Research ArticleValuation of Credit Derivatives with Multiple Time Scales inthe Intensity Model

Beom Jin Kim1 Chan Yeol Park2 and Yong-Ki Ma3

1 Department of Mathematics Yonsei University Seoul 120-749 Republic of Korea2 Korea Institute of Science and Technology Information (KISTI) 245 Daehak-ro Yuseong-gu Daejeon 305-806 Republic of Korea3 Department of Applied Mathematics Kongju National University Chungcheongnam-do 314-701 Republic of Korea

Correspondence should be addressed to Yong-Ki Ma ykmakongjuackr

Received 30 April 2014 Accepted 5 August 2014 Published 21 August 2014

Academic Editor Allan C Peterson

Copyright copy 2014 Beom Jin Kim et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose approximate solutions for pricing zero-coupon defaultable bonds credit default swap rates and bond options based onthe averaging principle of stochastic differential equations We consider the intensity-based defaultable bond where the volatilityof the default intensity is driven by multiple time scales Small corrections are computed using regular and singular perturbationsto the intensity of default The effectiveness of these corrections is tested on the bond price and yield curve by investigating thebehavior of the time scales with respect to the relevant parameters

1 Introduction

It is well-known that the methodology for modeling a creditrisk can be split into two primary models that attempt todescribe default processes the structural model and theintensity-based model Structural models assume the markethas complete information with respect to the underlyingfirmrsquos value process and knows the details of the under-lying firmrsquos capital structure In contrast intensity-based(or reduced-form) models have been developed under theassumption that the default is the first jump of an exogenouslygiven jump process Hence the underlying firmrsquos default timeis inaccessible and driven by a default intensity function ofsome latent state variables Because we are concerned withmodeling the default timewe adopt an intensity-basedmodelwith the fractional recovery assumption made by Duffie andSingleton [1] which can be written as follows

Z (119905 119879) = 119864lowast

[119890minusint

119879

119905119903119904+(1minus119877)120582119904119889119904 | F

119905] (1)

under a risk-neutral probability 119875lowast where 119905 is the currenttime 119879 is the time to maturity 119877 isin [0 1] is the recovery rateand F

119905is the filtration generated by the joint process of the

interest rate and intensity denoted by 119903119904and 120582

119904 respectively

Since the initial contribution to intensity-based model-ing given by Jarrow and Turnbull [2] who considered a con-stant Poisson intensity there have been many mathematicalstudies on credit risk Among them Lando [3] used a defaultterm described by a Cox process whereas Schonbucher [4]developed the term-structure model of defaultable interestrates using the Heath-Jarrow-Morton approach There arealso specific models for interest rate and intensity processesdepending on the macroeconomic environment For exam-pleDuffee [5] performed the estimation of an intensity-basedmodel using an extended Kalman filter approach andMa andKim [6] provided a pricing formula for the credit defaultswap (CDS) by modeling the intensities as jump-diffusionprocesses They provided affine processes for the interest rateand intensity steps supposing a zero correlation betweenthe two processes However several empirical papers havefound a nonzero correlation for the interest rate and intensityprocesses Tomodel this Tchuindjo [7] provided closed-formsolutions for pricing zero-coupon defaultable bonds

The aimof this paper is to investigate the effect ofmultipletime scales of default in pricing defaultable bonds This isbased on modeling the volatility of the default intensity viafast and slow time scales Papageorgiou and Sircar [8] studiedthe pricing of defaultable derivatives such as bonds bond

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 968065 12 pageshttpdxdoiorg1011552014968065

2 Journal of Applied Mathematics

options and CDS rates by intensity-based models under atwo-factor diffusion model for the default intensity Theirstudy was based on multiple time scales as developed byFouque et al [9] where the evolution of the stochastic defaultrate is subject to fast and slow scale variations and showedempirical evidence for the existence of multiple time scalesto price defaultable bonds However they do not assume aspecific process for the default intensity to derive approximatesolutions for the pricing of defaultable derivatives Also thecorrelated Hull and White model developed by Tchuindjodid not produce a hump-shaped yield curve that matches atypical yield curve for defaultable bonds as in the Mertonmodel [10] Thus we mathematically supplement Papageor-giou and Sircarrsquos model with the specific process for thedefault intensity and expand Tchuindjorsquos model bymodifyingmultiple time scales in the stochastic volatility of the intensityprocess For the analytic tractability of multiple time scaleswe use an asymptotic analysis developed by Fouque et al[11] for stochastic volatility in equity models and obtainapproximations of the pricing functions of zero-coupondefaultable bonds CDS rates and bond options Numericalexamples indicate that the multiple time scales have bothquantitative and qualitative effects and that the zero-coupondefaultable bond with a stochastic default intensity tends tobemispriced in terms of the relevant parameters In additionour results are compared to those in Fouque et al [12]who studied the price of defaultable bonds with stochasticvolatility using the structural model and a method to relaxthe drawbacks of the affine processes

The remainder of this paper is structured as follows InSection 2 we obtain a partial differential equation (PDE)withmultiple time scales to price zero-coupon defaultable bondsIn Section 3 approximate solutions for zero-coupon default-able bond prices are derived using an asymptotic analysisthat includes the behavior of multiple time scales in termsof the relevant parameters and some numerical examplesare provided Section 4 applies the results of Section 3 toCDS rates and bond options respectively We present ourconcluding remarks in Section 5

