1

2σ2)(τ − τM)+ ln(S)− ln(Kn )

σ√

τ − τM

+∫ τ−τM

0rKe−r(τ−τM−ξ ) ·N

(r−D0 − 12σ2)(τ − τM −ξ )+ ln(S)− ln(K

n )

σ√

τ − τM −ξdξ

−Me−r(τ−τM) ·N (r−D0 − 1


σ√

τ − τM

−∫ τ−τM

0rMe−r(τ−τM−ξ ) ·N

(r−D0 − 12σ2)(τ − τM −ξ )+ ln(S)− ln(Sp(τM +ξ ))

σ√

τ − τM −ξdξ


−K1S=Kn+M1S=Sp(τ), (B.54)

where

1S=Kn

=

12

S =Kn,

0 S <Kn,

(B.55)

1S=Sp(τ) =

12

S = Sp(τ),

1 S > Sp(τ).(B.56)

Appendix C


C.1 Appendix C.1Recall the PDE system (5.11)

−∂V∂τ

+12

σ2 ∂ 2V∂x2 +(r−D0 −

12

σ2)∂V∂x

− rV = 0,

V (x,0) = maxn1ex,Z,V (ln(Sr),τ) = F(τ),∂V∂x

(ln(Sr),τ) = Sr ·G(τ),

V (ln(Sc(τ)),τ) = n1 ·Sc(τ),∂V∂x

(ln(Sc(τ)),τ) = n1 ·Sc(τ),

(C.1)

and the definition of incomplete Fourier transform, Equation (5.10),

FV (x,τ)=∫ ln(Sc(τ))

ln(Sr)V (x,τ) · eiωxdx , V (ω,τ). (C.2)

Applying the incomplete Fourier transform on the PDE leads to

−F∂V∂τ

+ 12

σ2F∂ 2V∂x2 +(r−D0 −

12

σ2)F∂V∂x

− rFV= 0, (C.3)

and clearly we need to calculate every term involved to the above equation. Inparticular, we have

F∂V∂τ

(x,τ) =∫ ln(Sc(τ))

ln(Sr)

∂V∂τ

(x,τ) · eiωxdx

=∂

∂τ

∫ ln(Sc(τ))

ln(Sr)V (x,τ) · eiωxdx− S′c(τ)

Sc(τ)V (x,τ) · eiωx|ln(Sc(τ))

=∂V∂τ

(ω,τ)− S′c(τ)Sc(τ)

F(τ) · eiω ln(Sc(τ)), (C.4)

187

APPENDIX C. APPENDIX FOR CHAPTER 5 188

F∂V∂x

(x,τ) =∫ ln(Sc(τ))

ln(Sr)

∂V∂x

(x,τ) · eiωxdx

= V (x,τ) · eiωx|ln(Sc(τ))−V (x,τ) · eiωx|ln(Sr)− iω∫ ln(Sc(τ))

ln(Sr)V (x,τ) · eiωxdx

= n1Sc(τ) · eiω ln(Sc(τ))−F(τ) · eiω ln(Sr)− iωV (ω,τ), (C.5)

F∂ 2V∂x2 =

∫ ln(Sc(τ))

ln(Sr)

∂ 2V∂x2 (x,τ) · e

iωxdx

=∂V∂x

(x,τ) · eiωx|ln(Sc(τ))−∂V∂x

(x,τ) · eiωx|ln(Sr)− iω∫ ln(Sc(τ))

ln(Sr)

∂V∂x

(x,τ) · eiωxdx

= n1Sc(τ) · eiω ln(Sc(τ))−SrG(τ) · eiω ln(Sr)

−iω[n1Sc(τ) · eiω ln(Sc(τ))−F(τ) · eiω ln(Sr)− iωV (ω,τ)]

= (1− iω)n1Sc(τ) · eiω ln(Sc(τ))− [SrG(τ)− iωF(τ)] · eiω ln(Sr)−ω2V (ω,τ).

(C.6)

Combining all the equations above yields the ODE system (5.12).

C.2 Appendix C.2Equation (5.16) can be rewritten as

V (ω,τ) = V (ω,0) · e−B(ω)τ +∫ τ

0[F1(ω,ξ )−F2(ω,ξ )] · e−B(ω)(τ−ξ )dξ , (C.7)

with

B(ω) =12

σ2ω2 +(r−D0 −11

σ2)iω + r, (C.8)

F1(ω,τ) = [S′c(τ)Sc(τ)

− 12

σ2iω +(r−D0)] ·n1Sc(τ)eiω ln(Sc(τ)), (C.9)

F2(ω,τ) = [12

σ2G(τ)Sr −12

σ2iωF(τ)+(r−D0 −12

σ2)F(τ)] · eiω ln(Sr).(C.10)

In addition, the Fourier Inversion transform can be specified as

V (x,τ) = F−1V (ω,τ)= 12π

∫ ∞

−∞V (ω,τ) · e−iωxdω. (C.11)

Thus, if the Fourier inversion transform (C.11) is applied on Equation (C.7), we canobtain

