Multi-factor rational lognormal models for valuation

of credit swaptions and basket default swaps

Jean J. KONG and Yue Kuen KWOK∗

Department of MathematicsHong Kong University of Science and Technology

Clear Water Bay, Hong Kong

Abstract

We apply the multi-factor rational lognormal approach to model the

interest rate process and default intensity processes of risky obligors.

Under the rational lognormal framework, positivity of the default in-

tensities and interest rate are guaranteed. In our model, pairwise

correlation of the default processes is introduced through correlation

among the stochastic factors. For pricing of single-name credit default

swaps and swaptions, we manage to obtain analytic representation of

the fair swap premium and option price. For pricing of basket default

swaps, we derive the joint distribution of default times and the present

value of the cash flows up to the time of the kth-to-default under the

assumption of conditional independence. Numerical simulation exper-

iments were performed to demonstrate the dependence of the pricing

of basket default swaps on default correlation among the risky obligors

in the basket.

Keywords: Rational lognormal models, default intensities, default correla-tion, credit swaptions, basket default swaps

1 Introduction

Credit derivatives are customized financial agreements between two counter-parties on the transfer of credit exposures. The cash flows between the twocounterparties are linked to credit events on reference risky assets. Usuallyone party pays a stream of payments according to a reference rate (swap

∗Correspondence author; email: maykwok@ust.hk

1

premium) or single upfront payment and in return the other party providesthe contingent payment upon the occurrence of a specified credit event onthe reference assets. Credit derivatives are off-the-balance-sheet financial in-struments that allow the isolation and management of credit risk from othercomponents of risk. They have received wide acceptance in the financial mar-kets as effective tools of credit management since the 1992 Annual Meetingof the International Swap Dealers Association. The growth of the marketsof these products has been phenomenal in the past decade. Trading of creditdefault swaps become so liquid that we now have indices that track creditdefault swap spreads. For example, the Dow Jones iTraxx EUR 5 yr indexgives the average credit default swap spread for a portfolio of 125 invest-ment grade European companies. Correlation products whose value dependson the default correlation of the obligors in the reference basket, such asbasket default swaps and Collateralized Debt Obligations (CDOs) tranches,become more popular in recent years. As trading volumes increase, pricingand risk management of these complicated credit derivatives are becomingincreasingly important.

The pricing of a credit sensitive instrument requires modeling of the ar-rival time of default of the reference obligor(s). The reduced form approachmodels default as the arrival of a jump event – jumping from the state of“no-default” to the state of “default”. Default is a completely unpredictableprocess, and at all times there is a probability of occurrence of default. Allinformation and events that drive the creditworthiness of the obligor are hid-den behind and reduced to this probability of default. The instantaneouslikelihood of default is described by the default intensity process. In thispaper, the default process is modeled as a doubly stochastic process (Coxprocess), where the default intensity is also stochastic. A comprehensive re-view of the reduced form approach and Cox process can be found in the textby Schonbucher (2003). When we consider the pricing of multi-name creditderivatives, it is necessary to construct a correlation framework for modelingdefault dependencies among the reference risky obligors. Under the reducedform models, several approaches have been proposed to incorporate creditrisk correlation among the obligors.

The most popular approach is the copula method. Now, the Gaussiancopula model has become the standard market model for valuation of syn-thetic CDOs. Copulas were first introduced by Sklar (1959) and later appliedto default correlation studies by Li (2000). A copula function transformsmarginal probabilities into joint probabilities. Sklar (1959) shows that any

2

multi-factor distribution function can be expressed as a copula function. Li(2000) demonstrates that the industrial credit risk software CreditMetrics isin essence a normal copula function. To model default correlation amongobligors, one specifies a copula function for the default times of the obligors.Hull and White (2004) use the factor Gaussian copula model to price thespreads of various CDO tranches. In their copula model, the creditworthi-ness of an obligor is characterized by a random variable, which is a weightedsum of two random factors. One factor gives the common (systematic) riskwhile the other factor represents the firm specific (idiosyncratic) risk of theobligor. Obligor’s default occurs when the value of the credit variable fallsbelow some threshold level. Default correlations among the obligors are gen-erated through the common factor. To allow for simple implementation, theGaussian copula model assumes independence of defaults when conditionedon the common risk factor. Hull and White demonstrate in their numeri-cal experiments that the model is flexible enough to allow the fitting of a“double t-distribution” copula to the market values of traded Dow Jones in-dex tranches. When the distribution assumptions of these random factorschange, the CDS spreads are seen to vary quite significantly.

The second approach attempts to model the contagious effects of de-fault among firms that are commercially or financially related. Under theframework of interacting intensities, Jarrow and Yu (2001) model the defaultcontagion effect among related firms by introducing a positive jump in thedefault intensity of a non-defaulting when one of the counterparties defaults.Though the credit contagion model can be formulated using the Markov chainframework (Leung and Kwok, 2006), the computational complexity of calcu-lating the joint distribution of default times increases exponentially with thenumber of obligors.

The third approach introduces default correlation through the depen-dence of the stochastic intensity processes. Kijima (2000) assumes the defaultintensity processes of the obligors to follow the extended Vasicek model, withtheir stochastic random components being correlated. Based on the assump-tion of conditional independence (that is, default events are independent oncewe fix the realization of the underlying stochastic default intensities), he man-ages to derive the joint survival probabilities of default times. Though hismodel exhibits nice analytic tractability, the default intensities may assumenegative value since the default intensity processes under the Vasicek modelfollow the Gaussian type distribution. Also, the default correlation generatedthrough correlation among stochastic intensity processes is relatively weak,

3

as demonstrated by his calculation on the spread of the first-to-default basketdefault swap.