2 Pricing Equation for Zero-CouponDefaultable Bonds

Alizadeh et al [13] and Masoliver and Perello [14] haveshown that two volatility factors which not only control thepersistence of volatility but also revert rapidly to the meanand contribute to the volatility of volatility as pointed outby the empirical findings are in operation Fouque et al[11] also attempted to balance within a sing model involvedin different time scales That is empirical evidence suggeststhat there are two volatility factors To focus on the intensityprocess of intensity-based defaultable bonds we replace theconstant volatility of the default intensity with a stochasticterm denoted by

119905 The value of

119905is given by a bounded

smooth and strictly positive function 119891 such that 119905

=

119891(119910119905 119911

119905) with a fast scale-factor process 119910

119905and a slow scale-

factor process 119911119905 Note that leaving the choice of 119891 free

affords sufficient flexibility while our pricing functions are

unchanged by different choices Refer to Fouque et al [15]To model this we take the first factor 119910

119905driving the volatility

119905as a fast mean-reverting process For the analytic goal we

choose the following Ornstein-Uhlenbeck (OU) process

119889119910119905= 120572 (119898 minus 119910

119905) 119889119905 + 120573 119889119861

(119910)lowast

119905 (2)

where119861(119910)lowast

119905is a standard Brownianmotion in the risk-neutral

probability 119875lowast It is well-known that the solution of (2) is aGaussian process given by

119910119905= 119898 + (119910

0minus 119898) 119890

minus120572119905

+ 120573int119905

0

119890minus120572(119905minus119904)

119889119861(119910)lowast

119904(3)

and 119910119905sim 119873(119898 + (119910

0minus 119898)119890minus120572119905 (12057322120572)(1 minus 119890minus2120572119905)) leading to

an invariant distribution given by119873(119898 12057322120572) Note that wewould use the notation ⟨sdot⟩ called the solvability conditionfor the average with respect to the invariant distribution thatis for an arbitrage function 119896

⟨119896 (119910119905)⟩ =

1

radic2120587]2int+infin

minusinfin

119896 (119910) 119890minus(119910minus119898)

22]2

119889119910 (4)

where ]2 = 12057322120572 For the asymptotic analysis the rate ofmean reversion 120572 in the process 119910

119905is large and its inverse 120598 =

1120572 is the typical correlation of the process119910119905with 120598 gt 0 being

a small parameter We assume that the order of ]2 remainsfixed in scale as 120598 becomes zero Hence we get

120572 = O (120598minus1

) 120573 = O (120598minus12

) ] = O (1) (5)

From these assumptions we replace the stochastic differentialequation (SDE) (2) by

119889119910119905=

1

120598(119898 minus 119910

119905) 119889119905 +

]radic2

radic120598119889119861

(119910)lowast

119905 (6)

Note that the fast scale volatility factor has been consideredas a singular perturbation case Also the volatility

119905has a

slowly varying factor Here we choose as follows

119889119905= 119892 (

119905) 119889119905 + ℎ (

119905) 119889119861

(119911)lowast

119905 (7)

where 119861(119911)lowast

119905is a standard Brownian motion in the risk-

neutral probability 119875lowast We assume that Lipschitz and growthconditions for the coefficients 119892(119911) and ℎ(119911) are satisfiedrespectively For the asymptotic analysis we use anothersmall parameter 120575 gt 0 and then change time 119905 to 120575119905 in

119905

This means that 119911119905

119889

= 120575119905 so that this alteration replaces SDE

(7) by

119889119911119905= 120575119892 (119911

119905) 119889119905 + radic120575ℎ (119911

119905) 119889119861

(119911)lowast

119905 (8)

where 119861(119911)lowast

119905is another standard Brownianmotion in the risk-

neutral probability 119875lowast Note that the slow scale volatilityfactor has been considered as a regular perturbation situa-tion See for example Fouque et al [11] Consequently the

Journal of Applied Mathematics 3

dynamics of the interest rate process 119903119905 intensity process 120582

119905

and stochastic volatility process 119905are given by the SDEs

119889119903119905= (120579

119905minus 119886119903

119905) 119889119905 + 120590119889119861

(119903)lowast

119905

119889120582119905= (120579

119905minus 119886120582

119905) 119889119905 +

119905119889119861

(120582)lowast

119905

119905= 119891 (119910

119905 119911

119905)

119889119910119905=

1

120598(119898 minus 119910

119905) 119889119905 +

]radic2

radic120598119889119861

(119910)lowast

119905

119889119911119905= 120575119892 (119911

119905) 119889119905 + radic120575ℎ (119911

119905) 119889119861

(119911)lowast

119905

(9)

under the risk-neutral probability 119875lowast where 119886 119886 and 120590 areconstants 120579

119905and 120579

119905are time-varying deterministic functions

and the standard Brownian motions 119861(119903)lowast

119905 119861(120582)lowast

119905 119861(119910)lowast

119905 and

119861(119911)lowast

119905are dependent on each other with the correlation

structure given by

119889⟨119861(119903)lowast

119861(120582)lowast

⟩119905

= 120588119903120582119889119905 119889⟨119861

(119903)lowast

119861(119910)lowast

⟩119905

= 120588119903119910119889119905

119889⟨119861(119903)lowast

119861(119911)lowast

⟩119905

= 120588119903119911119889119905 119889⟨119861

(120582)lowast

119861(119910)lowast

⟩119905

= 120588120582119910119889119905

119889⟨119861(120582)lowast

119861(119911)lowast

⟩119905

= 120588120582119911119889119905 119889⟨119861

(119910)lowast

119861(119911)lowast

⟩119905

= 120588119910119911119889119905

(10)

The zero-coupon defaultable bond price with the frac-tional recovery assumption at time 119905 for an interest rateprocess level 119903