V (x,τ) = F−1V (ω,τ)

=1

2π

∫ ∞

−∞V (ω,0) · e−B(ω)τ · e−iωxdω


+1

2π

∫ ∞

−∞

∫ τ

0F1(ω,ξ ) · e−B(ω)(τ−ξ )dξ · e−iωxdω

− 12π

∫ ∞

−∞

∫ τ


, I1 + I2 − I3, (C.12)

implying that we need to compute these three integrals one by one. First, I1 can berearranged as

I1 =1

2π

∫ ∞

−∞V (ω,0) · e−B(ω)τ · e−iωxdω

=1

2π

∫ ∞

−∞V (ω,0) · e−[ 1


, 12π

∫ ∞

−∞V (ω,0) ·G(ω,τ) · e−iωxdω, (C.13)

where G(ω,τ) = e−[ 12 σ2ω2+(r−D0− 1

1 σ2)iω+r]τ , with the Fourier inversion of G(ω,τ)being calculated as

g(x,τ) = F−1G(ω,τ)= 12π

∫ ∞

−∞G(ω,τ) · e−iωxdω

=1

2π

∫ ∞

−∞e−[ 1


=e−rτ

2π

∫ ∞

−∞e−

12 σ2τω2−[(r−D0− 1

2 σ2)τ+x]iωdω

=e−rτ

2π

∫ ∞

−∞e−

12 σ2τ[ω+i

(r−D0−12 σ2)τ+x

σ2τ]2 · e−

[(r−D0−12 σ2)τ+x]2

2σ2τ dω

=e−rτ

2π· e−

[(r−D0−12 σ2)τ+x]2

2σ2τ ·√

2πσ2τ

=e−rτ

√2πσ2τ

· e−[(r−D0−

12 σ2)τ+x]2

2σ2τ , (C.14)

we can get the expression of I1 below

I1 =1

2π

∫ ∞

−∞V (ω,0) ·G(ω,τ) · e−iωxdω

= V (x,τ)∗g(x,τ)

=e−rτ

√2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+x−u]2

2σ2τ ·maxn1eu,Zdu, (C.15)

according to the convolution theorem.On the other hands, I2 and I3 can be respectively computed as

I2 =1

2π

∫ ∞

−∞

∫ τ



=1

2π

∫ ∞

−∞

∫ τ

0[S

′c(ξ )

Sc(ξ )− 1

2σ2iω +(r−D0)] ·n1Sc(ξ )eiω ln(Sc(ξ ))

· e−B(ω)(τ−ξ )dξ · e−iωxdω

=1

2π

∫ τ

0

∫ ∞

−∞[S

′c(ξ )

Sc(ξ )− 1

2σ2iω +(r−D0)] ·n1Sc(ξ )eiω ln(Sc(ξ ))

· e−[ 12 σ2ω2+(r−D0− 1

2 σ2)iω+r](τ−ξ ) · e−iωxdωdξ

=1

2π

∫ τ

0

∫ ∞

−∞[S′c(ξ )Sc(ξ )

+(r−D0)−12

σ2iω] · e−r(τ−ξ )

·n1Sc(ξ ) · e−12 σ2(τ−ξ )ω2−[(r−D0− 1

2 σ2)(τ−ξ )+x−ln(Sc(ξ ))]iωdωdξ

=∫ τ

0

n1Sc(ξ ) · e−r(τ−ξ )

2π

∫ ∞

−∞[S′c(ξ )Sc(ξ )

+(r−D0)−12

σ2iω]

· e−12 σ2(τ−ξ )ω2−[(r−D0− 1

2 σ2)(τ−ξ )+x−ln(Sc(ξ ))]iωdωdξ

=∫ τ

0

n1Sc(ξ ) · e−r(τ−ξ )

2π· e−

[(r−D0−12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ ) ·

√2π

σ2(τ −ξ )

· S′c(ξ )Sc(ξ )

+(r−D0)+12

σ2i2(r−D0 − 1

2σ2)(τ −ξ )+ x− ln(Sc(ξ ))σ2(τ −ξ )

dξ

=∫ τ

0

n1Sc(ξ ) · e−r(τ−ξ )√2πσ2(τ −ξ )

· e−[(r−D0−

12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ )


+(r−D0)−12(r−D0 − 1

2σ2)(τ −ξ )+ x− ln(Sc(ξ ))(τ −ξ )

dξ

=∫ τ

0

n1Sc(ξ ) · e−r(τ−ξ )√2πσ2(τ −ξ )

· e−[(r−D0−

12 σ2)(τ−ξ )+x−ln(Sc(ξ ))]2

2σ2(τ−ξ )


+(r−D0 +

12σ2)(τ −ξ )− x+ ln(Sc(ξ ))

2(τ −ξ )dξ , (C.16)

and

I3 =1

2π

∫ ∞

−∞

∫ τ


=1

2π

∫ ∞

−∞

∫ τ

0[12

σ2G(ξ )Sr −12

σ2iωF(ξ )+(r−D0 −12

σ2)F(ξ )]