The potential negativity of the credit spread generated by the Vasicektype default intensity process is undesirable. In the literature of interestrate modeling, the class of “positive interest rate models” have been con-structed that overcome negativity of interest rate. Flesaker and Hughston(1996) propose the representation of the bond prices by a family of positivemartingales under the real measure. Similar positive interest rate models canbe derived using the potential approach (Rogers, 1997) and the rational log-normal (RLN) approach (Rutkowski, 1997). As these positive interest ratemodels are specified under the real measure, the calibration of the parameterfunctions in the models can be performed directly from actual market data ofbond prices with varying maturities. A more recent survey and applicationsof the positive interest rate models can be found in Cairns’ paper (2004). Inour paper, we extend the RLN formulation to model the default intensityprocess so that positivity of the credit spread is guaranteed. Similar to thefactor copula model in Hull-White’s paper (2004), the state price density inour RLN model is taken to consist of two stochastic factors. This approach issimilar to the two-factor RLN model proposed by Nakamura and Yu (2000).Their two-factor model can provide flexibility to calibrate the correlation be-tween any two points in the interest rate term structures. Unlike Kijima’smodel, our two-factor RLN model provides sufficiently high level of defaultcorrelation among the default intensity processes. Like most reduced formdefault correlation models, we use similar assumption of conditional inde-pendence in order to achieve good analytic tractability in the joint survivalprobabilities calculations. As a result, we are able to obtain closed form an-alytic solution of the swap premium of a credit default swap. In addition,our RLN formulation guarantees the default intensity to stay positive.

This paper is organized as follows. In the next section, we present a briefreview of the RLN formulation of positive interest rate models. We thenshow how to extend the RLN formulation to model the default intensityprocess based on market data of the survival probability of a risky obligor.For multi-obligor models, we present the two-factor RLN formulation andillustrate how to model the default correlation among obligors. We also showhow the conditional independence assumption can be used to simplify thecalculation of the joint survival probabilities of the basket of risky obligors.In Section 3, we present analytic pricing calculations of single-name creditdefault swap and swaption. We also illustrate the numerical procedures in

4

the pricing calculations of the nth-to-default basket default swaps. In Section4, we present actual numerical calculations for single-name call swaptions andbasket default swaps. In particular, we explore the pricing behaviors of thesecredit instruments under various assumptions of the correlation structuresof the underlying state variables. The last section contains summary andconclusive remarks of the paper.

2 Rational lognormal formulation of default

intensity processes

In this section, we would like to show how to extend the rational lognormal(RLN) framework to formulate the default intensity process associated withthe arrival of default of a risky obligor. We start with a brief discussion of theinformation structure generated by the underlying state variables and defaultprocesses. The mathematical relation between the default intensity processand survival probability is presented. We then review the RLN formulationof positive interest rate models. In particular, we show how to relate thedefault free bond price process with the state price density. We then for-mulate the conditions under which the state price density and default freezero-coupon bond price can be expressed as sum of exponential martingalesin multi-factor RLN models. Mimicking the linkage between the bond priceprocess and the risk free short rate process, we relate the survival probabil-ity of a risky obligor to its default intensity process. We employ this formof analogy to define the RLN formulation of the default intensity process.To generate default correlations among the risky obligors, we specify thecorrelation structures among the stochastic factors in the default intensityprocesses. To enhance analytic tractability in the joint survival probabilitiescalculations, we adopt the conditional independence assumption. Numericalprocedures of calculating the joint survival probabilities is presented.

2.1 Information structure and default intensity pro-

cess

The probability setup of the uncertainty in the financial market is modeledby a filtered probability space (Ω,F , P ), where Ω is the set of possible statesof the world, P is the real probability measure (P may be replaced by an

5

equivalent martingale measure Q, depending on the context of the pricingproblem). The filtration Ft represents the flow of information over timeand F = σ(∪t≥0Ft) is a σ-algebra. Let X t denote a R

S-valued Markovprocess representing S economy-wide state variables, and let GX,t denote theinformation flow generated by X t. For the default processes of the I riskyobligors, we let N i

t denote the counting process which jumps from 0 to 1upon the default of obligor i, i = 1, 2, · · · , I. We denote the information flowgenerated by N i

t by GiN,t. The filtration Ft is obtained from the union of

information generated by both the economy-wide state variables and defaultstatus of the obligors, that is,

Ft = GX,t

∨G1

N,t

∨· · ·∨

GIN,t. (2.1)

Let τ i denote the default arrival time of obligor i. The indicator of thedefault event of obligor i up to time t is defined by

1τ i≤t =

1 τ i ≤ t0 otherwise

. (2.2)

The default intensity process λit is related to the counting process N i

t by therelation

dN it − λi

t dt = dM it , (2.3)

where M it is a martingale process. The survival probability of obligor i is

given by

P [N it = 0] = P [τ i > t] = E

[exp

(−

∫ t

0

λiu du

)]. (2.4)

In a loose sense, the default intensity λit can be interpreted as

P [τ ≤ t + ∆t|Ft] ≈ 1τ>tλt∆t. (2.5)

For a more rigorous definition of the default intensity, see the text by Lando(2004). If we denote the survival probability of obligor i from time t to timeT by si(t, T ), then

si(t, T ) = P [τ i > T |τ i > t] = E

[exp

(−

∫ T

t

λiu du

)∣∣∣∣Ft

]. (2.6)

6

2.2 Rational lognormal formulation for positive inter-

est rate models

We would like to give a brief review of the rational lognormal formulationfor the positive interest rate models. Let P be the real measure and let Gt

denote the information set of all available information up to and includingtime t. We assume that there exists a risk neutral measure Q equivalent toP such that the discounted price of a tradeable asset is a martingale underQ. Let Bt denote the money market account as defined by

Bt = exp

(∫ t

0

ru du

), (2.7)

where rt is the interest rate process. Let ξt denote the Radon-Nikodymderivative, which transforms the real measure P into the risk neutral measureQ. Note that ξt is a martingale under P so that