119905= 119903 an intensity process level 120582

119905= 120582 a

fast volatility level 119910119905= 119910 and a slow volatility level 119911

119905= 119911

denoted by 119875(119905 119903 120582 119910 119911 119879) is given by

119875 (119905 119903 120582 119910 119911 119879)

= 119864lowast

[119890minusint

119879

119905(119903119904+(1minus119877)120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(11)

and then using the four-dimensional Feynman-Kac formulawe obtain the Kolmogorov PDE

120597119875

120597119905+ (120579

119905minus 119886119903)

120597119875

120597119903+1

212059021205972119875

1205971199032+ (120579

119905minus 119886120582)

120597119875

120597120582

+1

21198912

(119910 119911)1205972119875

1205971205822+1

120598(119898 minus 119910)

120597119875

120597119910+]2

120598

1205972119875

1205971199102+ 120575119892 (119911)

120597119875

120597119911

+1

2120575ℎ

2

(119911)1205972119875

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119875

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119875

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119875

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119875

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119875

120597119910120597119911

minus 119903 + (1 minus 119877) 120582 119875 = 0

(12)

with the final condition 119875(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 Refer toOslashksendal [16]

3 Asymptotic Analysis of Zero-CouponDefaultable Bonds

In this section we employ an asymptotic analysis for thesolution of the PDE (12) and give an approximate solutionfor the zero-coupon defaultable bond price for the smallindependent parameters 120598 and 120575

The zero-coupon defaultable bond price function119875(119905 119903 120582 119910 119911 119879) consists of the Feynman-Kac formulaproblems

L119875 (119905 119903 120582 119910 119911 119879) = 0 119905 lt 119879

119875(119905 119903 120582 119910 119911 119879)1003816100381610038161003816119905=119879 = 1

(13)

where

L =1

120598L

0+

1

radic120598L

1+L

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

(14)

andL0L

1L

2M

1M

2 andM

3are satisfied by

L0= (119898 minus 119910)

120597

120597119910+ ]2

1205972

1205971199102

L1= radic2120590]120588

119903119910

1205972

120597119903120597119910+ radic2119891 (119910 119911) ]120588

120582119910

1205972

120597120582120597119910

L2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910 119911)1205972

1205971205822+ 120590119891 (119910 119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

M1= 120590ℎ (119911) 120588

119903119911

1205972

120597119903120597119911+ 119891 (119910 119911) ℎ (119911) 120588

120582119911

1205972

120597120582120597119911

M2= 119892 (119911)

120597

120597119911+1

2ℎ2

(119911)1205972

1205971199112

M3= radic2]ℎ (119911) 120588

119910119911

1205972

120597119910120597119911

(15)

Here 120572L0is the infinitesimal generator of the OU process

119910119905 L

1contains the mixed partial derivatives due to the

4 Journal of Applied Mathematics

correlation between 119903119905and 119910

119905and between 120582

119905and 119910

119905 L

2is

the operator of the correlated Hull and White model withconstant volatility at the volatility level 119891(119910 119911) M

1includes

the mixed partial derivatives due to the correlation between119903119905and 119911

119905and between 120582

119905and 119911

119905 M

2is the infinitesimal

generator with respect to 119911119905 and M

3contains the mixed

partial derivative due to the correlation between 119910119905and 119911

119905

Now we use the notation 119875119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 This choiceis somewhat simpler than the reverse ordering Hence weconsider an expansion of 119875120598120575

119875120598120575

= 119875120598

0+ radic120575119875

120598

1+ 120575119875

120598

2+ sdot sdot sdot (16)

119875120598

119896= 119875

0119896+ radic120598119875

1119896+ 120598119875

2119896+ 120598

32

1198753119896

+ sdot sdot sdot (17)

for 119896 = 0 1 2 The expansion (16) leads to the leading-order term 119875120598

0and the next-order term 119875120598

1given by the

solutions of the PDEs

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

0= 0 119905 lt 119879 (18)

with the terminal condition 119875120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 1 and

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

1= (M

1+

1

radic120598M

3)119875

120598

0 119905 lt 119879

(19)

with the terminal condition 119875120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively

31 Leading-Order Term 11987500 We insert 119896 = 0 into (17) as

119875120598

0= 119875

00+ radic120598119875

10+ 120598119875

20+ 120598

32

11987530

+ sdot sdot sdot (20)

Applying the expanded solution (20) to (18) leads to

1

120598L

011987500

+1

radic120598(L

011987510

+L111987500)

+ (L011987520

+L111987510

+L211987500)

+ radic120598 (L011987530

+L111987520

+L211987510) + sdot sdot sdot = 0

(21)

Theorem 1 The leading order term 11987500

of the expansion (17)with 119896 = 0 is independent of the fast scale variable 119910 andfurther it has the affine representation

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(22)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 0 where 119861(119905 119879) 119862(119905 119879)and 119865(119905 119879) are given by

119861 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119862 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119865 (119905 119879)

= ln11987500(0 119879)

11987500(0 119905)

minus 119862(119905 119879)120597

120597119904ln119875

00(0 119904)

100381610038161003816100381610038161003816100381610038161003816119904=119905

minus(119861(119905 119879) minus 119862(119905 119879))120597

120597119904lnW(0 119904)

10038161003816100381610038161003816100381610038161003816119904=119905

+1205902

41198863(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862B2 (119905 119879)

+120588119903120582120590120590 (119911)

119886119886 (119886 + 119886)(1 minus 119877)

times (119886 + 119886) (119879 minus 119905) minus (119886119861 (119905 119879) + 1) (119886119862 (119905 119879) + 1) + 1

+2

(119911)

41198863(1 minus 119877) [(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862

