Pricing FX Quanto Options under Stochastic...

Pricing FX Quanto Options underStochastic Volatility

A dissertation submitted to theWARWICK BUSINESS SCHOOLUNIVERSITY OF WARWICK

in partial fulfillment of the requirements for the degree ofMSc in Financial Mathematics

submitted byTANMOY NEOG (0850324)

supervised byProf. Dr. Nick Webber

7 September, 2009

All the work contained is my own unaided effort and conforms to the University guidelines on

plagiarism

Acknowledgment

I would like to begin by thanking my supervisor Prof. Nick Webber. I thank him for his patient

guidance and his enthusiasm to answer my questions. This dissertation introduced me to the

intricacies of derivative pricing. It was the enthusiasm of my supervisor that gave me the impetus

to try to improve my results. Hopefully I did not do very badly. I also thank the WBS authorities

who ensured that we had the adequate facilities to work efficiently.

I thank all the faculty members involved with the Financial Mathematics course. I could learn a

lot from the rigour involved in this course. I thank my batchmates Vineet Thakkar, Zenon, Ravi

Ganesan and Piyush Singh for several discussions related to Financial Mathematics. Coming

from a background where I had no knowledge of Computational Finance; these individuals

helped me to learn a lot of things in lesser time than I would have taken. Last but not the least

special thanks to my dear friend Vallu with whom there was never a dull moment.

ii

Contents

List of Figures vi

List of Tables viii

Abstract x

1 Literature Review 4

1.1 Bennett and Kennedy’s methodology for pricing the FX Quanto Option . . . . . 4

1.2 Pricing the FX quanto option under different frameworks . . . . . . . . . . . . . 7

2 Pricing FX Quanto Options in the Black Scholes Framework 14

2.1 The standard market practice in pricing FX Quanto Options . . . . . . . . . . . 14

2.1.1 Pricing using the Black Scholes Model . . . . . . . . . . . . . . . . . . . . 15

3 Pricing of FX Quanto under the Heston Model 17

3.1 Revisiting the Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Option Pricing under the Heston model . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 The Heston Pricer using Fast Fourier Transform . . . . . . . . . . . . . . 19

3.2.2 The Heston Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 The Heston model with jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

iii

3.4 Pricing the FX Quanto option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Pricing of FX Quanto under the GARCH Option Pricing Model 26

4.1 Duan’s GARCH Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Analytical Approximation of the GARCH Option Pricing Model . . . . . . . . . 28

5 Modelling the Dependence Structure Using a Copula 31

5.1 Archimedean Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Archimedean Copulas and their Measures of Dependence . . . . . . . . . 33

5.1.2 Identifying the Right Copula from the Archimedean Family to Model the

Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 The T Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.1 Calibration of the T Copula . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3 Pricing using the copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Data 41

6.1 Discount Factors, Implied Volatility quotes . . . . . . . . . . . . . . . . . . . . . 41

6.2 Recovering Strikes and Prices of FX vanilla options . . . . . . . . . . . . . . . . . 42

7 Implementation of Pricing Methods 45

7.1 Implementation of the Heston Stochastic Volatility Model with Jumps . . . . . . 45

7.1.1 Calibration of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.1.2 Finding Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.2 Implementation of the GARCH Option Pricing Model . . . . . . . . . . . . . . . 50

7.2.1 Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.3 Choosing a model to price the Quanto option . . . . . . . . . . . . . . . . . . . . 56

7.4 Results of pricing the quanto option . . . . . . . . . . . . . . . . . . . . . . . . . 56

iv

7.5 Implementing a copula to price the FX quanto option . . . . . . . . . . . . . . . 58

7.5.1 Parameter estimation for the T Copula . . . . . . . . . . . . . . . . . . . 59

7.5.2 Parameter estimation for the Frank Copula . . . . . . . . . . . . . . . . . 60

7.5.3 Calibration of marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.5.4 Numerical Results for pricing using copula . . . . . . . . . . . . . . . . . . 62

7.6 Monte Carlo Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 Conclusion 69

Bibliography 71

v

List of Figures

5.1 Frank Copula with ® = 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Clayton Copula with ® = 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Gumbel Copula with ® = 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.1 Time series observations of USD/JPY spot exchange rate . . . . . . . . . . . . . 46

7.2 Market and model volatility(in %) for USD/YEN call options with maturity 1

months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7.7 Market Implied Volatility Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.8 Heston with Jumps Implied Volatility Surface . . . . . . . . . . . . . . . . . . . . 53

7.9 Canonical Log likelihood function (Mashal and Zeevi) . . . . . . . . . . . . . . . 60

7.10 Market and model volatility (in %) for EUR/USD call options with maturity 1

months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

vi

7.11 Market and model volatility(in %) for EUR/USD call options with maturity 6

months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62


months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63


years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.14 Market and model volatility(in %) for EUR/JPY call options with maturity 1

months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.17 Market and model volatility(in %) for EUR/JPY call options with maturity 2 years 65

7.18 Frank Copula with ® = 0.0075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.19 T Copula with º = 14 and ½ = 0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 66

vii

List of Tables

5.1 Archimedean Copulas and their Generators . . . . . . . . . . . . . . . . . . . . . 33

5.2 Archimedean Copulas and Measures of Dependence . . . . . . . . . . . . . . . . . 34

6.1 Discount factors Di0,T corresponding to currency i and maturity T . . . . . . . . 42

6.2 EUR/USD implied volatility quotes of call options corresponding to standard

values of Black Delta and maturity T . . . . . . . . . . . . . . . . . . . . . . . . 42

6.3 EUR/JPY implied volatility quotes of call options corresponding to standard


6.4 USD/JPY implied volatility quotes of call options corresponding to standard


7.1 Parameters of Heston with jumps calibration on USD/JPY rates . . . . . . . . . 48

7.2 Parameters of GARCH calibration on USD/JPY rates . . . . . . . . . . . . . . . 53

7.3 Implied volatilities for maturity of 1 month . . . . . . . . . . . . . . . . . . . . . 54

7.4 Implied volatilities for maturity of 3 months . . . . . . . . . . . . . . . . . . . . . 54

7.5 Implied volatilities for maturity of 6 months . . . . . . . . . . . . . . . . . . . . . 55

7.6 Implied volatilities for maturity of 1 year . . . . . . . . . . . . . . . . . . . . . . 55

7.7 Implied volatilities for maturity of 2 years . . . . . . . . . . . . . . . . . . . . . . 55

7.8 Root mean squared errors for the implied volatility fits on USD/JPY FX rate . . 56

viii

7.9 Quanto prices for maturity of 1 month. Spot price is 0.0083 dollars. All the prices

are in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . . . . . . 57

7.10 Quanto prices for maturity of 3 months.Spot price is 0.0083 dollars. All the prices


7.11 Quanto prices for maturity of 6 months. Spot price is 0.0083 dollars. All the

prices are in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . . 58

7.12 Quanto prices for maturity of 1 year. Spot price is 0.0083 dollars. All the call

prices are in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . . 58

7.13 Quanto prices for maturity of 2 year. Spot price is 0.0083 dollars. All the prices


7.14 Parameters for the T copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.15 Parameters for the Frank copula . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.16 Parameters for EUR/USD and EUR/JPY after calibration . . . . . . . . . . . . 61

7.17 Root mean squared errors for the implied volatility fits on EUR/USD and EUR/JPY

FX rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.18 Relative Difference (RD)with Black Scholes for the models for maturity of 1

month. Values have been rounded off. . . . . . . . . . . . . . . . . . . . . . . . . 66

7.19 Relative Difference(RD)with Black Scholes for the models for maturity of 3 months.

Values have been rounded off. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.20 Relative Difference(RD)with Black Scholes for the models for maturity of 6 months.


7.21 Relative Difference(RD)with Black Scholes for the models for maturity of 1 year.


7.22 Average Standard Monte Carlo errors.Each error is an average of 25 simulations

for 25 different quanto options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

ix

Abstract

This dissertation looks at the pricing of FX quanto options using stochastic volatility models.

There are a lot of stochastic volatility models available. However we look only at the applicability

of the Heston model with jumps and the GARCH Option Pricing Model. By a no arbitrage

condition on FX rates we can view the FX quanto as a multi currency option. In a Black type

of model the pricing of the FX quanto depends on implied volatilities of three exchange rates.

This encourages the use of a copula function to model the dependency. We do so by pricing

the quanto with a T Copula; member of the elliptic family and the Frank Copula; a member of

the Archimedean Copula family with the marginals as Heston with jumps process. We observe

that under stochastic volatility there is a considerable difference in option prices from the Black

Scholes model. This is more so for options which have a low delta. Further given our data set

the Heston model with jumps fits the market behaviour for plain vanilla FX options better than

the GARCH Option Pricing model.

x

Introduction

In this dissertation we take up the problem of pricing a European style FX quanto option

under stochastic volatility. An FX quanto option has as its underlying an exchange rate with a

domestic and foreign currency. The payoff at maturity is converted into a third currency. This

third currency is called the quanto currency. An individual would buy a quanto call option if he

believes that the exchange rate would appreciate. Further he would expect the quanto currency

to appreciate more than the foreign currency for that particular exchange rate.

In the past various authors have priced these options in a non stochastic volatility framework.