ηt =dQ

dP

∣∣∣∣Gt

= EPt [ηT ] , t < T < ∞, (2.8)

where EPt denotes the expectation under P conditional on Gt. We define the

state price density At byAt = ξt/Bt. (2.9)

Let Yt be the price process of a tradeable asset. It is seen that AtYt is amartingale under P , that is,

AtYt = EPt [AT YT ], t < T < ∞. (2.10)

Furthermore, let p(t, T ) denote the time-t price of the zero coupon bondmaturing at time T , then

p(t, T ) = EQt

[Bt

BT

]= EP

t

[ξT

ξt

Bt

BT

]= EP

t

[AT

At

], (2.11)

where EQt denotes the expectation under Q conditional on Gt. It can be shown

that the term structure of zero coupon bonds as described by Eq. (2.11) isarbitrage free provided that At is a strictly positive diffusion process. Also, inorder to ensure positive interest rate, p(t, T ) must be decreasing in maturityT . As a consequence, At is a supermartingale under P .

7

Two-factor model

In the two-factor version of the rational lognormal model of the interest rate,the state price density At is assumed to take the form

At = G01(t)M

01 (t) + G0

2(t)M02 (t), (2.12)

where G01(t) and G0

2(t) are the coefficient functions of the two stochasticfactors M0

1 (t) and M02 (t), respectively. We impose the following conditions

on G0j (t) and M0

j (t), j = 1, 2.

C1. The coefficient functions G0j(t), j = 1, 2, must be strictly positive and

decreasing deterministic functions of t. Also, they are twice differen-tiable and tend to zero as t → ∞.

C2. The stochastic factors M0j (t), j = 1, 2, are uncorrelated lognormal pro-

cesses and they are strictly positive P -martingales. Hence, M0j (t) may

be expressed as

dM0j (t)

M0j (t)

= σ0j dZ0

j (t), j = 1, 2, (2.13)

where Z01(t) and Z0

2 (t) are uncorrelated standard P -Brownian pro-cesses. Also, we take M0

j (0) = 1, j = 1, 2.

Since G0j and M0

j , j = 1, 2, are strictly positive, so At is a strictly posi-tive diffusion. The supermartingale property of At follows directly from thedecreasing property of G0

j , j = 1, 2, since

EPt [AT ] = EP

t

[G0

1(T )M01 (T ) + G0

2(T )M02 (T )

]

= G01(T )M0

1 (t) + G02(T )M0

2 (t)

< G01(t)M

01 (t) + G0

2(t)M02 (t) = At. (2.14)

Consider the zero coupon bond price function

p(t, T ) = EPt

[G0

1(T )M01 (T ) + G0

2(T )M01 (T )

G01(t)M

01 (t) + G0

2(t)M02 (t)

]

=G0

1(T )M01 (t) + G0

2(T )M02 (t)

G01(t)M

01 (t) + G0

2(t)M02 (t)

; (2.15)

and since At is a supermartingale, we deduce that

p(t, T ) < 1, for t < T. (2.16)

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In addition, since limT→∞

G0j(T ) = 0, j = 1, 2, we obtain

limT→∞

p(t, T ) = 0, (2.17)

a property that must be observed by the bond price function. Thus thechoices of G0

j and M0j , j = 1, 2, satisfying conditions C1 and C2 would lead to

arbitrage free term structure of bond prices with guaranteed positive interestrate.

2.3 Two-factor rational lognormal model for default

intensity process

To construct the two-factor RLN model for the default process of a riskyobligor, we start with the formulation of its survival probability. Under thereal measure P , the survival probability si(t, T ) of obligor i is given by

si(t, T ) = EPt

[exp

(−

∫ T

t

λiu du

)]

= EPt

[Gi

1(T )M i1(T ) + Gi

2(T )M i2(T )

Gi1(t)M

i1(t) + Gi

2(t)Mi2(t)

]

=Gi

1(T )M i1(t) + Gi

2(T )M i2(t)

Gi1(t)M

i1(t) + Gi

2(t)Mi2(t)

, i = 1, 2, · · · I. (2.18)

Here, Gij(t) are the coefficient functions of the stochastic factors M i

j(t), j =1, 2 and i = 1, 2, · · · , I, where Gi

j and M ij satisfy conditions C1 and C2,

respectively. The above RLN formulation guarantees that 0 < si(t, T ) <

1 and si(t, T ) is decreasing in T . Writing gij(T ) =

d

dTGi

j(T ), the default

intensity λit can be expressed as

λit = −

∂

∂T

[ln si(t, T )

] ∣∣∣∣T=t

= −gi1(t)M

i1(t) + gi

2(t)Mi2(t)

Gi1(t)M

i1(t) + Gi

2(t)Mi2(t)

. (2.19)

Since gij(t) < 0 and M i

j(t) > 0, for all values of t, the default intensity is thenguaranteed to stay positive. Recall that in Vasicek type assumption for thestochastic default intensity process, λi

t may not always stay positive.

9

In subsequent discussion, only the real measure P will be used in allexpectation calculations. For notational simplicity, we suppress P in theexpectation operator, where Et implicitly means EP

t .