1198622

(119905 119879)

minus 119877 2119886 (119879 minus 119905) minus (119886119862 (119905 119879) + 3)

times (119886119862 (119905 119879) minus 1) minus 3 ]

(23)

respectively Here

120590 (119911) = ⟨119891⟩ (119911) =1

radic2120587]2int+infin

minusinfin

119891 (119910 119911) 119890minus(119910minus119898)

22]2

119889119910

2

(119911) = ⟨1198912

⟩ (119911) =1

radic2120587]2int+infin

minusinfin

1198912

(119910 119911) 119890minus(119910minus119898)

22]2

119889119910

(24)

and W(0 119904) is the price of a zero-coupon default-free bondaccording to Hull and White [17]

Proof Multiply (21) by 120598 and then let 120598 rarr 0 This gives thefirst two leading-order terms as

L011987500

= 0

L011987510

+L111987500

= 0(25)

Recall that because the infinitesimal operator L0is the

generator of the OU process 119910119905 the solution 119875

00of (25) must

be a constant with respect to the 119910 variable that is 11987500

=

11987500(119905 119903 120582 119911 119879) Similarly because 119875

00does not rely on the

119910 variable we get L111987500

= 0 and then L011987510

= 0 that is11987510

= 11987510(119905 119903 120582 119911 119879)Thismeans that the first two terms119875

00

and 11987510

do not depend on the current level 119910 of the fast scalevolatility driving the process 119910

119905 In this way we can continue

to eliminate the terms of order 1radic120598 120598 For the order-1term we getL

011987520

+L111987510

+L211987500

= 0This PDE becomes

L011987520

+L211987500

= 0 (26)

Journal of Applied Mathematics 5

because L111987510

= 0 This PDE is a Poisson equation for 11987520

with respect to the infinitesimal operatorL0 It is well-known

that a solution exists only ifL211987500

is centered with respect tothe invariant distribution of the stochastic volatility process119910119905 that is

⟨L211987500⟩ = 0 119905 lt 119879 (27)

with the terminal condition 11987500(119905 119903 120582 119911 119879)|

119905=119879

= 1 Because11987500

does not rely on the 119910 variable the solvability conditionbecomes

⟨L2⟩ 119875

00= 0 (28)

Here ⟨L2⟩ is a partial differential operator given by

⟨L2⟩ =

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582

+1

212059021205972

1205971199032+1

22

(119911)1205972

1205971205822+ 120590120590 (119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

(29)

Finally substituting (22) into (28) we directly obtain theresult of Theorem 1 using the results of Tchuindjo [7]

Note that in the stochastic volatility setting we get theleading order term 119875

00 which is a generalization of the

constant volatility result obtained by Tchuindjo [7] If 119891(119910 119911)is a constant function then our results (22) reduce to those inTchuindjorsquos study

32 First Perturbation Term 119875120598

10 In this subsection we

precisely calculate 119875120598

10by using the result of Theorem 1

Theorem 2 The correction term 11987510

is independent of thevariable 119910 and 119875120598

10= radic120598119875

10(119905 119903 120582 119911 119879) is given by

119875120598

10(119905 119903 120582 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(30)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119866(119879 119879) = 0 where119866(119905 119879) is given by

119866 (119905 119879) = minus int119879

119905

[119880120598

1(1 minus 119877)

2

119861 (119904 119879) 1198622

(119904 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119904 119879) 119862 (119904 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119904 119879)] 119889119904

119880120598

1=

]radic2

radic120572120588119903119910120590⟨120579

1015840

⟩ (119911) +]radic2

radic120572120588120582119910

⟨1198911205951015840

⟩ (119911)

119880120598

2=

]radic2

radic120572120588119903119910120590 ⟨120595

1015840

⟩ (z)

119880120598

3=

]radic2

radic120572120588120582119910

⟨1198911205791015840

⟩ (119911)

(31)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof The order-radic120598 term in (21) leads toL011987530

+ L111987520

+

L211987510

= 0 which is a Poisson equation for 11987530

with respectto the infinitesimal operator L

0whose solvability condition

is given by

⟨L111987520

+L211987510⟩ = 0 (32)

From (26) and (28) we obtain

11987520

= minusLminus1

0(L

2minus ⟨L

2⟩) 119875

00+ 119899 (119905 119903 120582 119911) (33)

for some function 119899(119905 119903 120582 119911) Inserting (33) into (32) wederive a PDE for 119875

10as follows

⟨L2⟩ 119875

10= ⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩ 119875

00 (34)

with the final condition 11987510(119905 119903 120582 119911 119879)|

119905=119879

= 0 Because ourfocus is only the first perturbation to 119875

00 we reset the PDE

(34) with respect to 119875120598

10(119905 119903 120582 119911 119879) = radic120598119875

10(119905 119903 120582 119911 119879) as

follows

⟨L2⟩ 119875

120598

10= A

120598

11987500

A120598

=1

radic120572⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩

(35)

From (15) and (29) we have

L2minus ⟨L

2⟩ =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))1205972

1205971205822

+ 120588119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

1205972

120597119903120597120582

(36)

Then we introduce functions 120579 1198772 rarr 119877 and 120595 1198772 rarr 119877

defined by the solutions of

L0120579 (119910 119911) =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))

L0120595 (119910 119911) = 120588

119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

(37)

respectively and obtain the operatorL120598 denoted by

L120598

= 119880120598

1

1205973

1205971199031205971205822+ 119880

120598

2

1205973

1205971199032120597120582+ 119880

120598

3

1205973

1205971205823 (38)

Hence we obtain (39) in the following form

120597119875120598

10

120597119905+ (120579

119905minus 119886119903)