The FX quanto is viewed as a multi asset option as volatility of the underlying exchange rate

is dependent on the volatilities of the quanto domestic and quanto foreign rates. We have a

closed form solution under the Black Scholes model for pricing FX quanto options. There have

been authors who have taken a different approach such as employing a copula to model the

dependence structure between exchange rates.(please refer to the literature review) The joint

distribution of asset prices is extracted from market implied volatilities. The price of the quanto

option is then evaluated as an integral involving the joint density of asset prices.

There have been expressions for quanto options under discrete time GARCH models which we

1

consider as well. While pricing a multi asset product like a quanto option one needs to take

into account multivariate models which can handle the co-movement of the underlying price

processes. The multivariate normal distribution is a very easy way of analyzing returns on

multiple assets. In a multivariate normal distribution the dependency between the margins is

measured by linear correlation. The actual association between the different assets may not

be so. The use of a copula is alternative measure of association between assets. The basic

idea of copulas is to separate the dependence structure between variables from their marginal

distributions.

Through the choice of copula, we can influence control on the association of certain parts of the

distribution.For instance at the tails. To give an example it may so happen that the returns of

two stocks might be correlated in the extreme tails, but not elsewhere in the distributions, and

there are copulas which can model this behaviour.

The aim of this dissertation would be to investigate as to how different quanto prices are under

a stochastic volatility framework. At first we do not incorporate a dependency between the

quanto, foreign and domestic rates. After this we do incorporate dependency through a copula.

It is unlikely that single copula family model can take care of the asymmetry of the underlying

asset process. If not then the answer could lie in perturbation of the copula family as used by

Bennett and Kennedy (2003) or mixture copulas.

I extend the standard FX quanto option pricing in two ways. First the marginals are assumed to

follow a Heston wih jumps process and the dependence is modelled using a copula. We choose a

t-copula from the elliptic family and the Frank copula from the Archimedean family. The choice

of a Frank copula is based on best fit to historical data.

2

When we model the individual FX rates as Geometric Brownian Motion as in the Black’s

formulae the log returns follow a Normal distribution. From an empirical point of view we

observe that the log returns of exchange rates we consider in this paper have a high excess

kurtosis of around 9. We need to replace GBM dynamics. Hence I take the marginals as non

linear GARCH processes. The GARCH process allows fatter tails. We use the Duan’s GARCH

Option Pricing Model for our purpose.

When using a pricing model it is important for the model to fit the market implied volatility

quotes for plain vanilla options. We observe that given our dataset the Heston with jumps model

gives us a much better fit to market quotes as compared to the GARCH model.

3

Chapter 1

Literature Review

Over the years valuation of contingent claims has been an area of extensive research. The

seminal work done by Black and Scholes (1973) and Cox et al. (1979) introduced us to risk

neutral pricing models. However there is a shortcoming of these models. The assumption of

a fixed volatility is violated in real time markets. The problem of pricing FX quanto options

in a non stochastic volatility model has been elaborately explained by Bennett and Kennedy

(2003). This paper is like a starting point for our dissertation. We give a brief review of the

methodology adopted by the authors in pricing the FX quanto options.

1.1 Bennett and Kennedy’s methodology for pricing the FX

Quanto Option

Benett and Kennedy express the payoff of a plain vanilla FX option in terms of two other

exchange rates. This is obtained from the triangular no arbitrage condition for currencies. The

4

price of this European multi asset option is written as an integral involving the joint density

of the asset prices at expiry. Hence the calculation of the price of the option involves the

integration of a joint density function of the two exchange rates. The authors then express the

joint density as a product of the marginal density functions of the individual exchange rates

and a copula function. This is done by using Sklar’s Theorem. The copula function determines

a joint density function of the two exchange rates. The use of the copula allows the authors

to separate the modelling of the marginal distributions from the modelling of the dependence

structure. The marginal densities are taken as mixture of lognormal distributions. The initial

calibration of marginal distributions to implied volatility quotes is done by a weighted non

linear least squares optimisation. This process of using option implied densities is similar to

Dupire (1994). A Gaussian copula is then used which is perturbed by a cubic spline to get a

dependence structure between the three currency pairs.The modification of the upper and lower

tail dependence characteristics through this perturbation is to allow calibration to a smile in

implied volatilities of the FX rates.

At the outset Bennett and Kennedy have considered a Gaussian copula. However to calibrate

the joint distribution to the implied volatility smile on the FX rates the dependence structure

associated with the Normal copula is perturbed. With this perturbation the tail dependence

characteristics are modified. The authors have used the following result which is due to Gen-

est(2000):

Theorem 1.1. Let ' : [0, 1] −→ [0, 1] be a continuous, twice differentiable concave function

such that '(0) = 0 and '(1) = 1. Then

C'(u, v) = '−1(C('(u), '(v))) (1.1)

5

is a copula if C(.,.) is a copula.

For Benett and Kennedy the starting point is a bivariate Normal distribution with correlation

parameter ½. A transformation function is then used to modify tail dependence. I will give a

review of how the authors obtained ' later. I proceed by giving a mathematical form of the

transformation function.

The Normal Copula has the density function as

CNormal(u, v∣½) = N½(N−1(u), N−1(v)) (1.2)

In equation (1.2) N½ denotes the standard bivariate normal distribution with correlation ½. N

denotes the standard univariate Normal distribution function. To obtain the density of the

transformed copula C' the partial derivatives of equation (1.1) with respect to both arguements

are taken. Bennett and Kennedy obtain c'(u, v), the density of the transformed copula as

c'(u, v) ='′(u)'′(v)'′(C'(u, v)

(c('(u), '(v))− '

′′(C'(u, v))['′(C'(u, v)]2

dC

du

dC

dv

)(1.3)

The final step which we can observe from equation (1.3) is to find the transformation function

'. From equation (1.1) we can say that the behaviour of '(x) near x = 1 controls the upper tail

dependence of the transformed copula. The lower tail dependence of the transformed copula can

be observed from the behaviour of '(x) near x = 0. The correlation ½ is fixed. The authors then

change the endpoints of the transformation which enables them to change the tail dependence

characteristics. Bennett and Kennedy observe that increasing the size of the second derivative

of ' at the end points can change the tail dependence characteristics. The authors specify ' as

a cubic spline with k + 1 suitable predefined knot points pi ∈ (−1, 1). The size of the second

derivative is an indicator of the increase in tail dependence. Hence the second derivative of the

6

spline at each knot point is first specified. The co-efficients of the spline are obtained by making

use of the requirements '(0) = 0 and '(1) = 1.Hence ' is found.

Hence if the knot points vector be denoted by p as p = (p0, p1, ......, pk)T with p0 = 0 and pk = 1.

The polynomial between the jtℎ and the (j + 1)tℎ nodes is given by

Sj(x) =3∑

i=0

ai,j(x− pj)i (1.4)

for x ∈ [pj , pj+1] and coefficients ai,j which are determined from an optimisation proceedure.

The cubic spline takes the value of the polynomial function Sj(x) between each of the knot

points. The final outcome is that ' is determined and the vanilla call options are priced. Once

the parameters from the plain vanilla calls are obtained; these parameters are then used to price

the quanto FX option.

The resulting quanto prices under a number of real scenarios is generally close to prices under

the Black formulae. The Black model often gives lower prices for the quanto call options and

higher for put options. The relative difference is occasionally large, that is in the region of 10

to 15 percent for standard strikes furthest away from at-the-money.

1.2 Pricing the FX quanto option under different frameworks

We have a standard approach to pricing quanto FX options which is based on the Black type

model. In this dissertation we will price the FX quanto option under two diffferent frameworks.

First we consider pricing the quanto in the continuous time Heston Stochastic Volatility Model

with jumps and then under the GARCH Option Pricing Model. Comparisons with the the Black

Type model is then done.

7

The shortcomings of the Black Scholes model in pricing foreign currency options has been

shown by Melino and Turnbull (1990) . It makes a strong assumption that returns are normally

distributed with known mean and variance. Stochastic volatility models have been used by Hull

and White (1989), Scott (1987). However the methodology adopted by them to price options

was computationally intensive in the absence of a closed form solution. Heston (1993) then

came up with a semi closed form solution to price European call options when the spot asset is

correlated with the volatility. We have a more elaborate discussion with the relevant equations

on the Heston model and the Heston model with jumps in Chapter 3.

Some other continuous time stochastic volatility models are the Stochastic Alpha Beta Rho

(SABR) model and the Variance Gamma model. The SABR model was developed by Hagan

et al. (2002). The inspiration behind the SABR model was that the behaviour of market smile

produced by local volatility models was opposite to the market. Hagan et al. (2002) say that due

to this discrepancy the delta and the vega hedges derived from local volatility models could be

unstable. In this two factor model the forward asset price F̂ and its volatility â are correlated.

dF̂ = âF̂ ¯dW1

dâ = ºâdW2

dW1dW2 = ½dt

Hagan et al. (2002) obtain the option prices from a perturbation technique and further provide

a closed form formula for the implied volatility as a function of the forward price F̂ and strike.

Madan et al. (1998) have used three parameter option pricing method termed as the Variance

Gamma (VG) Process. At any random time a Brownian motion with drift is evaluated by using

a gamma process. The drift of the Brownian motion and the volatility of time change form the

8

other two parameters.

In continuous time stochastic models it is difficult to obtain a volatility variable from a set of

discrete observations of spot prices. There has been a lot of research in estimating volatility in

continuous time models from historical observations. We can refer to Cvitanic et al. (2006).