· eiω ln(Sr) · e−[ 12 σ2ω2+(r−D0− 1

2 σ2)iω+r](τ−ξ )dξ · e−iωxdω

=1

2π

∫ τ

0

∫ ∞

−∞

12

σ2G(ξ )Sr · eiω ln(Sr) · e−[ 12 σ2ω2+(r−D0− 1


+1

2π

∫ τ

0

∫ ∞

−∞[(r−D0 −

12

σ2)− 12

σ2iω]F(ξ ) · eiω ln(Sr)

· e−[ 12 σ2ω2+(r−D0− 1


=∫ τ

0

12

σ2G(ξ )Sr ·e−r(τ−ξ )

2π

∫ ∞

−∞e−

12 σ2(τ−ξ )ω2−[(r−D0− 1

2 σ2)(τ−ξ )+x−ln(Sr)]iωdωdξ

+∫ τ

0F(ξ ) · e−r(τ−ξ )

2π

∫ ∞

−∞[(r−D0 −

12

σ2)− 12

σ2iω]


· e−12 σ2(τ−ξ )ω2−[(r−D0− 1

2 σ2)(τ−ξ )+x−ln(Sr)]iωdωdξ

=∫ τ

0

12

σ2G(ξ )Sr ·e−r(τ−ξ )

2π· e−

[(r−D0−12 σ2)(τ−ξ )+x−ln(Sr)]2

2σ2(τ−ξ ) ·

√2π

σ2(τ −ξ )dξ

+∫ τ

0F(ξ ) · e−r(τ−ξ )

2π· e−

[(r−D0−12 σ2)(τ−ξ )+x−ln(Sr)]2

2σ2(τ−ξ ) ·

√2π

σ2(τ −ξ )

· (r−D0 −12

σ2)+12

σ2i2(r−D0 − 1

2σ2)(τ −ξ )+ x− ln(Sr)

σ2(τ −ξ )dξ

=∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+x−ln(Sr)]2

2σ2(τ−ξ ) dξ

+∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+x−ln(Sr)]2

2σ2(τ−ξ )

· (r−D0 −12

σ2)−(r−D0 − 1

2σ2)(τ −ξ )+ x− ln(Sr)

2(τ −ξ )dξ

=∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+x−ln(Sr)]2

2σ2(τ−ξ ) dξ

+∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+x−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− x+ ln(Sr)

2(τ −ξ )dξ , (C.17)

where a simple identity is employed in the derivation process∫ ∞

−∞e−pω2−qω ·ωndω = (−1)n

√πp

∂ n

∂qn eq24p . (C.18)

Hence, we can finally arrive at the integral equation representations, (5.17), if weuse the original variable S to replace x.

C.3 Appendix C.3Firstly, we rewrite Equation (5.17) as

V (S,τ) =e−rτ

√2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+ln(S)−u]2


+∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√2πσ2(τ −ξ )

· e−[(r−D0−

12 σ2)(τ−ξ )+ln(S)−ln(Sc(ξ ))]2

2σ2(τ−ξ )


+(r−D0 +

12σ2)(τ −ξ )− ln(S)+ ln(Sc(ξ ))

2(τ −ξ )dξ


−∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ ) dξ

−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− ln(S)+ ln(Sr)

2(τ −ξ )dξ

, e−rτ√

2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+ln(S)−u]2


−∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ ) dξ

−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(S)−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− ln(S)+ ln(Sr)

2(τ −ξ )dξ

+I, (C.19)

where

I =∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√2πσ2(τ −ξ )

· e−[(r−D0−

12 σ2)(τ−ξ )+ln(S)−ln(Sc(ξ ))]2

2σ2(τ−ξ )


+(r−D0 +

12σ2)(τ −ξ )− ln(S)+ ln(Sc(ξ ))

2(τ −ξ )dξ . (C.20)

This demonstrates that the remaining task is to work out I. If we define

h(ξ ) ,[(r−D0 − 1

2σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))]2

2σ2(τ −ξ )

=1

2(τ −ξ )[(r−D0 − 1

2σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))σ

]2

=1

2(τ −ξ )[(r−D0 − 1

2σ2)τ + ln(S)σ

−(r−D0 − 1


]2

, 12(τ −ξ )

[y−P(ξ )]2, (C.21)

where y=(r−D0 − 1

2σ2)τ + ln(S)σ

and P(ξ )=(r−D0 − 1


, and notic-

ing the fact that P′(ξ ) =1σ[r−D0 −

12

σ2 +S′c(ξ )Sc(ξ )

], we can obtain

I =∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√2πσ2(τ −ξ )

· e−[(r−D0−

12 σ2)(τ−ξ )+ln(S)−ln(Sc(ξ ))]2

2σ2(τ−ξ )



+(r−D0 +

12σ2)(τ −ξ )− ln(S)+ ln(Sc(ξ ))

2(τ −ξ )dξ

=∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√2π(τ −ξ )

· e−(y−P(ξ ))2

2(τ−ξ )