2.4 Default correlation and joint survival probabilities

For each risky obligor, the default intensity in our two-factor RLN formu-lation is defined in terms of two stochastic factors. We follow the usualapproach of defining correlation among the stochastic factors in credit riskmodels, where one factor corresponds to common economy wide conditionswhile the other factor is obligor-specific. We also allow the interest rate pro-cess to be correlated with the default intensity processes. Recall that thestochastic factors M i

j(t), i = 0, 1, · · · , I, j = 1, 2 are chosen to be lognormallydistributed, where

dM ij(t)

M ij(t)

= σij dZ i

j(t). (2.20)

In our correlation structure, we assume the first stochastic factor of theinterest rate process and default intensity processes to be correlated suchthat

dZ i11 (t) dZ i2

1 (t) = ρi1,i2 dt, i1 = 0, 1, · · · , I, i2 = 0, 1, · · · , I, (2.21)

where ρi1,i2 ∈ [−1, 1] is the correlation coefficient. We argue that these firststochastic factors are influenced by a common set of economy wide statevariables, as characterized by the correlation structure defined in Eq. (2.21).Instead of taking M i

1(t) to be identical for all obligors, like that in Hull-White’s credit risk model (2004), we allow for greater flexibility in specifyingthe correlation among M i

1(t). The second stochastic factor of each individualdefault process or interest rate process is taken to be uncorrelated with anyother stochastic factor in the model, that is,

dZ i12 (t) dZ i2

2 (t) = 0, i1 6= i2, i1 = 0, 1, · · · , I, i2 = 0, 1, · · · , I (2.22a)

anddZ i1

1 (t) dZ i12 (t) = 0, i1 = 0, 1, · · · , I. (2.22b)

Under this setup, we allow the interest rate process or individual default in-tensity process to contain component that is obligor-specific and uncorrelatedwith the set of common economy wide state variables. The pairwise default

10

correlation of the risky obligors is introduced through correlation among thestochastic factors.

Joint survival probabilities

To derive the joint survival probabilities of the risky obligors, we follow theconditional independence assumption of the default times τ i, i = 1, 2, · · · , I.Given the realization FT , where T ≥ max(ti; i = 1, 2, · · · , I), we have

PT (τ 1 > t1, · · · , τ I > tI) =I∏

i=1

PT (τ i > ti). (2.23)

We take t < min(ti; i = 1, 2, · · · , I) ≤ max(ti; i = 1, 2, · · · , I) ≤ T and applythe law of iterated expectation. Under our two-factor RLN model, the jointsurvival probabilities of the obligors as observed at time t is given by

Pt(τ1 > t1, · · · , τ I > tI)

= Et

[I∏

i=1

Gi1(ti)M

i1(ti) + Gi

2(ti)Mi2(ti)

Gi1(t)M

i1(t) + Gi

2(t)Mi2(t)

]. (2.24)

After expanding the product of the binominal terms and observing M ij(ti)

to be Geometric Brownian processes with correlated structure defined byEqs. (2.21, 2.22a,b), it is possible to perform these expectation calculationsanalytically. Let I = 1, 2, · · · , I be the index set. For a typical term, it isnecessary to calculate

Et

∏

i1∈S

M i11 (ti1)

∏

i2∈SC

M i22 (ti2)

,

where S is a subset of I and SC is the complement of S. The details of theseanalytic calculations are relegated to the Appendix.

3 Single-name credit default swaps and bas-

ket default swaps

In this section, we would like to compute the fair swap premium of a single-name credit default swap (CDS) and the option price of a default swaption

11

under the two-factor RLN formulation. In their default swap model, Jarrowand Yildirim (2002) assume a relatively simple market and credit risk corre-lation in which the default intensity is taken to be a linear function of theshort rate. Our RLN model allows for greater flexibility of interest rate anddefault intensity correlation. Besides single-name credit derivatives, we alsoextend the analytic pricing approach to price various basket default swaps.

3.1 Single-name credit default swaps and swaptions

The single-name credit default swap (CDS) may be considered to be thesimplest form of a credit derivative. The protection buyer of the CDS paysperiodic swap premium payments so as to gain protection from the protectionseller against the loss upon the occurrence of the credit event (bankruptcy,obligation default, failure to pay, etc.) of the reference risky obligor. TheCDS is terminated either at the maturity of the swap or the arrival time ofthe credit event, whichever comes earlier. The pricing of a CDS amounts tothe determination of the fair swap premium rate such that the present valueof the premium payments and the credit contingent compensation paymentnet to zero value at initiation of the CDS.

Let τ denote the arrival time of the credit event. In our pricing model,we let δ(τ) denote the compensation payment paid by the protection sellerat τ . In real situation, the loss suffered by the protection buyer upon theoccurrence of the credit event will be determined by a third party after thesettlement period. The actual loss will not be known in advance, not even atthe time of default. Let Vp(t; T ) denote the time-t value of the compensationpayment (commonly called the protection leg). We then have (Lando’s text,p. 207; 2004)

Vp(t; T ) = Et

[∫ T

t

δ(u) exp

(−

∫ u

t

rs ds

)λu exp

(−

∫ u

t

λs ds

)du

]. (3.1)

Under our two-factor RLN model, we assume

exp

(−

∫ u

t

rs ds

)=

G01(u)M0

1 (u) + G02(u)M0

2 (u)

G01(t)M

01 (t) + G0

2(t)M0t (t)

, (3.2a)

exp

(−

∫ u

t

λs ds

)=

G1(u)M1(u) + G2(u)M2(u)

G1(t)M1(t) + G2(t)M2(t), (3.2b)

12

λu = −g1(u)M1(u) + g2(u)M2(u)

G1(u)M1(u) + G2(u)M2(u), (3.2c)

where g1(u) = G′1(u) and g2(u) = G′

2(u). By performing the expectationcalculations,, we obtain

Vp(t; T ) =

∫ T

t

δ(u)Q(t; u) du

M0(t)M(t)(3.3)

where

Q(t; u) = −[G01(u)g1(u)M0

1 (t)M1(t) exp(ρ01σ01σ1(u − t))

+ G01(u)g2(u)M0

1 (t)M2(t)

+ G02(u)g1(u)M0

2 (t)M1(t) + G02(u)g2(u)M0

2 (t)M2(t)], (3.4a)

M0(t) = G01(t)M

01 (t)+G0

2(t)M02 (t) and M(t) = G1(t)M1(t)+G2(t)M2(t),

(3.4b)σj is the volatility of Mj(t), j = 1, 2, and ρ01 is the correlation coefficientbetween M0