This problem is not inherent in discrete time autoregressive conditional heteroscedastic models

(ARCH) introduced by Engle (1982) and Bollerslev (1986). Engle (1982) was successfully able

to model properties of assets reurns having properties as fat tails and time varying variances.

There have been many extensions to Engle (1982) like the exponential GARCH model, the

non linear asymmetric GARCH models among others. The first attempt to price European

style options under the GARCH framework was by Duan (1995). He introduced the local risk

neutral valuation relationship for univariate GARCH processes. Duan (1996) successfully used

his GARCH option pricing model to fit the volatility smile and the term structure of implied

volatilities. Chaudhury and Wei (1996) did a simulation study of comparing Duan’s GARCH

model with the Black Scholes model. They found out that the GARCH model has least error

with the Black Scholes model when pricing out of the money options with a maturity of 30 days

or less.

For our situation we are interested in the applicability of the GARCH framework to the pricing

of options on foreign assets. The first work in this direction was done by Duan and Wei (1999).

In this paper the authors have modelled the foreign exchange rate and the underlying asset

process as a bivariate nonlinear asymmetric GARCH process. For our purpose of pricing the FX

quanto option the underlying is also a foreign exchange rate. The pricing framework incorporates

stochastic volatility, unconditional leptokurtosis and a correlation between the lagged return and

9

the conditional variance for both the exchange rates in question.

The inherent problem with Duan’s GARCH Option Pricing Formulae was that one had to use

a Monte Carlo scheme to price the option.There were people who worked around this problem.

We know that the discounted asset process is a martingale under the risk neutral measure. Duan

and Simonato (1999) examine that in a Monte Carlo simulation this property is violated. They

propose a technique called as the Empirical Martingale Simulation (EMS). This imposes the

martingale property on the collection of simulated sample paths. The EMS method generates

asset prices at future times t1, t2, ....., tn using the following system

S̃i(tj , n) = S0Zi(tj , n)

Z0(tj , n)(1.5)

where

Zi(tj , n) = S̃i(tj−1, n)ˆSi(tj)

ˆSi(tj−1)(1.6)

and

Z0(tj , n) =1

ne−rtj

n∑

i=1

Zi(tj , n) (1.7)

Here ˆSi(t) denotes the ith simulated asset price at time t before the EMS adjustment. S̃i(t0, n),

ˆSi(t) are set equal to S0. The steps followed in the simulation are as follows

1. The return from time period tj−1 to tj i.e. fromˆSi(tj)ˆSi(tj−1)

is first simulated.

2. Equation 1.7 is used to calculate the temporary asset price Zi(tj , n)

3. Equation 1.8 is used to calculate the discounted sample average.

4. Finally the EMS asset price at time tj is calculated by using equation 1.6.

Duan et al. (1999) then followed up the work with an analytical approximation for the GARCH

Option Pricing Model. This approximation uses the higher order moments of the distribution

10

of log returns of the asset.We will discuss this method and the GARCH Option Pricing Model

in detail in Chapter 4.

Around the same time Heston and Nandi (2000) introduced a closed form solution to the GARCH

model. They derived the risk neutral transformations of the parameters. The characteristic

function of the asset is taken to be in a log linear form. They derived recursions for the involved

terms and finally provided an analytic expression in the Fourier domain. A good study on the

capabilities and limitations of various GARCH option pricing models is availabe in Christoffersen

and Jacobs (2004).

In the later part of the dissertation we try to extend the Heston with Jumps option pricing

framework with a copula. A similar approach has been taken by Chiou and Tsay (2008) in the

pricing of index correlation options with Duan’s GARCH Option Pricing Framework. Chiou

and Tsay (2008) extend the univariate risk neutral pricing of Duan(1995) to a multivariate case

under the copula framework. As mentioned in the introduction we use a T Copula and a Frank

Copula. In the second part of the paper the copula based model is used to assess the risk of a

portfolio comprising of assets which have the underlying of the options as investment.The assets

in this case are investments on the NYSE and TAIEX index. The copula is used to measure

the tail dependence of the asset returns. The authors come to an interesting conclusion that

no matter what kind of dependence measure is used , holding a portfolio of both indices always

has a higher chance to gain over 5 per cent than to lose more than 5 per cent. The copula

based model is then used to calculate the Value-at-Risk of the portfolio. This paper illustrates

an interesting use of the copula based approach to pricing derivatives.

There are three methods by which we could have fit an copula to our bivariate exchange rate

11

data. They are the Exact Maximum Likelihood Method(EML), the Inference Function for

Margins Method (IFM) and the Canonical Maximum Likelihood Method (CML). In our case of

fitting the T Copula we use the CML method and its application by Mashal and Zeevi (2002).

We use this method because here because we do not have to make any assumption about the

distributional form of the marginals. We review this method in detail in Chapter 5. In the

EML method we have a n-dimensional vector µ of parameters to be estimated. We denote the

parameter space with Θ. We denote the log likelihood function of the observation at time t by

lt(µ). The log likelihood function l(µ) is hence

l(µ) = ΣTt=1lt(µ)

By a direct application of Sklar’s Theorem which enables the seperation of univariate margins

and the dependence structure we have

l(µ) = ΣTt=1 = ln c(F1(x

t1), ...., FN (x

tN )

)+

T∑

t=1

N∑

n=1

ln fn(xtn) (1.8)

Here c denotes the density function of the copula and Fi(; ) denotes the distribution function

of the marginals and f(; ) the density function of the marginals. xti denotes the time series

observation for the itℎ asset. The maximum likelihood estimator is defined as the vector µ̂ such

that

µ̂ =(

ˆµ1, ....., µ̂k

)= argmax {l(µ) : µ ∈ Θ}

The Inference Function for Margins method is based on equation (1.8). It is a two step fitting

proceedure in which one first finds the parameters of the univariate marginals ¯ and then the

vector of copula parameters ®̃.

12

1. At the first stage the EML method is used to find ¯ = (¯1, ......, ¯n) as

ˆ̄i = argmax

¯t

T∑

t=1

ln fi(xti;¯i)

2. The copula parameter vector ® is estimated using ˆ̄ =(ˆ̄1, ...., ˆ̄n

)

ˆ̃®IFM = argmax®̃

T∑

t=1

ln c(F1(x

t1;

ˆ̄1), ...., FN (x

tN ;

ˆ̄N ); ®̃

)

The IFM estimator is hence(ˆ̄, ˆ̃®IFM

).

13

Chapter 2

Pricing FX Quanto Options in the

Black Scholes Framework

2.1 The standard market practice in pricing FX Quanto Options

In this section we will look at the Black Scholes model used in pricing FX quanto options. Our

currency set is given by A = {f, d, q}. Here f denotes the foreign currency, d the domestic

currency and q the quanto currency. We take i , j , k ∈ A Let us denote by Xi,jt the exchange

rate between currency i and currency j . We begin with some notations:

Dit,T : The time t value in currency i of a zero coupon bond with maturity T

Qi : Equivalent martingale measure associated with the numeraire Dit,T

Xi,jt : The value in currency i of one unit of currency j at time t.

K : The strike price of the Quanto FX Option in consideration

14

· denotes the quanto conversion factor which is predetermined. In all our calculations we will

take this value to be 1.

M i,jt,T =Djt,TX

i,jt

Dit,Tis the forward FX rate.

In an arbitrage free economy the following relations hold:

Xi,jt = (Xj,it )

−1

Xi,jt =Xk,jtXk,it

Bennett and Kennedy (2003) consider a triangular no arbitrage condition to price FX quanto

options. The numeraire considered will be in the quanto currency. In the pricing formula we

take the risk neutral expectation with respect to the numeraire in the quanto currency. The

payoff of the FX quanto call option is ·[Xi,jt −K]+. The price of the FX quanto call option at

initial time:

Cquanto0 = ·Dq0,TE

qQ[X

i,jt −K]+ (2.1)

Here we use q because we want to denote that the discount factor Dq0,T and the risk neutral

measure Qq is with respect to the quanto currency. Hence i refers to the foreign currency and

j to the domestic currency.

2.1.1 Pricing using the Black Scholes Model

We assume that the forward FX rates follow correlated Brownian motions

dM i,jt,T = ¾i,jMi,jt,TdW

i,jt (2.2)

15

Here ¾ij is the implied volatility of a vanilla FX option on the particular exchange rate with the

particular option parameters as the quanto FX call.

dW k,it dWk,jt = ½dt (2.3)

½ =¾2k,i + ¾

2k,j − ¾2ij

2¾k,i¾k,j(2.4)

Here ½ is the implied correlation which we recover from the implied volatilities of vanilla FX

options on the particular exchange rate. After evaluating the expectation in equation (2.1)

Bennett and Kennedy (2003) find the price of the quanto FX option under the Black Scholes

Model as

Cquanto0 = ·Dk0,T [MÁ(d̃1)−KÁ(d̃2)] (2.5)

M = M i,j0,T e(¾2k,i−½¾k,i¾k,j)T

d̃1 =ln(M i,j0,T )/K

¾ij√T

+1

2¾i,j

√T

d̃2 = d1 − ¾i,j√T

We use equation (2.5) to price FX quanto options under the Black Scholes model in the disser-

tation.