120597119875120598

10

120597119903+ (120579

119905minus 119886120582)

120597119875120598

10

120597120582+1

212059021205972119875120598

10

1205971199032

+1

22

(119911)1205972119875120598

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120598

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120598

10

= minus [119880120598

1(1 minus 119877)

2

119861 (119905 119879) 1198622

(119905 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119905 119879) 119862 (119905 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119905 119879)] 11987500

(39)

with the terminal condition 119875120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (30) into (39) we obtain the result of Theorem 2by direct computation

6 Journal of Applied Mathematics

33 First Perturbation 119875120575

01 Using similar arguments to those

in Section 32 we will derive the first perturbation term 119875120575

01

We insert 119896 = 1 in (17) as

119875120598

1= 119875

01+ radic120598119875

11+ 120598119875

21+ 120598

32

11987531

+ sdot sdot sdot (40)

Substituting the expansions (20) and (40) into (19) we get

1

120598L

011987501

+1

radic120598(L

011987511

+L111987501)

+ (L011987521

+L111987511

+L211987501)

+ radic120598 (L011987531

+L111987521

+L211987511) + sdot sdot sdot

=1

radic120598M

311987500

+ (M111987500

+M311987510)

+ radic120598 (M111987510

+M311987520) + sdot sdot sdot

(41)

Theorem 3 The correction term 11987501

does not depend on thevariable 119910 and 119875120575

01= radic120575119875

01(119905 119903 120582 119911 119879) is given by

119875120575

01(119905 119903 120582 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(42)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119867(119879 119879) = 0 where119867(119905 119879) is given by

119867(119905 119879) = minusint119879

119905

119881120575

1119861 (119904 119879) + 119881

120575

2(1 minus 119877)119862 (119904 119879) 119889119904

119881120575

1= radic120575120588

119903119911120590ℎ (119911)

119881120575

2= radic120575120588

120582119911ℎ

119911(119911) ⟨119891⟩ (119911) + ℎ (119911) ⟨119891⟩

119911(119911)

(43)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof Multiply (41) by 120598 and let 120598 rarr 0 Then we find thefirst two leading-order terms as follows

L011987501

= 0

L011987511

+L111987501

= M311987500

(44)

Recall that because L0is the infinitesimal generator of the

OU process 119910119905 the solution 119875

01of (44) must be a constant

with respect to the 119910 variable that is 11987501

= 11987501(119905 119903 120582 119911 119879)

The next order also generates L011987511

= 0 Because M3has

a derivative with respect to the 119910 variable and 11987500

does notrely on the 119910 variable we get M

311987500

= 0 and L111987501

= 0respectively as remarked above Then 119875

11does not depend

on the 119910 variable that is 11987511

= 11987511(119905 119903 120582 119911 119879) This means

that the first two terms 11987501

and 11987511

do not depend on thecurrent level 119910 of the fast scale volatility driving the process119910119905 In this way we can continue to eliminate terms of order

1radic120598 120598 For the order-1 term we haveL011987521

+L111987511

+

L211987501

= M111987500

+ M311987510 This PDE becomes L

011987521

+

L211987501

= M111987500 because L

111987511

= 0 and M311987510

= 0This PDE is a Poisson equation for 119875

21with respect to the

infinitesimal operator L0 It is well-known that a solution

exists only if L211987501

= M111987500

is centered with respect to

the invariant distribution of the stochastic volatility process119910119905 Because 119875

00and 119875

01do not depend on the 119910 variable we

obtain

⟨L2⟩ 119875

01= A

120575

11987500 119905 lt 119879 (45)

with the terminal condition 11987501(119905 119903 120582 119911 119879)|

119905=119879

= 0 where

A120575

= 120590ℎ119911(119911) 120588

119903119911

120597

120597119903+ 120588

120582119911(ℎ

119911(119911) ⟨119891⟩ (119911)

+ℎ (119911) ⟨119891⟩119911(119911))

120597

120597120582

⟨119891⟩119911(119911) =

120597

120597119911⟨119891⟩ (119911) ℎ

119911(119911) =

120597

120597119911ℎ (119911)

(46)

respectively Because we concentrate on only the first per-turbation to 119875

00 we reset the PDE (45) with respect to

119875120575

01(119905 119903 120582 119911 119879) = radic120575119875

01(119905 119903 120582 119911 119879) as follows

⟨L2⟩ 119875

120575

01= radic120575A

120575

11987500 119905 lt 119879 (47)

with the final condition 119875120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 We thenobtain the operatorL120575 expressed as

L120575

= 119881120575

1

120597

120597119903+ 119881

120575

2

120597

120597120582 (48)

Hence we obtain (45) as follows

120597119875120575

10

120597119905+ (120579

119905minus 119886119903)

120597119875120575

10

120597119903+ (120579

119905minus 119886120582)

120597119875120575

10

120597120582+1

212059021205972119875120575

10

1205971199032

+1

22

(119911)1205972119875120575

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120575

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120575

10

= minus 119881120575

1119861 (119905 119879) + 119881

120575

2(1 minus 119877)119862 (119905 119879) 119875

00

(49)

with the terminal condition 119875120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (42) into (49) we obtain the result of Theorem 3by direct computation

Note that the group parameters 119880120598

1 119880120598

2 119880120598

3 119881120575

1 and 119881120575

2

which contain model parameters are needed for the pricingof the zero-coupon defaultable bond and simplify the estima-tion procedure See Fouque et al [15] for a general reference

In summary from (16) the asymptotic analysis of thezero-coupon defaultable bond price gives