1 (t) and M1(t).Provided that τ > T , the protection buyer pays cash amount α∆t at

times t1, t2, · · · , tN = T within the life of the CDS. Here, α is the swap rate,T is the maturity date of the CDS and ∆t denotes the uniform time intervalbetween successive payment dates. The CDS premium payments cease afterthe occurrence of the credit event. Let Vf(t; T, α) denotes the time-t value ofthese swap premium payments under swap rate α (commonly called the feeleg), t < t1. Under our two-factor RLN formulation, we obtain

Vf(t; T, α)

= Et

[N∑

n=1

α∆t exp

(−

∫ tn

t

ru du

)1τ>tn

]

= α∆tN∑

n=1

Et

[exp

(−

∫ tn

t

(ru + λu) du

)](3.5a)

13

= α∆tN∑

n=1

Et

[G0

1(tn)M01 (tn) + G0

2(tn)M02 (tn)

G01(t)M

01 (t) + G0

2(t)M02 (t)

G1(tn)M1(tn) + G2(tn)M2(tn)

G1(t)M1(t) + G2(t)M2(t)

]

=α∆t

∑N

n=1 F (t; tn)

M0(t)M(t),

where

F (t; tn) = G01(tn)G1(tn)M0

1 (t)M1(t) exp(ρ01σ01σ1(tn − t))

+ G01(tn)G2(tn)M0

1 (t)M2(t) + G02(tn)G1(tn)M0

2 (t)M1(t)

+ G02(tn)G2(tn)M0

2 (t)M2(t). (3.5b)

The fair swap rate α(t; T ) of the T -maturity CDS is obtained by setting Vf

equal Vp at time t. This gives

α(t; T ) =

∫ T

t

δ(u)Q(t; u) du

∆tN∑

n=1

F (t; tn)

. (3.6)

Credit default swaptions

We consider the pricing of a credit default swaption under the two-factorRLN model. A European credit default swaption (CDS option) gives theholder the right to buy or sell a CDS protection on a reference risky obligorover a future time period (swaption’s life) for a specified swap rate Kα onswaption’s maturity date TS. The CDS option is knocked out if the referenceobligor defaults during the life of the option. More details on the productnature of credit default swaptions and an alternative pricing approach of theswaption can be found in Hull-White’s paper (2003).

Consider a call swaption, provided that the obligor has not defaulted atTS, the option holder chooses to exercise the swaption if the market rateof the underlying CDS is higher than the swap spread Kα. For notationalconvenience, we write t0 = TS and tN = T . The swaption maturity datet0 is ∆t time period before the first swap payment date t1 and the CDSmaturity date tN is the last payment date, where ∆t = tn − tn−1 (valid

14

for n = 1, 2, · · ·N) is the uniform time interval between successive paymentdates.

Let V (t; t0, tN) denote the time-t price of the t0-maturity call swaption.The life of the underlying CDS covers the period [t0, tN ], t < t0. Under thetwo-factor RLN model, we obtain

V (t; t0, tN) = Et

[max(Vp(t0; tN ) − Vf (t0; tN , Kα), 0)

exp

(−

∫ t0

t

ru du

)1τ>t0

]

= Et

max

∫ tN

t0

δ(u)Q(t0; u) du

M0(t0)M(t0)−

Kα∆tN∑

n=1

F (t0; tn)

M0(t0)M(t0), 0

M0(t0)

M0(t)

M(t0)

M(t)

]. (3.7)

There are four stochastic quantities involved in the above expectationcalculations. They are the exponential martingales M0

1 (t0), M02 (t0), M1(t0)

and M2(t0). Since the functional dependence on these exponential martin-gales is quite complicated, the Black-Scholes type closed form formula forswaption price V (t; t0, tN) cannot be obtained. Note that other exponentialmartingales M0

1 (ti), M02 (ti), M1(ti), M2(ti), i = 1, 2, · · · , N , do not appear in

Eq. (3.7). This is expected since the swaption possesses the optionality fea-ture over the time interval (t, t0) only. The underlying asset of the swaptionis the CDS, which is a non-option product.

3.2 Basket default swaps

We consider the valuation of the kth-to-default credit default swap whosereference portfolio consists of I risky assets. Similar to the single-name CDS,the protection buyer pays periodic premium payments until the kth defaultoccurs or the default swap matures, whichever comes earlier. In our pricingmodel, we assume that the protection seller pays at the time of the kth defaultthe loss of the kth defaulting obligor.

First, we would like to price a basket default swap with the first-to-defaultfeature. Following a similar approach as that of the single-name CDS, we

15

compute the time-t value of the fee leg and protection leg payments. Let τbe the arrival time of the first default among the I risky obligor. The time-tfee leg value is given by [see Eq. (3.4a)]

Vf (t; T, α) = α∆t

N∑

n=1

Et

[exp

(−

∫ tn

t

ru du

)1τ>tn

]. (3.8)

Using the conditional independence assumption of the default times andapplying the RLN formulation, we obtain [see Eq. (2.24)]

Vf(t; T, α) = α∆tN∑

n=1

Et

[I∏

i=0

Mi(tn)

Mi(t)

], (3.9a)

whereMi(t) = Gi

1(t)Mi1(t) + Gi

2(t)Mi2(t). (3.9b)

The evaluation of the expectation of the product in Eq. (3.9a) has beendiscussed in Sec. 2.4 (also see the Appendix).