16

Chapter 3

Pricing of FX Quanto under the

Heston Model

The Heston (1993) Stochastic Volatility model relaxes the assumption of constant volatility in the

classical Black Scholes model. An instantaneous short term variance process is incorporated. In

this section we will look at the key aspects adopted while pricing the options under the Heston

stochastic volatility model. However we price our FX quanto options by incorporating jump

processes in the Heston model.

3.1 Revisiting the Heston model

We shortly formalise the model to take care of the notations. The dynamics of the stock process

{St, t ≥ 0} are given as :dStSt

= (r − q)dt+√vtdWtS0 ≥ 0 (3.1)

17

The instantaneous variance process vt is taken as a mean reverting square root stochastic process

which is also known as Cox Ingersoll Ross (CIR) process. The SDE is given by

dvt = ·(´ − vt)dt+ ¸√vtdW̃tv0 = ¾20 ≥ 0 (3.2)

Here W = Wt where t ≥ 0 and W̃ = W̃t where t ≥ 0 are two correlated standard Brownian

motions with correlation ½. The parameters in the equation are: initial volatility, ¾0 > 0, mean

reversion rate · > 0, the long run variance ´ > 0, the volatility of variance ¸ > 0. The variance

is always positive and by the Feller condition it has been shown that it cannot reach zero if

2·´ > ¸2. The variance process vt is non centrally Chi-Square distributed.

3.2 Option Pricing under the Heston model

As shown in the Black and Scholes model (Black and Scholes (1973))the value of a derivative

is dependent on the underlying tradeable assets. As the assets are tradeable the option can be

hedged by trading in the underlying. When this happens we say that the market is complete.

This means that every derivative can be replicated.

In the Heston model the price of a derivative would depend on both the randomness of the

asset process (St, t ≥ 0) and its volatility (Vt, t ≥ 0).The volatility is not a tradeable asset and

hence under the Heston model we do not work in a complete market setting. An implication of

this fact to option pricing under the Heston model is that we do not find a unique equivalent

martingale measure (EMM). Under the Heston model, the value of any option U(St, Vt, t, T )

18

must satisfy the partial differential equation

1

2V S2

∂2U

∂S2+½¾V S

∂2U

∂S∂V+

1

2¾2V

∂2U

∂V 2+ rS

∂U

∂S+·[µ − V ]− Λ(S, V, T )¾

√V∂U

∂V− rU + ∂U

∂t= 0

(3.3)

Λ(S, V, T ) is called the market price of volatility risk. In his paper Heston makes the assumption

that the market price of volatility risk is proportional to volatility.

Heston has derived a closed form solution which makes it easier for practitioners to price options

using stochastic volatility. However for our purpose of pricing we will refrain from using the

closed form solution given by Heston. Instead we will use the approach used by Carr and Madan

(1998). Carr and Madan have considered a transformation of the option pricing formulae and

then applied Fourier inversion techniques to compute option prices. We discuss this method as

follows.

3.2.1 The Heston Pricer using Fast Fourier Transform

We consider an asset with value Xt at time t and an option written on the asset with strike K.

Let xt = ln(Xt) ; the logarithm of the underlying asset value at time t. For our purpose the

asset is a foreign exchange rate. Let K̃ =ln (K); the logarithm of the strike price. Then the

value of a European Call Option with maturity T as a function of K̃ is given by

C(T, K̃) = e−rT∫ ∞k

(exT − eK̃)fT (xT )dxT (3.4)

Here fT (x) is the risk neutral density of x. r denotes the riskfree interest rate. For our FX option

it is the riskfree interest rate in the foreign currency. We observe from the above equation that

C(T, K̃) tends to the initial spot of the underlying as K̃ tends to −∞. Hence the call pricing

19

function is not square integrable. As such Carr and Madan modify the call price as

C̃(T, K̃) = e®kC(T, K̃)k ≥ 0 (3.5)

Carr and Madan then obtain an analytical expression for the Fourier transform of C̃(T, K̃) as

³(u) =

∫ ∞−∞

eiuk̃C̃(T,K)dK̃. (3.6)

In terms of the characteristic function Á of the logarithm of the stock price as

³(u) =e−(r−q)TÁ(u− (®+ 1)i)®2 + ®− u2 + i(2®+ 1)u. (3.7)

Here r denotes the risk free rate in foreign currency and q the riskfree rate in the foreign currency.

This is followed by the numerical computation of the call prices using the inverse transform

C(T, K̃) =e−®K̃

2¼

∫ ∞−∞

e−iuK̃Ã(u)du. (3.8)

The Fast Fourier Transform(FFT) is an efficient algorithm to evaluate summations of the form

P (e) = Σj=Nj=1 e−i 2¼

N(j−1)(e−1)x(j) (3.9)

Carr and Madan note that (3.8) can be integrated using the FFT algorithm. The FFT algorithm

is as follows

(a) We discretize (3.8) using the Trapezoid Rule and set uj = ´(j − 1). This gives us

C(T, K̃) ≈ e−®K̃

¼ΣNj=1e

−iujK̃³(uj)´ (3.10)

(b) We take a regular grid for K̃ of size ¸ to obtain K̃u = −b+¸(s−1) for s = 1, 2, ...., N giving

log strike levels from -b to b where b = N¸2 .

20

(c) We substitute step (b) in step (a) to obtain

C(T, K̃u) =e−®K̃

¼

j=N∑

j=1

e−i2¼N

(j−1)(s−1)eibuj³(uj)´

3(3 + (−1)j − ±j−1). (3.11)

Here ±n is the Kronecker delta function.

(d) Finally we use the FFT to compute the call prices.

Carr and Madan provide the optimum parameter values as N = 4096, ´ = 0.25 and ® = 1.5.

As we see to evaluate ³ we need to use the characteristic function Á of the logarithm of the

underlying asset.

In the Heston framework we need to take special care while evaluating this characteristic func-

tion.

3.2.2 The Heston Trap

In this section we talk about the use of an alternative characteristic function as given by Al-

brecher et al. (2007). For the log asset price distribution the characteristic function is given

by

Á(u, t) = E[exp(iu ln(St))∣S0, ¾20

](3.12)

However there are two formulas for the characteristic function. We can find the first formula in

Heston (1993). It is

Á1(u, t) = exp(A)× exp(B)× exp(C). (3.13)

21

where

A = iu(lnS0 + (r − q)t).

and

B =

(´·¸−2(·− ½¸iu+ d)t− 2 ln

(1− g1edt1− g1

)).

and

C =

(¾20¸

−2(·− ½¸iu+ d)(1− edt)1− g1edt

).

To explain notations further

d =√(½¸ui− ·)2 + ¸2(iu+ u2).

g1 = (·− ½¸iu+ d)/(·− ½¸iu− d).

The second one is as discussed in Albrecher et al. (2007).

Á2(u, t) = exp(A)× exp(D)× exp(E). (3.14)

where

D =

(´·¸−2(·− ½¸iu− d)t− 2 ln

(1− g1e−dt1− g1

)).

E =

(¾20¸

−2(·− ½¸iu+ d)(1− e−dt)1− g2e−dt

).

where

g2 =1

g1.

22

We note that the difference between B and D is that we have a negative sign before dt in D while

a positive sign in B. Albrecher et al. (2007) have shown that the options are mispriced when

using the value of Á1 in the Carr Madan Formulae for option pricing. Our Heston parameter

space has to be restricted to use Á1 as a characteristic function. However Á2 can be applied to

the whole unrestricted parameter space and is a better choice.

3.3 The Heston model with jumps

One of the primary requisites of option pricing is that we can fit our model prices to the market

smile. While the Heston stochastic variance model fits the long term behaviour of the asset price

it does not adequately describe the short term behaviour as shown by Weron et al. (2004). As

such we extend the Heston stochastic volatility model with jumps in the asset price process as

by Bates (1993) and Bakshi et al. (1993).This model is a jump diffusion model. Carr and Wu

(2004) say that in this model the desired smile is created for short maturities by jumps while at

longer maturities the effect is created by stochastic volatility.

dStSt

= (r − q − ¸¹J)dt+ ¾tdWt + JtdNt. (3.15)

Here N = {Nt, t ≥ 0} is an independent Poisson process with parameter ¸ ≥ 0. Jt is the

percentage jump size which is assumed to be lognormally and identically distributed with time

and with unconditional mean ¹J . ln(1 + Jt) is normally distributed with mean ln(1 + ¹J)− ¾2J2

and variance ¾2J . As far as our pricing methodology goes the only change with the Heston model

would be in the characteristic function. The characteristic function of the Heston with jumps is

actually a product of the characteristic function of Heston Á2 and the characteristic function of

23

the jump process, ÁJ . This is given in Schoutens et al. (2005).

ÁHJ2 (u, t) = exp(A)× exp(D)× exp(E)× exp(F )× exp(G). (3.16)

where

F = −¸¹J iut+ ¸t((1 + ¹J)iu.

G = ¾2J(iu/2)((iu− 1))− 1).

We can see that

ÁJ = exp(F )× exp(G).