119875120598120575

(119905 119903 120582 119910 119911 119879)

asymp 11987500

(119905 119903 120582 119911 119879) + 119875120598

10(119905 119903 120582 119911 119879) + 119875

120575

01(119905 119903 120582 119911 119879)

119905 lt 119879

(50)

with the final condition 119875120598120575(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 and

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120598

10(119905 119903 120582 119911 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120575

01(119905 119903 120582 119911 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(51)

Journal of Applied Mathematics 7

Table 1 Effect of the recovery rate

Recovery rate (119877) 01 02 03 05 07 08Fast scale factor 035 009 minus011 minus029 minus027 minus020Slow scale factor minus043 minus026 minus010 024 057 073Multiple scale factors minus010 minus020 minus021 minus003 034 058

where 11987500(119905 119903 120582 119911 119879) 119875120598

10(119905 119903 120582 119911 119879) and 119875120575

01(119905 119903 120582 119911 119879)

are given by (22) (30) and (42) respectively In terms ofaccuracy of the approximation we refer to the error estimateobtained by Papageorgiou and Sircar [8]

34 Numerical Results In this subsection we conclude ourpaper with some sensitivity analyses of themodel parametersThe parameter values used to obtain the results shown inTable 1 are 119886 = 02 119886 = 03119880120598

1= 001119880120598

2= minus003 119880120598

3= 004

119881120575

1= 002 119881120575

2= minus003 119905 = 0 and 119879 = 1 We calculate

the magnitude of mispricing with respect to the zero-coupondefaultable bond as a percentage of the face value of the bondThat is we show that the zero-coupon defaultable bond tendsto be overpriced or underpriced in terms of the parametersinvolvedThe mispricing as a percentage of the zero-coupondefaultable bond price caused by changes in the value ofthe recovery rate is shown in Table 1 The second row showsresults when only the fast scale factor of the intensity (119881

120575

1=

119881120575

2= 0) is considered and the third row shows results

when only the slow scale factor (119880120598

1= 119880

120598

2= 119880

120598

3= 0) is

considered The fourth row shows the results using multiplescale factors We can see from Table 1 that the mispricing ofthe zero-coupon defaultable bondwith a fast scale factor doesnot behave monotonically as the value of the recovery rateincreases but the mispricing of the zero-coupon defaultablebond with a slow scale factor monotonically increases withthe value of the recovery rate Additionally the mispricing ofthe zero-coupon defaultable bond with multiple scale factorsdoes not behave monotonically as the value of the recoveryrate increases From the above investigation we note that themispricing of the zero-coupon defaultable bond takes variousforms depending on the recovery rate parameters involved

Figures 1 and 2 show results for parameter values of 119886 =

02 119886 = 03 120590 = 01 120590(119911) = 015 2(119911) = 018 120588119903120582

= minus02119880120598

1= minus007 119880120598

2= 001 119880120598

3= minus006 119881120575

1= 001 119881120575

2= minus004

119903 = 015 120582 = 013 119905 = 0 119879 = 10 and 119877 = 04 These figuresillustrate the defaultable bond price and the correspondingyield curve respectively Figures 1 and 2 have two typesof curve Case 1 corresponds to constant volatility in theintensity process whereas Case 2 contains both fast andslow scale factors Figure 1 shows that the defaultable bondprices with stochastic intensity become higher than thosewith constant volatility as the time to maturity increasesIn addition the hump-shaped yield curve which matchesthose in structural models [10] appears in Case 2 in Figure 2That is the multiple time scales have both quantitative andqualitative effects Hence Figures 1 and 2 show the significanteffect of multiple time scales in the stochastic intensity onboth defaultable bond prices and yields

Time to maturity (years)

Def

aulta

ble b

ond

pric

es

0 2 4 6 8 100

02

04

06

08

1

Case 1Case 2

Figure 1 Defaultable bond prices (source KIS Pricing)

Time to maturity (years)0 2 4 6 8 10

00025

0003

00035

0004

00045

0005

00055

0006

Yield

s (

)

Case 1Case 2

Figure 2 Yield curves (source KIS Pricing)

4 Credit Default Swap and Bond Option

In this section we derive a formula for the CDS rate andobtain another formula for options when the underlyingasset is a zero-coupon defaultable bond using the results ofSection 3

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

options and CDS rates by intensity-based models under atwo-factor diffusion model for the default intensity Theirstudy was based on multiple time scales as developed byFouque et al [9] where the evolution of the stochastic defaultrate is subject to fast and slow scale variations and showedempirical evidence for the existence of multiple time scalesto price defaultable bonds However they do not assume aspecific process for the default intensity to derive approximatesolutions for the pricing of defaultable derivatives Also thecorrelated Hull and White model developed by Tchuindjodid not produce a hump-shaped yield curve that matches atypical yield curve for defaultable bonds as in the Mertonmodel [10] Thus we mathematically supplement Papageor-giou and Sircarrsquos model with the specific process for thedefault intensity and expand Tchuindjorsquos model bymodifyingmultiple time scales in the stochastic volatility of the intensityprocess For the analytic tractability of multiple time scaleswe use an asymptotic analysis developed by Fouque et al[11] for stochastic volatility in equity models and obtainapproximations of the pricing functions of zero-coupondefaultable bonds CDS rates and bond options Numericalexamples indicate that the multiple time scales have bothquantitative and qualitative effects and that the zero-coupondefaultable bond with a stochastic default intensity tends tobemispriced in terms of the relevant parameters In additionour results are compared to those in Fouque et al [12]who studied the price of defaultable bonds with stochasticvolatility using the structural model and a method to relaxthe drawbacks of the affine processes