To compute the time-t value of the protection leg payment, we observethat

Vp(t; T ) =I∑

i=1

Et

[∫ T

t

δi(u) exp

(−

∫ u

t

rs ds

)

PT

[u < τ i ≤ u + du, τ j > u for all j 6= i

]]. (3.10)

Under our two-factor RLN formulation, we have [see Eq. (2.19) and (2.23)]

PT [u < τ i ≤ u + du, τ j > u for all j 6= i]

= −gi1(u)M i

1(u) + gi2(u)M i

2(u)

Gi1(t)M

i1(t) + Gi

2(t)Mi2(t)

I∏

j = 1

j 6= i

Mj(u)

Mj(t)du (3.11a)

so that

Vp(t; T ) =I∑

i=1

Et

∫ T

t

−δi(u)gi1(u)M i

1(u) + gi2(u)M i

2(u)

Gi1(u)M i

1(u) + Gi2(u)M i

2(u)

I∏

j = 0

j 6= i

Mj(u)

Mj(t)du

.

(3.11b)

16

The expectation calculation required in Eq. (3.11b) can be performed in asimilar manner. Once the present value of the fee leg and protection leg pay-ments are available, we can obtain the fair swap rate α by equating Vf(t; T, α)and Vp(t; T ).

Next, we would like to price a basket default swap with the kth-to-defaultfeature. For simplicity of presentation, we make the assumption that all riskyobligors are homogeneous in the sense that their default processes share thesame set of parameter values and the losses upon default are identical. Weassume that obligor i1 defaults first, obligor i2 defaults next, · · · , and the kth-to-default is obligor ik, k < I. Under the assumption of conditional indepen-dence of the default times and the two-factor RLN formulation, the probabil-ity of these default events occurring in (u1, u1+du], (u2+du2], · · · , (uk+duk],where t < u1 < u2 < · · · < uk < T conditional on other obligors survive be-yond uk is given by

fk(u1, u2, · · · , uk) du1 du2 · · · duk

= PT [u1 < τ i1 ≤ u1 + du1, u2 < τ i2 ≤ u2 + du2, · · · , uk < τ ik ≤ uk + duk,

τ i > uk for all i, where i 6= i1, · · · , ik]

=

∏

iq=i1,··· ,ik

[−

giq1 (uq)M

iq1 (uq) + g

iq2 (uq)M

iq2 (uq)

Giq1 (t)M

iq1 (t) + G

iq2 (t)M

iq2 (t)

]

I∏

j = 1

j 6= i1, · · · , ik

Mj(uk)

Mj(t)

du1 du2 · · ·duk. (3.12)

We have considered one particular set of k defaulting firms that defaultin a pre-set order. The total number of choices of k obligors from I obligorsthat default in a specified order is I!/(I − k)!. At the default time uk of thekth defaulting obligor, the compensation payment made by the protectionseller is δik(uk). Taking advantage of the homogeneity assumption of the

17

risky obligors, the time-t value of the protection leg payment is given by

V (k)p (t; T ) =

I!

(I − k)!Et

[∫ T

t

∫ T

u1

· · ·

∫ T

uk−1

δik(uk)M0(uk)

M0(t)

]

fk(u1, u2, · · · , uk) duk · · · du2 du1

]. (3.13)

Let V(k)f (t; T, α) denote the time-t value of the fee leg payments of the kth-

to-default basket default swap. We would like to develop a recursive relationbetween V

(k)f and V

(k−1)f , with V

(1)f given by Eq. (3.9a). At each payment

time tn, n = 1, 2, · · · , N , it may occur that the cash amount α∆t is paid asthe fee leg payment in the kth-to-default swap but not in the (k − 1)th-to-default swap. Such scenario appears when there are exactly k − 1 defaultsbefore tn. Using the homogeneity assumption of obligors, we obtain

V(k)f (t; T, α) = V

(k−1)f (t; T, α) + α∆t

N∑

n=1

Et

[exp

(−

∫ tn

t

rs ds

)

1exactly k − 1 defaults before tn

]

= V(k−1)f (t; T, α) + α∆t

I!

(I − k + 1)!∫ tn

t

∫ tn

u1

· · ·

∫ tn

uk−2

N∑

n=1

Et

[M0(tn)

M0(t)fk−1(u1, u2, · · · , uk−1)

duk−1 · · · du2 du1

], (3.14)

where

fk−1(u1, u2, · · · , uk−1)

=∏

iq=i1,··· ,ik−1

[

−g

iq1 (uq)M

iq1 (uq) + g

iq2 (uq)M

iq2 (uq)

Giq1 (t)M

iq1 (t) + G

iq2 (t)M

iq2 (t)

]

I∏

j = 1

j 6= i1, · · · , ik−1

Mj(tn)

Mj(t). (3.15)

Lastly, the fair swap rate α is obtained by equating V(k)p (t; T ) and V

(k)f (t; T, α).

18

4 Pricing calculations of swaptions and bas-

ket default swaps

In this section, we would like to illustrate the applicability of the two-factorRLN approach to price two common types of credit derivatives, namely,single-name credit default swaptions and kth-to-default basket default swaps.First of all, one has to specify the coefficient functions Gi

j(t), j = 1, 2, i =1, 2, · · · , I, of the stochastic factors. We chose the class of exponential func-tions as the coefficient functions in our numerical calculations, where

Gij(t) = e−vt, (4.1)

for some constant v. Drawing the analogy between the short rate in a riskfreebond and the default intensity in a risky bond, the parameter v in the co-efficient functions can be viewed as the credit deterioration rate of the riskyobligor.

For ease of numerical implementation in our sample pricing calculations ofcredit derivatives, we take the homogeneity assumption where the stochasticfactors share the same set of parameter values and all underlying bondsin the reference portfolio have maturity beyond the expiration date of thecredit derivatives. Also, the compensation payment is chosen to take theform: δ(u) = 0.4022e0.04u.

4.1 Single-name credit default swaptions

Under our two-factor RLN model, the pricing calculation of a credit defaultswaption amounts to the evaluation of the expectation of the optionality terminvolving the exponential martingales as shown in Eq. (3.7). Once the cor-relation structure among the four exponential martingales, M0

1 (t0), M02 (t0),

M1(t0), M2(t0), has been specified, it becomes quite straightforward to applystandard Monte Carol simulation method to implement the expectation cal-culations. In our sample calculation, we consider an one-year swaption ona 5-year swap with yearly fee payments so that we set t = 0, t0 = 1, T = 6and N = 5. We use 10, 000 paths in our Monte Carlo simulation. Otherparameter values in the model used in the calculations are: v = 0.022 andρ = −0.6.