· P′(ξ )−r−D0 − 1

2σ2

σ+

(r−D0 +12σ2)(τ −ξ )− ln(S)+ ln(Sc(ξ ))

2σ(τ −ξ )dξ

=∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√2π(τ −ξ )

· e−(y−P(ξ ))2

2(τ−ξ )

· P′(ξ )−(r−D0 − 1

2σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))2σ(τ −ξ )

+σ2dξ

=∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√2π(τ −ξ )

· e−(y−P(ξ ))2

2(τ−ξ ) · P′(ξ )− y−P(ξ )2(τ −ξ )

+σ2dξ

=∫ τ

0n1Sc(ξ ) ·

e−r(τ−ξ )√

2π· e−

(y−P(ξ ))22(τ−ξ )

· (P′(ξ )+σ)

√τ −ξ − (y−P(ξ )+σ(τ −ξ )) 1

2√

τ−ξ

(√

τ −ξ )2dξ

=∫ τ

0−n1Sc(ξ ) ·

e−r(τ−ξ )√

2π· e

2(y−P(ξ ))σ(τ−ξ )+σ2(τ−ξ )22(τ−ξ )

· e−(y−P(ξ )+σ(τ−ξ ))2

2(τ−ξ ) · ∂∂ξ

(y−P(ξ )+σ(τ −ξ )√

τ −ξ)dξ

=∫ τ

0−n1Sc(ξ ) · e−r(τ−ξ ) · e(r−D0)(τ−ξ )+ln(S)−ln(Sc(ξ ))

· ∂∂ξ

N ((r−D0 +

12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

=∫ τ

0−n1S · e−D0(τ−ξ ) · ∂

∂ξN (

(r−D0 +12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ .(C.22)

Using the integration by parts yields

I =∫ τ

0D0n1Se−D0(τ−ξ ) · ∂

∂ξN (

(r−D0 +12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

−n1Se−D0(τ−ξ ) ·N ((r−D0 +

12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))√

σ2(τ −ξ ))|τ0

=∫ τ

0D0n1Se−D0(τ−ξ ) · ∂

∂ξN (

(r−D0 +12σ2)(τ −ξ )+ ln(S)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

−n1S ·LS=Sc(τ)+n1Se−D0τ ·N ((r−D0 +

12σ2)τ + ln(S)− ln(Sc(τ))√

σ2τ), (C.23)


where

LS=Sc(τ) =

0, S < Sc(τ),12 , S = Sc(τ).

(C.24)

Here, N (·) is the cumulative distribution function of the standard normal distribu-tion. This has completed the proof.

C.4 Appendix C.4To prove Proposition 5.3.1, we start with the PDE system (5.7) and make thetransform τ = T − t, which yields

−∂V∂τ

+12

σ2S2 ∂ 2V∂S2 +(r−D0)S

∂V∂S

− rV = 0, (C.25)

with the initial condition as V (S,0) = maxn1S,Z. Considering the continuity ofV (S,τ) and V (S,0) = maxn1S,Z, for any S1 < Z/n1, there always exists a smallnumber δ > 0, such that

V (S1,τ)> n1S1, ∀ τ < δ , (C.26)

which further leads toSc(0) = lim

τ→0+Sc(τ)> S1. (C.27)

As a result, the arbitrariness of S1 gives

Sc(0)≥ Z/n1. (C.28)

On the other hand, if we assume Sc(0)> Z/n1, there exists S such that Sc(0)> S >

Z/n1. This implies V (S,0) = n1S, as a result of n1S > Z, and thus

∂V∂τ

(S,0) =−D0n1S < 0, (C.29)

after the substitution of V (S,0) into (C.25). In this case, there exists τ > 0 beingsmall enough such that

V (S,τ)< n1S, (C.30)

which contradicts to V ≥ n1S. Therefore,

Sc(0)≤ Z/n1. (C.31)

Combining Equation (C.28) and (C.31) yields the desired result.


C.5 Appendix C.5To prove Proposition 5.3.2, we only need to show that there exists τ∗, s.t. G(τ)< 0

for any 0 < τ < τ∗, as G(τ) it is the first-order derivative of the bond price, ∂V∂S

(S,τ),at S = Sr.

Recall Equation (5.21)

12

F(τ) =e−rτ

√2πσ2τ

∫ ln(Sc(0))

ln(Sr)e−

[(r−D0−12 σ2)τ+ln(Sr)−u]2


+n1Sr · e−D0τ ·N ((r−D0 +

12σ2)τ + ln(Sr)− ln(Sc(0))√

σ2τ)

+∫ τ

0D0n1Sr · e−D0(τ−ξ ) ·N (

(r−D0 +12σ2)(τ −ξ )+ ln(Sr)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

−∫ τ

0

12


2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(Sr)−ln(Sr)]2

2σ2(τ−ξ ) dξ

−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

[(r−D0−12 σ2)(τ−ξ )+ln(Sr)−ln(Sr)]2

2σ2(τ−ξ )

·(r−D0 − 1

2σ2)(τ −ξ )− ln(Sr)+ ln(Sr)