The remainder of this paper is structured as follows InSection 2 we obtain a partial differential equation (PDE)withmultiple time scales to price zero-coupon defaultable bondsIn Section 3 approximate solutions for zero-coupon default-able bond prices are derived using an asymptotic analysisthat includes the behavior of multiple time scales in termsof the relevant parameters and some numerical examplesare provided Section 4 applies the results of Section 3 toCDS rates and bond options respectively We present ourconcluding remarks in Section 5

2 Pricing Equation for Zero-CouponDefaultable Bonds

Alizadeh et al [13] and Masoliver and Perello [14] haveshown that two volatility factors which not only control thepersistence of volatility but also revert rapidly to the meanand contribute to the volatility of volatility as pointed outby the empirical findings are in operation Fouque et al[11] also attempted to balance within a sing model involvedin different time scales That is empirical evidence suggeststhat there are two volatility factors To focus on the intensityprocess of intensity-based defaultable bonds we replace theconstant volatility of the default intensity with a stochasticterm denoted by

119905 The value of

119905is given by a bounded

smooth and strictly positive function 119891 such that 119905

=

119891(119910119905 119911

119905) with a fast scale-factor process 119910

119905and a slow scale-

factor process 119911119905 Note that leaving the choice of 119891 free

affords sufficient flexibility while our pricing functions are

unchanged by different choices Refer to Fouque et al [15]To model this we take the first factor 119910

119905driving the volatility

119905as a fast mean-reverting process For the analytic goal we

choose the following Ornstein-Uhlenbeck (OU) process

119889119910119905= 120572 (119898 minus 119910

119905) 119889119905 + 120573 119889119861

(119910)lowast

119905 (2)

where119861(119910)lowast

119905is a standard Brownianmotion in the risk-neutral

probability 119875lowast It is well-known that the solution of (2) is aGaussian process given by

119910119905= 119898 + (119910

0minus 119898) 119890

minus120572119905

+ 120573int119905

0

119890minus120572(119905minus119904)

119889119861(119910)lowast

119904(3)

and 119910119905sim 119873(119898 + (119910

0minus 119898)119890minus120572119905 (12057322120572)(1 minus 119890minus2120572119905)) leading to

an invariant distribution given by119873(119898 12057322120572) Note that wewould use the notation ⟨sdot⟩ called the solvability conditionfor the average with respect to the invariant distribution thatis for an arbitrage function 119896

⟨119896 (119910119905)⟩ =

1

radic2120587]2int+infin

minusinfin

119896 (119910) 119890minus(119910minus119898)

22]2

119889119910 (4)

where ]2 = 12057322120572 For the asymptotic analysis the rate ofmean reversion 120572 in the process 119910

119905is large and its inverse 120598 =

1120572 is the typical correlation of the process119910119905with 120598 gt 0 being

a small parameter We assume that the order of ]2 remainsfixed in scale as 120598 becomes zero Hence we get

120572 = O (120598minus1

) 120573 = O (120598minus12

) ] = O (1) (5)

From these assumptions we replace the stochastic differentialequation (SDE) (2) by

119889119910119905=

1

120598(119898 minus 119910

119905) 119889119905 +

]radic2

radic120598119889119861

(119910)lowast

119905 (6)

Note that the fast scale volatility factor has been consideredas a singular perturbation case Also the volatility

119905has a

slowly varying factor Here we choose as follows

119889119905= 119892 (

119905) 119889119905 + ℎ (

119905) 119889119861

(119911)lowast

119905 (7)

where 119861(119911)lowast

119905is a standard Brownian motion in the risk-

neutral probability 119875lowast We assume that Lipschitz and growthconditions for the coefficients 119892(119911) and ℎ(119911) are satisfiedrespectively For the asymptotic analysis we use anothersmall parameter 120575 gt 0 and then change time 119905 to 120575119905 in

119905

This means that 119911119905

119889

= 120575119905 so that this alteration replaces SDE

(7) by

119889119911119905= 120575119892 (119911

119905) 119889119905 + radic120575ℎ (119911

119905) 119889119861

(119911)lowast

119905 (8)

where 119861(119911)lowast

119905is another standard Brownianmotion in the risk-

neutral probability 119875lowast Note that the slow scale volatilityfactor has been considered as a regular perturbation situa-tion See for example Fouque et al [11] Consequently the

Journal of Applied Mathematics 3

dynamics of the interest rate process 119903119905 intensity process 120582

119905

and stochastic volatility process 119905are given by the SDEs

119889119903119905= (120579

119905minus 119886119903

119905) 119889119905 + 120590119889119861

(119903)lowast

119905

119889120582119905= (120579

119905minus 119886120582

119905) 119889119905 +

119905119889119861

(120582)lowast

119905

119905= 119891 (119910

119905 119911

119905)

119889119910119905=

1

120598(119898 minus 119910

119905) 119889119905 +

]radic2

radic120598119889119861

(119910)lowast

119905

119889119911119905= 120575119892 (119911

119905) 119889119905 + radic120575ℎ (119911

119905) 119889119861

(119911)lowast

119905

(9)

under the risk-neutral probability 119875lowast where 119886 119886 and 120590 areconstants 120579