3.4 Pricing the FX Quanto option

Let Xa,bt denote the exchange rate which follows the asset process in the Heston with jumps

equation. Here a denotes the foreign currency and b denotes the domestic currency. The price of

a plain vanilla European option with the exchange rate as underlying,strike K, time to maturity

K under an EMM Q is given by

Cvanilla0 = e−raTEQ[Xa,bT −K]+ (3.17)

Here ra denotes the risk free rate in the foreign currency i. The price of an FX quanto option

is given by

Cquanto0 = e−rcTEQ[Xa,bT −K]+ (3.18)

Here rc denotes the risk free rate in the quanto currency c. This is the formulae we will be

using while pricing the FX plain vanilla and the quanto options without a copula. Later we

24

also incorporate a copula. This is discussed in section 5.3. Once the prices of the plain vanilla

options are evaluated we back out the Black Scholes implied volatilities which are then used for

the calibration process. The parameters we get after the calibration of the plain vanilla options

are then used to price the quanto with the copula.

25

Chapter 4

Pricing of FX Quanto under the

GARCH Option Pricing Model

4.1 Duan’s GARCH Option Pricing Model

In this section we will discuss the methodology adopted in pricing the quanto option using the

GARCH Option Pricing Model. This has been developed by Duan (1995). Duan and Wei (1999)

have further extended the GARCH Option Pricing Model for valuation of Foreign Exchange

Options. The conditional variance ¾t follows a non linear asymmetric GARCH(1,1) model. As

before our currency set is given by A in section 2.1. Let us denote by Xi,jt the exchange rate

between currency i and currency j where i,j ∈ A. The asset process of Xi,jt is governed by the

probability law ℙ with respect to information filtration Ft. We take the measure ℙ with respect

to the domestic currency.

26

We have the following process for the exchange rate Xi,jt .

ln

[Xi,jt+Δ

Xi,jt

]= (rf − rd) + ¸¾t+Δ − 1

2¾2t+Δ + ¾t+Δ²t+Δ (4.1)

²t+Δ ∼ N(0, 1)

under ℙ

¾2t+Δ = ¯0 + ¯1¾2t + ¯2¾

2t (²t − µ)2 (4.2)

Here rf denotes the foreign risk free rate and rd denotes the domestic risk free rate. The

parameter ¸ represents the unit risk premium for the exchange rate.The parameter µ represents

a leverage parameter. The leverage effect in GARCH models comes from the empirical evidence

that increase in volatility is larger when the returns are negative than when they are positive.

Duan (1995) comes to the conclusion that if µ is positive there is a negative correlation between

the innovations of the asset return and its conditional volatility. As we can see ¾2t+Δ is expressed

in terms of Ft measurable random variables. The volatility process is hence predictable or known

at time t.

Under the locally risk neutralized probability measure ℚ the exchange rate dynamics given by

Duan(1995) is

ln

[Xi,jt+Δ

Xi,jt

]= (rf − rd)− 1

2¾2t+Δ + ¾t+Δ²̃t+1 (4.3)

²̃t+1 = ²t+1 + ¸ ∼ N(0, 1)

under ℚ

¾2t+Δ = ¯0 + ¯1¾2t + ¯2¾

2t (²̃t − µ − ¸)2

The condition for first order stationarity is ¯1 + ¯2[1+ (µ+¸)2] < 1. Under the risk neutralized

system the volatility remains as an NGARCH process. The leverage parameter here is µ +

27

¸. When we try to fit the prices from the model to market prices we have to calibrate four

parameters. They are ¯0, ¯1, ¯2 and µ + ¸.

4.2 Analytical Approximation of the GARCH Option Pricing

Model

Duan et al. (1999) has derived an analytical approximation to the option pricing problem under

GARCH. We give a review of this method in this section. The option price is determined in

terms of the moments of the cummulative asset return ½T = ln

(Xi,j0Xi,jT

). We denote the standard

normal distribution function as N and the density function as n(). The formulae for the price

of a European FX call option with spot price Xi,j0 , strike K, time to maturity T is given as

Capprox = C + ·3A3 + (·4 − 3)A4 (4.4)

where

C = Xi,j0 Á(d̃)−Ke−(rd−rf )TN(d̃− ¾½T )

The constants A3, A4 are given as

A3 =1

3!Xi,j0 ¾½T

[(2¾½T − d̃)n(d̃)− ¾2½TN(d̃)

]

A4 =1

4!Xi,j0 ¾½T

{[d̃2 − 1− 3¾½T (d̃− ¾½T )]n(d̃) + ¾3½TN(d̃)

}

with

d̃ = d+ ±

where

d =ln(Xi,j0 /K

)+((rd − rf )T + 12¾2½T

)

¾½T

28

and

± =¹½T − (rd − rf )T + 12¾2½T

¾½T

Here

¹½T = Eℚ0

[½T

]

and

¾½T =

√V arℚ0

[½T

]

We use Eℚ0 to denote conditional expectation under ℚ with respect to F0.

·3 and ·4 represent the third and fourth cumulants of the normalized cummulative return zT

which is given by

zT =½T − ¹½T

¾½T

We note that

Xi,jT = Xi,j0 exp(¹½T + ¾½T zT ) (4.5)

The price of an European FX Call option with strike K is given by

exp(−(rd − rf )T )Eℚ0[max(Xi,jT −K, 0)

]

Hence

Xi,jT ≥ K ⇔ −zT ≤lnXi,j0

K + ¹½T¾½T

= K̃ (4.6)

Now

Eℚ0[max(Xi,jT −K, 0)

]= (4.7)

∫ K̃−∞

[Xi,j0 exp(¹½T − ¾½T z)−K

]g(z)dz

29

To evaluate the above expression we observe from equation (4.5) that we need the true density

function g(z) of zT . Duan et al. (1999) follow the method adopted by Jarrow and Rudd (1982). It

involves the approximation of a probability distribution by an arbitrary distribution.The density

function is expressed in terms of an expansion which involves second and higher order moments

of the arbitrary distribution. In this case the approximation was given as

g(z) = n(z)

[1− ·3

3!(z3 − 3z) + ·4 − 3

4!(z4 − 6z2 + 3)

](4.8)

With this approximation Duan et al. (1999) uses equation (4.7) to evaluate the price of the

option. Equation (4.4) gives the approximated price of the option after the approximation.

Duan et al. (1999) have also provided explicit expressions for ¹½T ,¾½T , ·3 and ·4 in terms of the

parameters involved in the NGARCH volatility process which enables the computation of the

option prices.

30

Chapter 5

Modelling the Dependence Structure

Using a Copula

The previous two sections spoke about pricing the FX quanto option using the Heston Model

and the GARCH Option Pricing Model. Let us denote by Xi,jt the exchange rate with i as the

foreign currency and j as the domestic currency. By principle of no arbitrage we can say

Xi,jt =Xk,jt

Xk,it(5.1)

The payoff of the FX quanto option with Xi,jt as underlying, maturity T , strike K is given by

P quantoT = max

[Xk,jt

Xk,it−K, 0

](5.2)

Such an approach could be useful if for instance Xi,jt was a less frequently traded, illiquid asset.

And both Xk,jt and Xk,it were more liquid in the market as compared to the original currency. In

the context of our pricing problem there is another motivation for doing so. In the Black model

31

the implied volatility quote of a quanto on an exchange rate, take for instance the USD/JPY

exchange rate depends on the implied volatilities of the EUR/USD and the EUR/JPY rates.

Here EUR is the quanto currency. Since the forward rate and the exchange rate process is linked

only by a scaler factor we can say that the volatility of the USD/JPY rate is influenced by the

EUR/USD and the EUR/JPY rates. We have not accounted for this in the GARCH or Heston

model with jumps to price the FX quanto option. We need to model the dependence structure

of the EUR/USD and the EUR/JPY rates. One way of doing this could be by using a copula.

The marginals are free to be chosen depending on the calibration of plain vanilla quotes.

We can now view the FX quanto option as a multivariate contingent claim. In such a situation

it is valid to think that the payoff of the option would depend on the co-movement between the

two exchange rate returns. We will use a copula to measure the association between the two

assets.We can use a wide choice of dependent structures using a copula, for instance linear, non

linear or tail dependent.

We will formalise the definition of a copula. This is given in Frees and Valdez (1977).

Let us consider p uniform random variables u1, u2, ...., up. We need not assume that they are inde-

pendent. The dependence relationship is given by a joint distribution function C(u1, u2, ..., up) =

Prob(U1 ≤ u1, U2 ≤ u2, ...., Up ≤ up))

We select arbitrary marginal distribution functions F1(x1), F2(x2), ...., Fp(xp). Then the function

C[F1(x1), F2(x2), ...., Fp(xp)

]= F

(x1, x2, ...., xp

)(5.3)

defines a multivariate distribution function evaluated at x1, x2, ...., xp with marginal distributions

F1, F2, F3..., Fp. We call the function C a copula.

32

In our case of modelling the dependence between the exchange rates we will use the T-copula

from the Elliptic distribution and a copula from the Archimedean family. Archimedean copulas

are simpler to apply. It allows us to reduce the study of a multivariate copula to a single

univariate family.

5.1 Archimedean Copulas

We first formalise the definition of Archimedean copulas.

Definition 5.1. Let Á be a convex, decreasing function with with domain (0, 1] and range [0,∞)

such that Á(1) = 0. Then the function CÁ(u, v) = Á−1(Á(u) + Á(v)) for u,v ∈ (0, 1] is said to be

an Archimedean Copula. Á is called the generator of the copula.

It is worthwhile to mention that a generator uniquely determines an Archimedean Copula. We

give below a table of Archimedean Copulas and their generators in table 5.1.