2(τ −ξ )dξ . (C.32)

With Sc(τ)|τ=0 = Sr, the first term involved in the above equation is eliminated. Ifwe rearrange the equation by moving the integral involving G(τ) to the left handside while moving all the other terms to the right hand side, we can obtain

∫ τ

0

12


2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 dξ

= n1Sr · e−D0τ ·N ((r−D0 +

12σ2)

√τ

σ)

+∫ τ

0D0n1Sr · e−D0(τ−ξ ) ·N (

(r−D0 +12σ2)(τ −ξ )+ ln(Sr)− ln(Sc(ξ ))√

σ2(τ −ξ ))dξ

−12

F(τ)−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 ·

r−D0 − 12σ2

2dξ .(C.33)

It should be noted that the pricing domain is the range of the underlying asset of theresettable bond before conversion or reset takes place; once either of these actionshas taken place, the bond value is known already. Thus, we have Sr ≤ S ≤ Sc(τ), orsimply Sc(τ)≥ Sr, from which it is not difficult to obtain

∫ τ

0

12


2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 dξ


≤ I − 12

F(τ)−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 ·

r−D0 − 12σ2

2dξ ,(C.34)

where

I = n1Sr ·e−D0τ ·N ((r−D0 +

12σ2)

√τ

σ)+

∫ τ

0D0n1Sr ·e−D0(τ−ξ ) ·N (

(r−D0 +12σ2)

√τ −ξ

σ)dξ .

In fact, by making use of the integration by parts, I can be computed through

I = n1Sr · e−D0τ ·N ((r−D0 +

12σ2)

√τ

σ)+n1Sr · e−D0(τ−ξ ) ·N (

(r−D0 +12σ2)

√τ −ξ

σ)|τ0

+∫ τ

0

n1Sr · e−D0(τ−ξ )

2√

2π(τ −ξ )e−

(r−D0+12 σ2)2(τ−ξ )2σ2 ·

r−D0 +12σ2

σdξ

=12

n1Sr +∫ τ

0


2√

2π(τ −ξ )e−

(r−D0+12 σ2)2(τ−ξ )2σ2 ·

r−D0 +12σ2

σdξ . (C.35)

Thus, Equation (C.34) can further lead to

∫ τ

0

12


2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 dξ

≤ 12

n1Sr +∫ τ

0


2√

2π(τ −ξ )· e−

(r−D0+12 σ2)2(τ−ξ )2σ2 ·

r−D0 +12σ2

σdξ

−12

F(τ)−∫ τ

0F(ξ ) · e−r(τ−ξ )√

2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 ·

r−D0 − 12σ2

2dξ .

(C.36)

Now, if we make the transformation of x =√

τ −ξ for the right hand side, we obtain

∫ τ

0

12


2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 dξ

≤ 12

n1Sr +∫ √

τ

0

n1Sr · e−D0x2

√2π

· e−(r−D0+

12 σ2)2x2

2σ2 ·r−D0 +

12σ2

σdx

−12

F(τ)−∫ √

τ

0

F(τ − x2) · e−rx2

√2π

· e−(r−D0−

12 σ2)2x2

2σ2 ·r−D0 − 1

2σ2

σdx

≤ 12

n1Sr +∫ √

τ

0

n1Sr · e−D0x2

√2π

· e−(r−D0+

12 σ2)2x2

2σ2 · |r−D0 +

12σ2

σ|dx

−12

F(τ)+∫ √

τ

0

F(τ − x2) · e−rx2

√2π

· e−(r−D0−

12 σ2)2x2

2σ2 · |r−D0 − 1

2σ2

σ|dx

≤ 12

n1Sr +n1Sr ·√

τ√2π

· |r−D0 +

12σ2

σ|


−12

F(τ)+maxF(ξ )|ξ ∈ [0,τ] ·√

τ√2π

· |r−D0 − 1

2σ2

σ|, (C.37)

From the definition of the function F(τ), we know that F(τ)≥ n2Sr > n1Sr holds forany τ , and thus we have

∫ τ

0

12


2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 dξ

<12(n1 −n2)Sr +maxF(ξ )|ξ ∈ [0,τ] 2

√τ√

2π·max|

r−D0 − 12σ2

σ|, |

r−D0 +12σ2

σ|.

Considering the fact that there exists τ0, s.t.

√τ ·maxF(ξ )|ξ ∈ [0,τ]<

√π(n2 −n1)Sr

2√

2max| r−D0− 12 σ2

σ |, | r−D0+12 σ2

σ |(C.38)

for any 0 < τ < τ0, it is straightforward that

∫ τ

0

12


2πσ2(τ −ξ )· e−

(r−D0−12 σ2)2(τ−ξ )2σ2 dξ < 0 (C.39)

for any 0 < τ < τ0. Therefore, one can certainly reach the conclusion that thereexists 0 < τ∗ ≤ τ0, s.t. G(τ)< 0 for any 0 < τ < τ∗. This has completed the proof.