In Table 1, we tabulate the swaption price V (0; 1, 6) (rounded to thenearest basis points) with varying values of strike swap rate Kα and volatility

19

parameter σ of the exponential martingales. The forward price at time 0of the underlying credit default swap is also shown. The forward price isobtained by taking it to be the value of Kα such that

Et

∫ tN

t0

δ(u)Q(t0; u) du− Kα∆t

N∑

n=1

F (t0; tn)

M0(t)M(t)

= 0. (4.2)

If the strike swap rate Kα is set at the forward price, then the swaptionbecomes at-the-money. In our sample calculations, we have chosen Kα to beless than the forward price so that the swaptions are all in-the-money.

The numerical results of swaption prices shown in Table 1 reveal thatthe option price shows almost a linear relationship with the moneyness ofthe strike rate. This price-strike linearity property is consistent with othercalculations of swaption prices using different pricing approach (Hull andWhite, 2003). We may expect to observe convexity in the price-strike relationwhen Kα moves further away from the forward price (deeper-in-the-money).Since most credit default swaptions are traded near-the-money, it would beinterested to observe this linear price-strike phenomenon when Kα is closeto the forward price. Also, as revealed form Table 1, the swaption price isseen to be an increasing function of the volatility level of the exponentialmartingales in the stochastic factors. This is consistent with usual propertyof impact of volatility on option price functions.

4.2 Multi-name kth-to-default basket default swap

We would like to present the numerical results of applying our two-factorRLN model to price basket default swaps with the kth-to-default feature. Thenumerical procedure of finding the fair swap rate using a recursive scheme hasbeen formulated in Sec. 3.2 [see Eqs. (3.13-3.15)]. Note that the calculationsdo not involve Monte Carlo simulation. Model parameter values used inthe calculations are: σ = 0.3, v = 0.018, δ(u) = 0.4022e0.04u (or otherwisespecified). In Tables 2 and 3, we tabulate the swap rate for the 5-yearprotection for the kth-default, k = 1, 2, 3, 4, from a basket of 10 obligors.As shown in Table 2, the swap rate values are seen to be decreasing withrespect to the correlation coefficient ρ between the stochastic factors. This

20

is consistent with the market observations on most CDS spreads where theswap rate increases with lower correlation among the obligors’ credit risks.For the first-to-default basket CDS, the chance of arrival of the first default ishigher when the obligors’ risk are uncorrelated. Our numerical results showthat the same phenomenon of increasing swap rate with lower correlationpersists with the general kth-to-default basket CDS. The numerical swap ratevalues in Table 3 reveal the obvious fact that the swap rate increases withhigher default probability of the obligor, as proxied by higher value of thecredit deterioration rate v. If one seeks protection on the occurrence of adefault that occurs after multiple earlier defaults, one expects to pay a lowerprotection fee (swap rate) for this less likely event. This explains why theswap rate of the kth-to-default feature decreases with increasing value of k.

5 Conclusion

We have presented the multi-factor rational lognormal approach to model theriskfree interest rate process and default intensity processes of risky obligors.Similar to other default risk models, we relate the default intensity processto the term structure of survival probabilities. One may use observed termstructures of default free bond prices and prices of defaultable bonds issuedby the risky firm to calibrate the survival probabilities. Under the rationallognormal framework, positivity of the values of the default intensities andinterest rate are guaranteed. This avoids the drawback in Vasicek type modelof stochastic intensity processes that intensity value may become negative.Correlation structure is introduced between the stochastic factors in our two-factor rational lognormal framework. In each of the intensity processes, onestochastic factor is obligor-specific while the other is correlated to a commonset of economy wide state variables. Default correlation among obligors isgenerated through correlation between these stochastic factors.

To enhance the analytic tractability of our pricing model, we take thestochastic factors to be lognormally distributed and follow the assumptionof conditional independence of the default times. We manage to derive theanalytic procedure to compute the joint distribution of default times of abasket of risky obligors. As a result, we have been successful to find closedform analytic formula for the fair swap premium of single-name credit defaultswaps under the rational lognormal model. Though closed form formulacannot be obtained for the price function of the credit default swaption, we

21

illustrate how to calculate the numerical value of the swaption price usingMonte Carlo simulation. Pricing behaviors of swaptions under varying valuesof strike swap rate and volatility have been examined. Furthermore, weconsidered the pricing of the kth-to-default basket default swaps. We manageto devise an effective recursive scheme for the calculation of the swap rate,thus avoiding the computationally time consuming Monte Carlo simulation.Our numerical calculations show that the swap rate decreases with increasingcorrelation among the stochastic factors and increases with increasing creditdeterioration rate.

In summary, we model the riskfree interest rate and default intensitiesunder the unified rational lognormal framework. Taking the usual lognor-mality assumption of the stochastic factors and conditional independence ofthe default times, our pricing approach exhibits nice analytic tractability inpricing single-name and multi-name credit derivatives. With the advantageof ease of numerical implementation, the rational lognormal approach maybe proved to be useful in pricing other exotic correlation products traded inthe financial markets.

References

[1] Cairns, A. 2004. A family of term-structure models for long-term riskmanagement and derivative pricing. Mathematical Finance, vol. 14, no.3, 415-444.

[2] Flesaker, B., L. Hughston. 1996. Positive interest. Risk, vol. 9, no. 1,46-49.

[3] Hull, J., A. White. 2003. The valuations of credit default swap options.Journal of Derivatives, vol. 10, issue 3, 40-52.