Family Generator Á(t) Domain(®) CÁ(u, v)

Clayton t−® − 1 ® > 1 (u−® + v−® − 1)−1/®

Gumbel(− ln t)® ® ≥ 1 exp{-[(-lnu)® + (− ln v)®]1/®}

Frank ln e®t−1e®−1 −∞ < ® < ∞ 1® ln

(1 + (e

®u−1)(e®v−1)e®−1

)

Table 5.1: Archimedean Copulas and their Generators

5.1.1 Archimedean Copulas and their Measures of Dependence

There is an important result due to Schweizer and Wolf (1981). They have established that if

f1 and f2 were strictly increasing but arbitrary functions over the range of the random variables

33

X1, X2 then f1(X1) and f2(X2) have the same copula as X1 and X2. This ensures that the

co-movement between the two random variables X1, X2 is captured by the copula. Schweizer

and Wolf (1981)have also shown that two standard non parametric correlation measures, the

Spearman’s Rank Correlation and the Kendall’s correlation coefficient can be expressed in terms

of the copula function.

The first one is the Spearman’s Rank Correlation Co-efficient defined as

½(X1, X2) = 12

∫ ∫ {C(u, v)− uv}dudv

The second one is the Kendall’s correlation coefficient defined by

¿(X1, X2) = 4

∫ ∫C(u, v)dC(u, v)− 1

For the Archimedean copula family both the Spearman’s Rho and Kendall Tau correlation can

be expressed in terms of the parameter ®. This is summarised in the table 5.2.

Family Kendall’s ¿ Spearman’s ½

Clayton ®®+2 No closed form

Gumbel 1− ®−1 No closed form

Frank 1− 4®{D1(−®)− 1} 1− 12® {D2(−®)−D1(−®)}

Table 5.2: Archimedean Copulas and Measures of Dependence

where for the Frank Copula we have

Dk(x) =k

xk

∫ x0

tk

et − 1dt

Dk(−x) = Dk(x) + kxx+ 1

34

5.1.2 Identifying the Right Copula from the Archimedean Family to Model

the Dependence

We will use the proceedure used by Genest and Rivest (1993) to identify the right copula to

model the dependence structure between the two exchange rates. We start with a bivariate set of

observations{(X1i, Y2i)

}ni=1

which in our case are the log returns of exchange rate. We assume

that the distribution function F of the bivariate data has an associated Archimedean Copula

CÁ. Let Zi = F (X1i, X2i) be a random variable which has the distribution function K(z) =

Prob(Zi ≤ z). It can be proved that the distribution function is related to the generator of an

Archimedean copula through the expression

K = t− Á(t)Á′(t)

The following steps are taken to find Á. This is the method followed by Genest and Rivest

(1993).

1. The Kendall correlation coefficient is estimated using the historical bivariate exchange rate

data. This is given as

¿n =∑

i

3. We now construct a parametric estimate of K using

KÁ(z) = t− Á(z)Á′(z)

We select Á and hence the particular copula from the Archimedean family such that the para-

metric estimate most closely resembles the non parametric estimate. We choose the copula for

what the value of d is the lowest where d is given by

d =

∫ [KÁn(z)−Kn(z)

]dKn(z) (5.5)

Once we know the copula to choose we will proceed by first finding the Kendall Tau Rank

Correlation from the historical data. The parameter of the copula ® is then estimated from the

Kendall Tau Rank Correlation from Table 5.2.

Our copula fititng technique is based on the historical spot exchange rate data. The use of a

member of the Archimedean Copula Family simplifies the fitting procedure a lot. Apart from

the independent copula we have three classes of members of the Archimedean family which have

distinct forms of the generator function. We have covered all three families by picking one from

each. The Frank, Gumbel and Clayton copulas belong to three distinct Archimedean families.

These three copulas can model various kinds of dependencies. The Clayton Copula exhibits

greater negative tail dependence than positive dependency. The Gumbel Copula on the other

hand exhibits more positive dependence than negative. The Frank Copula is symmetric. It has

lighter tails than a Normal Distribution. We generate a copula from the Archimedean family

with the proceedure given by Nelsen (2006). This is shown in Figures 5.1,5.2 and 5.3.

36

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

u

v

Random sample from Frank Copula

Figure 5.1: Frank Copula with ® = 1.5

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

u

v

Random sample from Clayton Copula

Figure 5.2: Clayton Copula with ® = 1.5

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

u

v

Random Sample from Gumbel Copula

Figure 5.3: Gumbel Copula with ® = 1.5

5.2 The T Copula

We begin by formalising the definition of the T Copula

Definition 5.2. Let R be a symmetric, positive definite matrix with diag (R)= 1. Let TR,º be

the standardized multivariate Student’s t distribution with correlation matrix R and º degrees

of freedom. Then the multivariate Student’s t copula is defined as C(u1, u2, ......, un;R, º) =

TR,º(t−1º (u1), t−1º (u2), ....., t−1º (un)

)Here t−1º (u) denotes the inverse of the Student’s t cumulative

distribution function.

37

5.2.1 Calibration of the T Copula

Mashal and Zeevi(2002) have shown that the empirical fit of a T Copula is better than a

Gaussian copula. Financial data sometimes have heavy tails. Log returns of exchange rates we

have taken have high kurtosis of 9. Mashal and Zeevi(2002) say that a T Copula captures better

dependent extreme values. In the copula function the choice of marginals and the dependency

structure is independent. In the T Copula we estimate the correlation matrix R and the degrees

of freedom º. The methodology we will adopt to calibrate the T Copula to the log returns of

the spot exchange rates is due to Mashal and Zeevi (2002). In this method there is no prior

assumption on the distribution form of the marginals. It is a Canonical Maximum Likelihood

Estimate(CML) method. We take the observation vector (of historical log returns of exchange

rates)X = (X1t, X2t, .....XNt)Tt=1 and and approximate the parametric marginals

ˆFn(.) as

ˆFn(x) =1

T

T∑

t=1

1{Xnt≤x}

We first give a result due to Lindskog et al. (2001).

Theorem 5.3. Let X ∼ EN (º,∑

) where for i,j ²{1, 2, ...., N}, Xi and Xj are continuous. Then,

Γ(Xi, Xj) =2

¼arcsinRi,j (5.6)

where EN (º,∑

) denotes the N-dimensional elliptical distribution with parameters (º,∑

). Γ(Xi, Xj)

and Ri,j denote the Kendall’s Tau and Pearson’s linear correlation coefficient for the random

variables (Xi, Xj).

The procedure we adopt to calibrate the T Copula is as follows

38

1. We start with a historical sample X of the exchange rate data. We then transform the initial

data set into a set of uniform variates U using an empirical marginal transformation. Given the

data set we use the empirical marginal distribution. For t = 1, ......, T let

ût = (ût1, ût2, û

tN ) =

[ ˆF1(X1t), ...., ˆFN (XNt)]

2. We use equation 5.6 to estimate the correlation matrix RCML.

3. We then find the estimate º̃ of the degrees of freedom by maximizing the log likelihood

function of the Student’s t-copula density.

º̃ = argmaxº∈Θ

ΣTt=1 ln cstudent(ût1, û

t2, ...., û

tN ;R

CML, º)

5.3 Pricing using the copula

We use a Monte Carlo simulation to price options with the copula. Let us denote the number

of samples as M and the number of timesteps as m.We generate pairs (ui, vi) from the copula

where i = 1, 2, ....,m. We take the marginals which are exchange rates to follow the Heston

with jumps process as discussed in Section 3.0.6. We denote Δt = Tm where T denotes the

time to maturity of the option.The Brownian motions (dW 1t , dW2t ) driving each of the marginal

processes are constructed as follows

dW 1t = N−1(ui)

√Δt

dW 2t = N−1(vi)

√Δt

39

Here N denotes the distribution function of a standard normal random variable. We also know

that the Brownian motions driving the variance process and asset process for the first exchange

rate have a correlation ½ (which we obtain from the calibration of the marginals). We construct

the Brownian motion for the variance process as

dW̃ 1t = xi√Δt

where

xi = N−1(ui)½+

√1− ½2N−1(ai)

Here ai is the third uniform variate which is chosen independently from the first two. The first

two ui and vi are generated from the copula.

We follow the same procedure for the second exchange rate as well. In this way we simulate the

asset process and the variance process. The price of the FX Quanto option is evaluated by the

usual procedure followed in a Monte Carlo.

40

Chapter 6

Data

6.1 Discount Factors, Implied Volatility quotes

In this section we will present the data which will be used to price the FX quanto options. We

have obtained the data from Bennett and Kennedy (2003) The source of data in the paper is

Meryll Lynch United Kingdom. The data comprises of

a. The discount factors of three currencies - United States Dollar (USD), Euros(EUR) and

Japanese Yen(JPY) pertaining to maturities of 1 month, 3 months, 6 months, 1 year and 2 year

as on 7 July 2001. Hence we price our options on 7 July 2001.

b. The implied volatilities of plain vanilla FX options written on EUR/USD, EUR/JPY and

USD/JPY exchange rates on 7 July 2001. The implied volatilities are given in terms of the

Black Delta (Δ) of the option.