Appendix D


D.1 Appendix D.1Recall the FDE in our system

∂V (n)i, j

∂τ= a jδxxV

(n)i, j +b jδvvV

(n)i, j + c jδxvV

(n)i, j +[d j +λ j]δxV

(n)i, j + e jδvV

(n)i, j − rV (n)

i, j , (D.1)

with the implicit Euler scheme applied to the time derivative function, we can obtain

V (n+1)i, j −V (n)

i, j

∆τ= a jδxxV

(n+1)i, j +b jδvvV

(n+1)i, j +c jδxvV

(n+1)i, j +[d j+λ j]δxV

(n+1)i, j +e jδvV

(n+1)i, j −rV (n+1)

i, j .

(D.2)If we define

A0 = ∆τ · c jδxv, (D.3)A1 = ∆τ[a jδxx +(d j +λ j)δx −

r2I], (D.4)

A2 = ∆τ[b jδvv + e jδv −r2I], (D.5)

then the PDE can be derived

[I− (A0 +A1 +A2)]V(n+1)i, j =V (n)

i, j +O((∆τ)2). (D.6)

Similarly, if the explicit Euler scheme used instead of the implicit one, we obtain

[I+(A0 +A1 +A2)]V(n)i, j =V (n+1)

i, j +O((∆τ)2). (D.7)

Therefore, the weighted average of these two scheme can be displayed

[I−ϕ(A0 +A1 +A2)]V(n+1)i, j = [I+(1−ϕ)(A0 +A1 +A2)]V

(n)i, j +O((∆τ)2). (D.8)

198

APPENDIX D. APPENDIX FOR CHAPTER 6 199

It should be noted that when ϕ equals to zero and one, the above equation is assame as the explicit Euler scheme and the implicit Euler scheme, respectively. Inaddition, the Crank-Nicolson scheme is derived when ϕ equals to 1/2.

Now, adding ϕ 2A1A2V (n+1)i, j to both sides of the above equation

[I−ϕA0 −ϕA1 −ϕA2 +ϕ 2A1A2]V(n+1)i, j

= [I+(1−ϕ)A0 +(1−ϕ)A1 +(1−ϕ)A2]V(n)i, j +ϕ 2A1A2V (n+1)

i, j +O((∆τ)2)

⇒ [I−ϕA0 −ϕA1 −ϕA2 +ϕ 2A1A2]V(n+1)i, j

= [I+(1−ϕ)A0 +(1−ϕ)A1 +(1−ϕ)A2 +ϕ 2A1A2]V(n)i, j +ϕ 2A1A2(V

(n+1)i, j −V (n)

i, j )+O((∆τ)2)

⇒ [I−ϕA0 −ϕA1 −ϕA2 +ϕ 2A1A2]V(n+1)i, j

= [I+(1−ϕ)A0 +(1−ϕ)A1 +(1−ϕ)A2 +ϕ 2A1A2]V(n)i, j +O((∆τ)2)

⇒ [I−ϕA1 −ϕA2 +ϕ 2A1A2]V(n+1)i, j

= [I+A0 +(1−ϕ)A1 +(1−ϕ)A2 +ϕ 2A1A2]V(n)i, j +ϕA0(V

(n+1)i, j −V (n)

i, j )+O((∆τ)2)

⇒ [I−ϕA1 −ϕA2 +ϕ 2A1A2]V(n+1)i, j

= [I+A0 +(1−ϕ)A1 +(1−ϕ)A2 +ϕ 2A1A2]V(n)i, j +O((∆τ)2)

⇒ (I−ϕA1)(I−ϕA2)V(n+1)i, j

= [I+A0 +(1−ϕ)A1 +A2]V(n)i, j − (I−ϕA1)ϕA2V (n)

i, j , (D.9)

where two mergers appear since ϕ 2A1A2(V(n+1)i, j −V (n)

i, j )∼O((∆τ)3) and ϕA0(V(n+1)i, j −

V (n)i, j )∼ O((∆τ)2).In summary, the linear operators A0, A1 and A2 at the (n+1)th time step are

A0 = ∆τ · c jδxv

= ∆τ · [ρσv j −σ2v jξ j]δxv, (D.10)A1 = ∆τ[a jδxx +(d j +λ j)δx −

r2I]

= ∆τ[(12

v j +12

σ2v jξ 2j −ρσv jξ j)δxx

+(−12

v j +12

σ2v jξ j −12

σ2v jβ j + r−D0 −κ(η − v j)ξ j +λ j)δx −r2I], (D.11)

A2 = ∆τ[b jδvv + e jδv −r2I]

= ∆τ[12

σ2v jδvv +κ(η − v j)δv −r2I], (D.12)

where

ξ j =δv(ϕS(n+1)

f ( j)+(1−ϕ)S(n)f ( j))

ϕS(n+1)f ( j)+(1−ϕ)S(n)f ( j)

, (D.13)


β j =δvv(ϕS(n+1)

f ( j)+(1−ϕ)S(n)f ( j))

ϕS(n+1)f ( j)+(1−ϕ)S(n)f ( j)

, (D.14)

λ j =S(n+1)

f ( j)−S(n)f ( j)

(ϕS(n+1)f ( j)+(1−ϕ)S(n)f ( j))∆τ

. (D.15)

D.2 Appendix D.2AYj = Pj +Bx j, (D.16)

with

A=

1+ϕ( 2a j∆τ(∆x)2 + r∆τ

2 ) −ϕ( a j∆τ(∆x)2 +

d′j∆τ2∆x )

−ϕ( a j∆τ(∆x)2 −

d′j∆τ2∆x ) 1+ϕ( 2a j∆τ

(∆x)2 + r∆τ2 ) −ϕ( a j∆τ

(∆x)2 +d′j∆τ2∆x )

. . . . . . . . .