[4] Hull, J., A. White. 2004. Valuation of a CDO and an nth to default CDSwithout Monte Carlo simulation. Journal of Derivatives, vol. 12, issue 2,8-23.

[5] Jarrow, R., F. Yu. 2001. Counterparty risk and the pricing of defaultablesecurities. Journal of Finance, 56(5), 1765-1799.

22

[6] Kijima, M. 2000. Valuation of a credit swap of the basket type. Reviewof Derivatives Research, 4, 81-97.

[7] Lando, D. 2004. Credit risk modeling, Princeton University Press, Prince-ton, USA.

[8] Leung, K.S., Y.K. Kwok. 2006. Credit contagion via interacting intensi-ties: Markov chain framework. Working paper of Hong Kong Universityof Science and Technology.

[9] Li, D. 2000. On default correlation: a copula function approach. Journalof Fixed Income, 9(4), 43-54.

[10] Nakamura, N., F. Yu. 2000. Interest rate, currency and equity deriva-tives: Valuation using the potential approach. International Review ofFinance, 1:4, 269-294

[11] Rogers, C. 1997. The potential approach to the term structure of interestrates and foreign exchange. Mathematical Finance, 7, 157-176.

[12] Rutkowski, M. 1997. A note on the Flesaker-Hughston model of the termstructure of interest rates. Applied Mathematical Finance, 4, 151-163.

[13] Schonbucher, P. 2003. Credit derivatives pricing models: Models, pricingand implementation, Wiley, West Sussex, UK.

[14] Sklar, A. 1959. Fonctions de repartition a n dimensions et leurs marges,Publ. Inst. Stat. Univ. Paris 8, 229-231.

Acknowledgement

This research was supported by the Research Grants Council of Hong Kong,HKUST6425/05H.

23

Appendix – Analytic calculations of the joint survival probabilities

Let PI be the power set of I = 1, 2, · · · , I. Take a subset S of PI , say,S = i1, · · · , ik, where i1, · · · , ik are distinctive positive integers belonging

to the set I. The notation∏

i∈S

Gi1(ti)M

i1(ti) means the product

[Gi1

1 (ti1)Mi11 (ti1)

]· · ·[Gik

1 (tik)Mik1 (tik)

].

The joint survival probabilities defined in Eq. (2.24) can be expressed as

Pt(τ1 > t1, · · · , τ I > tI)

=Et

[∑S∈PI

∏i1∈S

Gi11 (ti1)M

i11 (ti1)

∏i2∈SC Gi2

2 (ti2)Mi22 (ti2)

]∏I

i=1 [Gi1(t)M

i1(t) + Gi

2(t)Mi2(t)]

=

∑S∈PI

∏i1∈S

Gi11 (ti1)

∏i2∈SC Gi2

2 (ti2)Et

[∏i1∈S

M i11 (ti1)

]∏i2∈SC Et

[M i2

2 (ti2)]

∏I

i=1 [Gi1(t)M

i1(t) + Gi

2(t)Mi2(t)]

.

Here, we have made use of the correlation structure that the second stochasticfactors M i

2 are uncorrelated with any other stochastic factor in the model.The product of the two expectation terms can be expressed as

Et

[∏

i1∈S

M i11 (ti1)

]∏

i2∈SC

Et

[M i2

2 (ti2)]

= Et

[exp

(∑

i1∈S

ln M i11 (ti1)

)]

∏

i2∈SC

M i22 (t)

= exp

(Et

[∑

i1∈S

ln M i11 (ti1)

]+ var

(∑

i1∈S

ln M i11 (ti1)

))

∏

i2∈SC

M i22 (t)

.

Since ln M i11 (ti1) are Brownian processes, it is quite straightforward to com-

pute the mean and variance of the sum of these Brownian processes. Weobtain

Et

[∑

i∈S

ln M i1(ti)

]

=∑

i∈S

[lnM i

1(t) − (σi1)

2(ti − t)]

var

(∑

i∈S

ln M i1(ti)

)=∑

i∈S

∑

k∈S

cov(ln M i

1(ti), ln Mk1 (tk)

),

24

where

cov(ln M i

1(ti), lnMk1 (tk)

)=

ρikσ

i1σ

k1 [min(ti, tk) − t], i 6= k

(σi1)

2(ti − t), i = k.

25

σ Forward price Kα V (0; 1, 6)103 18

0.2 104 102 34100 67103 22

0.3 104 102 38100 70103 26

0.4 105 102 41100 72

Table 1 We tabulate the swaption price V (0; 1, 6) (rounded to the near-est basis point) with varying values of strike swap rate Kα and volatilityparameter σ. The forward price of the underlying CDS is also shown as abenchmark. Parameter values used in the calculations are: v = 0.022, ρ =−0.6, δ(u) = 0.422e0.04u, t = 0, t0 = 1, tN = 6, and yearly swap payment isassumed.

ρ = 0 ρ = 0.3 ρ = 0.61st-to-default 872 669 3512nd-to-default 213 193 883rd-to-default 45 41 144th-to-default 7 5 1

Table 2 We tabulate the swap rate (rounded to the nearest basis point) ofa 5-year kth-to-default basket default swap referencing 10 obligors with vary-ing values of correlation coefficient ρ between the stochastic factors. Otherparameter values used in the calculations are: v = 0.018, σ = 0.3, δ(u) =0.422e0.04u and yearly swap payment is assumed.

26

v = 0.01 v = 0.018 v = 0.031st-to-default 357 669 11862nd-to-default 70 193 4463rd-to-default 9 41 1394th-to-default 1 5 31

Table 3 We tabulate the swap rate (rounded to the nearest basis point) of a5-year kth-to-default basket default swap referencing 10 obligors with varyingvalues of the credit deterioration rate v. The other parameter values are setto be: ρ = 0.3, σ = 0.3, δ(u) = 0.4022e0.04u, and yearly swap payment isassumed.

27