41

Maturity EUR USD JPY

1/12 0.996243 0.996661 0.999948

1/4 0.988734 0.990054 0.999805

1/2 0.978343 0.980503 0.999566

1 0.958074 0.959798 0.998965

2 0.916243 0.911070 0.997404

Table 6.1: Discount factors Di0,T corresponding to currency i and maturity T

Maturity 90%(Δ) 80%(Δ) 50%(Δ) 20%(Δ) 10%(Δ)

1/12 0.1289 0.123 0.1215 0.126 0.134

1/4 0.1277 0.121 0.119 0.1235 0.1319

1/2 0.1272 0.1203 0.118 0.1223 0.1305

1 0.1269 0.1196 0.117 0.1216 0.1303

2 0.1249 0.1176 0.115 0.1196 0.1283

Table 6.2: EUR/USD implied volatility quotes of call options corresponding to standard values

of Black Delta and maturity T

6.2 Recovering Strikes and Prices of FX vanilla options

For calibration of plain vanilla FX options we need to price options using a particular model.

For this purpose we need to recover strikes (K)for FX call options from the implied volatility

quotes. We use the formulae given by Carr and Wu (2004).

K = F exp

[1

2¾2T − ¾

√TÁ−1(erfTΔ)

]

42

Maturity 90%(Δ) 80%(Δ) 50%(Δ) 20%(Δ) 10%(Δ)

1/12 0.1467 0.1441 0.148 0.1606 0.1747

1/4 0.1466 0.1372 0.1385 0.1487 0.1667

1/2 0.1444 0.1329 0.132 0.1414 0.1597

1 0.1443 0.132 0.13 0.138 0.1557

2 0.1508 0.1358 0.1308 0.1358 0.1508

Table 6.3: EUR/JPY implied volatility quotes of call options corresponding to standard values


Maturity 90%(Δ) 80% (Δ) 50%(Δ) 20%(Δ) 10%(Δ)

1/12 0.1109 0.1015 0.095 0.0945 0.099

1/4 0.1156 0.1045 0.0985 0.0995 0.1066

1/2 0.1178 0.1055 0.1 0.1024 0.1122

1 0.1186 0.1065 0.101 0.1048 0.1166

2 0.1235 0.1105 0.106 0.1105 0.1235

Table 6.4: USD/JPY implied volatility quotes of call options corresponding to standard values


where

F = Xi,j0 e(rd−rf )T

Here Á denote the distribution function of a standard normal variable, Xi,j0 denotes the spot

exchange rate,rd the domestic risk free interest rate, rf the foreign risk free interest rate, T the

time to maturity and ¾ the implied volatility quote.

43

We price an FX plain vanilla FX call option using the following formulae used by Carr and Wu

(2004).We will use this when we have to calculate implied volatility using the bisection method.

This will be discussed in Chapter 7.

Cvanilla0 = erfTXi,j0 Á(d+)− erdTKÁ(d−) (6.1)

where

d+ =ln(F/K)

¾√T

+1

2¾√T

d− =ln(F/K)

¾√T

− 12¾√T

44

Chapter 7

Implementation of Pricing Methods

In this section we will look at pricing the FX quanto option under two different models-the

Heston model with jumps (HJ) and the GARCH Option Pricing Model with the analytical

approximation.

7.1 Implementation of the Heston Stochastic Volatility Model

with Jumps

7.1.1 Calibration of the Model

Before pricing the FX quanto option we would like to fit our pricing model to the market implied

volatilities. Carr and Wu (2004) have carried out an analysis on the suitability of the Heston

model to fit the market smile. They find that for extremely short and longer maturities the

Heston model does not give a good fit. They say that the Heston model with jumps fits the

45

market smile better. We will look to model the dynamics of the USD/JPY exchange rate with

the Heston with jumps model. As we mentioned in the data section we intend to price the

options as on 7 July 2001. Before proceeding to applying a model we take a look at the pattern

of the historical spot USD/JPY exchange rate. We download data of the spot rates from 1

January 1998 to 7 July 2001 (from Datastream)

0 200 400 600 800 1000 12006.5

7

7.5

8

8.5

9

9.5

10x 10

−3

Observations

USD/

JPY

Exch

ange

Rat

e

Time series plot of USD/JPY Exchange Rate

Figure 7.1: Time series observations of USD/JPY spot exchange rate

Before we price a FX quanto option on an exchange rate we will look to calibrate plain vanilla FX

options. The purpose of our calibration is to obtain values of parameters of the Heston Model

with jumps that accurately describes current market implied volatilities. There is a reason why

we choose to calibrate with market implied volatilities (or market prices of options)of the option

instead of the asset prices of the underlying. If we calibrate to asset prices we obtain parameters

which correspond to the true process of the asset and not the risk neutral process. We would

the need to calculate the market risk premium associated with exposure to volatility changes by

estimating returns on options that are being used to hedge against volatility.

We take up the pricing of vanilla FX options on the USD/JPY rate (please refer to Data for all

46

values).In the calibration process we minimize the least squares error function given by

Min(Error) = Σni=1(Xi − X̃i)2P

where X and X̃ denote market and model implied volatilities and P denotes the parameter

vector. The parameters involved in the Heston model with jumps are are ¾0, the initial volatility,

·, the mean reversion rate of volatility, ´ the long run variance,¸ the volatility of volatility, µ,the

frequency of jumps in a year, ¹ the percentage size of the jump and ¾J the variance of the jump

process and ½, the correlation between the Brownian motions driving the asset price process and

the variance process. The constraint

2·´ > µ2

is imposed to ensure that the variance is always positive. We see that the calibration problem for

the Heston model with jumps is thus a general non linear optimization problem. We calibrate to

all maturities using an in-built function called fminsesarch in MATLAB. This function uses the

Nelder and Mead (1965) Algorithm which is an unconstrained non linear optimization algorithm.

Here we add the constraints externally. We set the error function to a very high number if the

constraints are violated. Since fminsearch is a local optimization algorithm we have to take care

while choosing the initial parameters We choose the same values of parameters as by Schoutens

et al. (2005) while pricing options using the Heston with jumps model.

(a) ¾0 = 0.05

(b) · = 0.5

(c) ´ = 0.06

47

(d) ¸ = 0.22

(e) ½ = -0.9

(f) µ = 0.13

(g) ¹ = 0.17

(h) ¾J = 0.13

After the calibration we obtain the following results of the optimum parameters.

Parameter Value

¾0 0.1182

· 1.5288

´ 0.0481

¸ 0.3705

½ -0.8102

µ 0.120

¹ 0.005

¾J 0.01137

Table 7.1: Parameters of Heston with jumps calibration on USD/JPY rates

The Carr Madan formula with the FFT pricer is most effective for ATM options. Hence we

would expect that the relative error between OTM model and market volatilities is higher. For

options which are in the money region we expect more accurate results.We have checked our

pricer using the Carr Madan formula with the results given by Schoutens et al. (2005).

48

In the calibration excercise we vary the number of iterations from 50 to 2000. We observe that

the parameters converge to the optimum in approximately 1000 iterations.

We would like our model results to be coherent with the market. This can be examined by

comparing the market smile with the smile obtained from our model. We obtained our prices

from the Heston with Jumps model and then inverted them to get the implied volatilities during

the calibration proceedure.

7.1.2 Finding Implied Volatility

During our calibration process we obtain call prices from our Heston with jumps model. To find

the corresponding implied volatility we need to invert the Black Scholes equation. In the Black

Scholes case there is no analytic solution for the Implied Volatility. Instead one has to use a

numerical proceedure or approximation.In our case we use the bisection method to back out the

implied volatilities.The bisection method relies on the fact that option prices increase when the

volatility increases.

The fits to the market smile are given in Figures 7.2 - 7.6 for varying maturities.

We do a piecewise linear interpolation to construct both the volatility surface of the Heston

model with jumps and the market volatility surface for USD/JPY quotes.We observe from the

surface of Heston with jumps (Figure 7.8) that as the delta of the option increases from 0

to 0.50 (roughly in the money region) the implied volatility decreases. As we go from at the

money region to the out of money regions the implied volatility increases. This is similar to the

behaviour of the market implied volatilities (Figure 7.7).

49

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08

0.09

0.1

0.11

0.12

0.13

Delta

Vola

tilit

y

Market and model volatilities for USD/JPY FX options with maturity 1 month

Heston with jumps volatilityMarket volatility

Figure 7.2: Market and model volatility(in %) for USD/YEN call options with maturity 1

months

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08

0.09

0.1

0.11

0.12

0.13

Delta

Vola

tilit

y

Market and model volatilities for USD/JPY FX options with maturity 3 months

Heston with jumps VolatilityMarket Volatility


months

7.2 Implementation of the GARCH Option Pricing Model

Our aim is to price the FX quanto options with USD/JPY exchange rate as underlying using

the GARCH Option Pricing Model with a Monte Carlo simulation. We first calibrate to the

implied volatility quotes on the USD/JPY rates. In the calibration proceedure we use the same

50

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08

0.09

0.1

0.11

0.12

0.13

Delta

Vola

tilit

y

Market and model volatilities for USD/JPY FX options with maturity 6 months



months

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08

0.09

0.1

0.11

0.12

0.13

Delta

Vola

tilit

y

Market and model volatilities for USD/JPY FX options with maturity 1 year



months

set of innovations during each optimisation step. We calibrate to all maturities. The use of

the GARCH Option Pricing Model in pricing the quanto would be justifiable if we can fit the

market smile using this model.

We follow the same optimisation technique as in the calibration of the Heston model with

51

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08

0.09

0.1

0.11

0.12

0.13

Delta

Vola

tilit

y

Market and model volatilities for USD/JPY FX options with maturity 2 years



months

jumps.The parameters we have to calibrate are ¯0, ¯1, ¯2 and µ + ¸.