−ϕ( a j∆τ(∆x)2 −

d′j∆τ2∆x ) 1+ϕ( 2a j∆τ

(∆x)2 + r∆τ2 ) −ϕ( a j∆τ

(∆x)2 +d′j∆τ2∆x )

−ϕ( a j∆τ(∆x)2 −

d′j∆τ2∆x ) 1+ϕ( 2a j∆τ

(∆x)2 + r∆τ2 )

Nx−1×Nx−1

,

(D.17)Yj = (Y1, j,Y2, j, · · · ,YNx−1, j)

T , (D.18)Pj = (P1, j,P1, j, · · · ,PNx−1, j)

T , (D.19)

Bx j =

ϕ( a j∆τ

(∆x)2 −d′

j∆τ2∆x )Y0, j

...ϕ( a j∆τ

(∆x)2 +d′

j∆τ2∆x )YNx, j

Nx−1×1

, (D.20)

Pi, j = [I+A0 +(1−ϕ)A1 +A2]V(n)i, j

=V (n)i, j +

c j∆τ4∆x∆v

(V (n)i+1, j+1 −V (n)

i+1, j−1 −V (n)i−1, j+1 +V (n)

i−1, j−1)

+(1+ϕ)(a j∆τ(∆x)2 −

d′j∆τ

2∆x)V (n)

i−1, j +(1−ϕ)(−a j∆τ(∆x)2 −

r∆τ2

)V (n)i, j

+(1−ϕ)(a j∆τ(∆x)2 +

d′j∆τ

2∆x)V (n)

i+1, j +(b j∆τ(∆v)2 −

e j∆τ2∆v

)V (n)i, j−1

+(−b j∆τ(∆v)2 −

r∆τ2

)V (n)i, j +(

b j∆τ(∆v)2 +

e j∆τ2∆v

)V (n)i, j+1, (D.21)

where d′j = d j +λ j.

D.3 Appendix D.3CV (n+1)

i = Qi +Bvi, (D.22)


with

C=

1+ϕ( 2b j∆τ

(∆v)2 + r∆τ2 ) −ϕ( b j∆τ

(∆v)2 +e j∆τ2∆v )

−ϕ( b j∆(∆v)2 −

e j∆τ2∆v ) 1+ϕ( 2b j∆τ

(∆v)2 + r∆τ2 ) −ϕ( b j∆τ

(∆v)2 +e j∆τ2∆v )

. . . . . . . . .−ϕ( b j∆

(∆v)2 −e j∆τ2∆v ) 1+ϕ( 2b j∆τ

(∆v)2 + r∆τ2 ) −ϕ( b j∆τ

(∆v)2 +e j∆τ2∆v )

−ϕ( b j∆(∆v)2 −

e j∆τ2∆v ) 1+ϕ( 2b j∆τ

(∆v)2 + r∆τ2 )

Nv×Nv

,

(D.23)V (n+1)

i = (V (n+1)i,1 ,V (n+1)

i,2 , · · · ,V (n+1)i,Nv−1)

T , (D.24)

Q(n+1)i = (qi,1,qi,2, · · · ,qi,Nv−1)

T , (D.25)

Bvi =

ϕ(b j∆τ

∆v − e j∆τ2∆v )V

(n+1)i,0

...ϕ( b j∆τ

(∆v)2 +e j∆τ2∆v )V

(n+1)i,Nv

Nv×1

, (D.26)

qi, j = Yi, j −ϕ(b j∆τ(∆v)2 −

e j∆τ2∆v

)V (n)i, j−1 −ϕ(−2

b j∆τ(∆v)2 −

r∆τ2

)V (n)i, j −ϕ(

b j∆τ(∆v)2 +

e j∆τ2∆v

)V (n)i, j+1.