Duan (1996) had fit this model to the market smile with the help of a Monte Carlo simula-

tion.Duan et al. (1999) have shown that the difference between the Monte Carlo price and the

analytical approximation is very low. The analytical approximation is considered to be precise

by the authors. Our Monte Carlo was computationally very extensive. It would have been better

to use the analytical approximation. Our choice of initial parameters is based on a parameter

set used by Duan et al. (1999) to price options.

(a) ¯0 = 1E-5.

(b) ¯1 = 0.1.

(c) ¯2 = 0.7

(d) µ + ¸ =0.5

After the calibration we obtain the following results of the optimum parameters as shown in

52

0.1 0.20.3 0.4

0.5 0.60.7 0.8

0

0.5

1

1.5

20.09

0.1

0.11

0.12

0.13

0.14

Delta

Market Volatility Surface

Time to Matutity

Impl

ied

Vol

atili

ty

Figure 7.7: Market Implied Volatility Surface

0.10.2

0.30.4

0.50.6

0.7

0

0.5

1

1.5

20.08

0.1

0.12

0.14

Delta

Heston with Jumps Volatility Surface

Time to Matutity

Impl

ied

Vol

atili

ty

Figure 7.8: Heston with Jumps Implied

Volatility Surface

Table 7.2.

Parameter Value

¯0 1.23E-5

¯1 0.231

¯2 0.859

µ + ¸ 0.935

Table 7.2: Parameters of GARCH calibration on USD/JPY rates

7.2.1 Calibration Results

Finally the values of the implied volatilities obtained from the GARCH Model, Heston Model

with Jumps (HJ) as compared to the market volatilities are given in Tables 7.3- 7.7. We observe

that the volatility we obtain from the GARCH model does not behave in similar way to the

53

market. Actually the behaviour is somewhat opposite to the market. Duan (1996) has fit the

same model to FTSE 100 index options. It could be the case that for this particular set of data

the GARCH model was not well applicable. However our Heston with jumps model fits the

market behaviour a lot better.

Delta Market volatility GARCH volatility HJ volatiltiy

0.10 0.0990 0.0959 0.1044

0.20 0.0945 0.1078 0.0934

0.5 0.095 0.1221 0.0950

0.8 0.1015 0.1286 0.1004

0.9 0.1109 0.1242 0.1036

Table 7.3: Implied volatilities for maturity of 1 month


0.10 0.1066 0.0907 0.1088

0.20 0.0995 0.1068 0.0997

0.50 0.0985 0.1248 0.1015

0.80 0.1045 0.1306 0.1069

0.90 0.1156 0.1185 0.1086

Table 7.4: Implied volatilities for maturity of 3 months

54


0.10 0.1122 0.0989 0.1153

0.20 0.1024 0.1153 0.1098

0.50 0.1 0.1320 0.1123

0.80 0.1055 0.1343 0.1168

0.90 0.1178 0.1102 0.1191

Table 7.5: Implied volatilities for maturity of 6 months


0.10 0.1166 0.1124 0.1168

0.20 0.1048 0.1266 0.1126

0.50 0.101 0.1380 0.1162

0.80 0.1065 0.1299 0.1209

Table 7.6: Implied volatilities for maturity of 1 year


0.10 0.1235 0.1211 0.1082

0.20 0.1105 0.1295 0.0971

0.50 0.106 0.1288 0.102

0.80 0.1105 0.1310 0.1114

0.90 0.1235 0.1342 0.1243

Table 7.7: Implied volatilities for maturity of 2 years

55

7.3 Choosing a model to price the Quanto option

We obtain the Root Mean Squared Errors (RMSE) for the GARCH and the HJ model for the

implied volatility fits.The RSME is defined as

RSME =

√∑(Market volatility−Model volatility)2

Number of options

Method RMSE

GARCH 0.31

HJ 0.089

Table 7.8: Root mean squared errors for the implied volatility fits on USD/JPY FX rate

We can observe from table 7.8 that the error for the HJ model is substantially lower. We can

say that the HJ model would suit our purpose of pricing the FX quanto option better. However

we price the quanto option using both the models.

7.4 Results of pricing the quanto option

We now proceed to price the FX quanto option on the USD/JPY underlying with EUR as the

quanto currency. We denote the Heston model with jumps price as HJ Quanto,the GARCH

option pricing model price as the GARCH price and the Black Scholes price as BS Quanto.The

prices are in tables 7.9, 7.10, 7.11,7.12 and 7.13. Relative difference (RD) of the HJ model as

compared to the Black Scholes model is calculated as

RD = (Black Scholes Price-Model Price)/(Black Scholes Price).

56

A negative value of the relative difference would mean that the price under stochastic volatility

is more than under constant volatility and vice versa.

Strikes BS Quanto HJ Quanto GARCH Quanto Rel.difference(HJ) Rel.difference(GARCH)

0.00864 1.12E-05 1.65E-05 9.88E-06 -47.95% 11.81%

0.00852 2.52E-05 3.58E-05 3.45E-05 -42.31% -40%

0.00833 9.01E-05 1.05E-04 1.15E-04 -16.49% -28.5%

0.00812 2.28E-04 2.30E-04 2.48E-04 -1.57% -8.39%

0.00798 3.47E-04 3.39E-04 3.53E-04 2.18% -1.7%

Table 7.9: Quanto prices for maturity of 1 month. Spot price is 0.0083 dollars. All the prices

are in the Euro currency.Strike prices in dollars.


0.0089 2.09E-05 2.105E-05 1.08E-05 -0.56% 48.44%

0.00868 4.60E-05 5.20E-05 5.48E-05 -12.72% -19.17%

0.00832 1.61E-04 1.75E-04 2.05E-04 -8.158% -27.03%

0.00796 4.07E-04 4.04E-04 4.39E-04 0.79% -8.01%

0.00771 6.2E-04 6.04E-04 6.30E-04 3.71% -7.36%

Table 7.10: Quanto prices for maturity of 3 months.Spot price is 0.0083 dollars. All the prices


57


0.00923 3.13E-05 2.61E-05 1.88E-05 16.653% 39.78%

0.00885 6.73E-05 7.09E-05 8.98E-05 -5.17% -33.52%

0.00832 2.33E-04 2.49E-04 3.08E-04 -6.81% -32.39%

0.00780 5.84E-04 5.83E-04 6.35E-04 0.74% -8.7%

0.00743 9.12E-04 8.86E-04 9.1 E-04 3.5% -0.10%

Table 7.11: Quanto prices for maturity of 6 months. Spot price is 0.0083 dollars. All the

prices are in the Euro currency.Strike prices in dollars.

Strikes BS Quanto HJ Quanto GARCH Quanto Rel.difference(HJ) Rel. difference(HN)

0.00970 4.68E-05 3.51E-05 4.07E-05 25% 12.88%

0.00911 9.92E-05 1.05E-04 1.55E-04 -6.51% -56.40%

0.00832 3.40E-04 3.7E-04 4.62E-04 -8.46% -36.30%

0.00755 8.59E-04 8.67E-04 9.14E-04 -0.03% -6.4%

0.00698 1.3E-03 1.34E-03 1.32E-03 1.72% 3.3%

Table 7.12: Quanto prices for maturity of 1 year. Spot price is 0.0083 dollars. All the call

prices are in the Euro currency.Strike prices in dollars.

7.5 Implementing a copula to price the FX quanto option

In sections 5.1.2 we discussed the methodology by which we would choose our copula from the

Archimedean family. We follow it to obtain the Frank Copula to model the dependence. The

58

Strikes BS Quanto HJ Quanto GARCH Quanto Rel. difference(HJ) Rel.difference(GARCH)

0.01047 7.43E-05 4.47E-05 6.85E-05 40% 7.8%

0.00950 1.57E-04 1.54E-04 2.26E-04 1.9% -44.20%

0.00826 5.39E-04 5.57E-04 6.40E-04 -3.3% -18.84%

0.00702 1.4E-03 1.38E-03 1.29E-03 1.85% 8.28%

0.00570 2.65E-03 2.67E-03 2.12E-03 -0.57% 20%

Table 7.13: Quanto prices for maturity of 2 year. Spot price is 0.0083 dollars. All the prices


Copulas will be used to model the dependence between the EUR/USD and EUR/JPY rates.

7.5.1 Parameter estimation for the T Copula

We used the Mashal and Zeevi (2002) approach discussed in Chapter 5 to find the optimal

degrees of freedom º, the correlation ½T and the log likelihood value L for the T Copula. We

get the following results as in Table 7.14. We get the plot for the log likelihood function as in

Parameter Value

º 14

½T 0.6

L 246.52

Table 7.14: Parameters for the T copula

Figure7.9.

59

10 12 14 16 18 20 22 24 26 28 30232

234

236

238

240

242

244

246

248Cannonical Maximum Likelihood Estimation (Mashal and Zeevi)

Degrees of Freedom

Va

lue

of L

og

−L

ike

liho

od

Fu

nct

ion

Figure 7.9: Canonical Log likelihood function (Mashal and Zeevi)

7.5.2 Parameter estimation for the Frank Copula

For the Frank Copula we get the Kendall Tau Rank Correlation (Γ) and ®

Pricing FX Quanto Options under Stochastic...

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