(D.27)

D.4 Appendix D.4

A0 = ∆τ · c jδxr

= ∆τ · [σρξ√r j −ξ 2r jζ j]δxr, (D.28)A1 = ∆τ[a jδxx +(d j +λ j)δx −

r j

2I]

= ∆τ[(12

σ2 +12

ξ 2r jζ 2j −σρξ√r jζ j)δxx

+(−12

σ2 +12

ξ 2r jζ 2j −

12

ξ 2r jβ + r j −D0 −κ(η − r j)ζ j +λ j)δx −r j

2I],(D.29)

A2 = ∆τ[b jδrr + e jδr −r j

2I]

= ∆τ[12

ξ 2r jδrr +κ(η − r j)δv −r j

2I], (D.30)

where

ζ j =δr(ϕS(n+1)

c ( j)+(1−ϕ)S(n)c ( j))

ϕS(n+1)c ( j)+(1−ϕ)S(n)c ( j)

, (D.31)

β j =δrr(ϕS(n+1)

c ( j)+(1−ϕ)S(n)c ( j))

ϕS(n+1)c ( j)+(1−ϕ)S(n)c ( j)

, (D.32)

λ j =S(n+1)

c ( j)−S(n)c ( j)

(ϕS(n+1)c ( j)+(1−ϕ)S(n)c ( j))∆τ

. (D.33)


D.5 Appendix D.5AYj = Pj +Bx j, (D.34)

with

A=

1+ϕ( 2a j∆τ(∆x)2 + r∆τ

2 ) −ϕ( a j∆τ(∆x)2 +

d′j∆τ2∆x )

−ϕ( a j∆τ(∆x)2 −

d′j∆τ2∆x ) 1+ϕ( 2a j∆τ

(∆x)2 + r∆τ2 ) −ϕ( a j∆τ

(∆x)2 +d′j∆τ2∆x )

. . . . . . . . .

−ϕ( a j∆τ(∆x)2 −

d′j∆τ2∆x ) 1+ϕ( 2a j∆τ

(∆x)2 + r∆τ2 ) −ϕ( a j∆τ

(∆x)2 +d′j∆τ2∆x )

−ϕ( a j∆τ(∆x)2 −

d′j∆τ2∆x ) 1+ϕ( 2a j∆τ

(∆x)2 + r∆τ2 )

Nx−1×Nx−1

,

(D.35)Yj = (Y1, j,Y2, j, · · · ,YNx−1, j)

T , (D.36)Pj = (P1, j,P1, j, · · · ,PNx−1, j)

T , (D.37)

Bx j =

ϕ( a j∆τ

(∆x)2 −d′

j∆τ2∆x )Y0, j

...ϕ( a j∆τ

(∆x)2 +d′

j∆τ2∆x )YNx, j

Nx−1×1

, (D.38)

Pi, j = [I+A0 +(1−ϕ)A1 +A2]V(n)i, j

=V (n)i, j +

c j∆τ4∆x∆r

(V (n)i+1, j+1 −V (n)

i+1, j−1 −V (n)i−1, j+1 +V (n)

i−1, j−1)

+(1+ϕ)(a j∆τ(∆x)2 −

d′j∆τ

2∆x)V (n)

i−1, j +(1−ϕ)(−a j∆τ(∆x)2 −

r∆τ2

)V (n)i, j

+(1−ϕ)(a j∆τ(∆x)2 +

d′j∆τ

2∆x)V (n)

i+1, j +(b j∆τ(∆r)2 −

e j∆τ2∆r

)V (n)i, j−1

+(−b j∆τ(∆r)2 −

r∆τ2

)V (n)i, j +(

b j∆τ(∆r)2 +

e j∆τ2∆r

)V (n)i, j+1, (D.39)

where d′j = d j +λ j.

D.6 Appendix D.6CV (n+1)

i = Qi +Bri, (D.40)

with

C=

1+ϕ( 2b j∆τ

(∆r)2 + r∆τ2 ) −ϕ( b j∆τ

(∆r)2 +e j∆τ2∆r )

−ϕ( b j∆(∆r)2 −

e j∆τ2∆r ) 1+ϕ( 2b j∆τ

(∆r)2 + r∆τ2 ) −ϕ( b j∆τ

(∆r)2 +e j∆τ2∆r )

. . . . . . . . .−ϕ( b j∆

(∆r)2 −e j∆τ2∆r ) 1+ϕ( 2b j∆τ

(∆r)2 + r∆τ2 ) −ϕ( b j∆τ

(∆r)2 +e j∆τ2∆r )

−ϕ( b j∆(∆r)2 −

e j∆τ2∆r ) 1+ϕ( 2b j∆τ

(∆r)2 + r∆τ2 )

Nr×Nr

,


(D.41)V (n+1)

i = (V (n+1)i,1 ,V (n+1)

i,2 , · · · ,V (n+1)i,Nr−1)

T , (D.42)

Q(n+1)i = (qi,1,qi,2, · · · ,qi,Nr−1)

T , (D.43)

Bvi =

ϕ(b j∆τ

∆r − e j∆τ2∆r )V

(n+1)i,0

...ϕ( b j∆τ

(∆r)2 +e j∆τ2∆r )V

(n+1)i,Nr

Nr×1

, (D.44)

qi, j = Yi, j −ϕ(b j∆τ(∆r)2 −

e j∆τ2∆r

)V (n)i, j−1 −ϕ(−2

b j∆τ(∆r)2 −

r∆τ2

)V (n)i, j −ϕ(

b j∆τ(∆r)2 +

e j∆τ2∆r

)V (n)i, j+1.

(D.45)

A study on quantitatively pricing various convertible bonds

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