119905and 120579

119905are time-varying deterministic functions

and the standard Brownian motions 119861(119903)lowast

119905 119861(120582)lowast

119905 119861(119910)lowast

119905 and

119861(119911)lowast

119905are dependent on each other with the correlation

structure given by

119889⟨119861(119903)lowast

119861(120582)lowast

⟩119905

= 120588119903120582119889119905 119889⟨119861

(119903)lowast

119861(119910)lowast

⟩119905

= 120588119903119910119889119905

119889⟨119861(119903)lowast

119861(119911)lowast

⟩119905

= 120588119903119911119889119905 119889⟨119861

(120582)lowast

119861(119910)lowast

⟩119905

= 120588120582119910119889119905

119889⟨119861(120582)lowast

119861(119911)lowast

⟩119905

= 120588120582119911119889119905 119889⟨119861

(119910)lowast

119861(119911)lowast

⟩119905

= 120588119910119911119889119905

(10)

The zero-coupon defaultable bond price with the frac-tional recovery assumption at time 119905 for an interest rateprocess level 119903

119905= 119903 an intensity process level 120582

119905= 120582 a

fast volatility level 119910119905= 119910 and a slow volatility level 119911

119905= 119911

denoted by 119875(119905 119903 120582 119910 119911 119879) is given by

119875 (119905 119903 120582 119910 119911 119879)

= 119864lowast

[119890minusint

119879

119905(119903119904+(1minus119877)120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(11)

and then using the four-dimensional Feynman-Kac formulawe obtain the Kolmogorov PDE

120597119875

120597119905+ (120579

119905minus 119886119903)

120597119875

120597119903+1

212059021205972119875

1205971199032+ (120579

119905minus 119886120582)

120597119875

120597120582

+1

21198912

(119910 119911)1205972119875

1205971205822+1

120598(119898 minus 119910)

120597119875

120597119910+]2

120598

1205972119875

1205971199102+ 120575119892 (119911)

120597119875

120597119911

+1

2120575ℎ

2

(119911)1205972119875

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119875

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119875

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119875

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119875

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119875

120597119910120597119911

minus 119903 + (1 minus 119877) 120582 119875 = 0

(12)

with the final condition 119875(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 Refer toOslashksendal [16]

3 Asymptotic Analysis of Zero-CouponDefaultable Bonds

In this section we employ an asymptotic analysis for thesolution of the PDE (12) and give an approximate solutionfor the zero-coupon defaultable bond price for the smallindependent parameters 120598 and 120575

The zero-coupon defaultable bond price function119875(119905 119903 120582 119910 119911 119879) consists of the Feynman-Kac formulaproblems

L119875 (119905 119903 120582 119910 119911 119879) = 0 119905 lt 119879

119875(119905 119903 120582 119910 119911 119879)1003816100381610038161003816119905=119879 = 1

(13)

where

L =1

120598L

0+

1

radic120598L

1+L

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

(14)

andL0L

1L

2M

1M

2 andM

3are satisfied by

L0= (119898 minus 119910)

120597

120597119910+ ]2

1205972

1205971199102

L1= radic2120590]120588

119903119910

1205972

120597119903120597119910+ radic2119891 (119910 119911) ]120588

120582119910

1205972

120597120582120597119910

L2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910 119911)1205972

1205971205822+ 120590119891 (119910 119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

M1= 120590ℎ (119911) 120588

119903119911

1205972

120597119903120597119911+ 119891 (119910 119911) ℎ (119911) 120588

120582119911

1205972

120597120582120597119911

M2= 119892 (119911)

120597

120597119911+1

2ℎ2

(119911)1205972

1205971199112

M3= radic2]ℎ (119911) 120588

119910119911

1205972

120597119910120597119911

(15)

Here 120572L0is the infinitesimal generator of the OU process

119910119905 L

1contains the mixed partial derivatives due to the

4 Journal of Applied Mathematics

correlation between 119903119905and 119910

119905and between 120582

119905and 119910

119905 L

2is

the operator of the correlated Hull and White model withconstant volatility at the volatility level 119891(119910 119911) M

1includes

the mixed partial derivatives due to the correlation between119903119905and 119911

119905and between 120582

119905and 119911

119905 M

2is the infinitesimal

generator with respect to 119911119905 and M

3contains the mixed

partial derivative due to the correlation between 119910119905and 119911

119905

Now we use the notation 119875119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 This choiceis somewhat simpler than the reverse ordering Hence weconsider an expansion of 119875120598120575

119875120598120575

= 119875120598

0+ radic120575119875

120598

1+ 120575119875

120598

2+ sdot sdot sdot (16)

119875120598

119896= 119875

0119896+ radic120598119875

1119896+ 120598119875

2119896+ 120598

32

1198753119896

+ sdot sdot sdot (17)

for 119896 = 0 1 2 The expansion (16) leads to the leading-order term 119875120598

0and the next-order term 119875120598

1given by the

solutions of the PDEs

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

0= 0 119905 lt 119879 (18)

with the terminal condition 119875120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 1 and

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

1= (M

1+

1

radic120598M

3)119875

120598

0 119905 lt 119879

(19)

with the terminal condition 119875120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively

31 Leading-Order Term 11987500 We insert 119896 = 0 into (17) as

119875120598

0= 119875

00+ radic120598119875

10+ 120598119875

20+ 120598

32

11987530

+ sdot sdot sdot (20)

Applying the expanded solution (20) to (18) leads to

1

120598L

011987500

+1

radic120598(L

011987510

+L111987500)

+ (L011987520

+L111987510

+L211987500)

+ radic120598 (L011987530

+L111987520

+L211987510) + sdot sdot sdot = 0

(21)

Theorem 1 The leading order term 11987500

of the expansion (17)with 119896 = 0 is independent of the fast scale variable 119910 andfurther it has the affine representation

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(22)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 0 where 119861(119905 119879) 119862(119905 119879)and 119865(119905 119879) are given by

119861 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119862 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119865 (119905 119879)

= ln11987500(0 119879)

11987500(0 119905)

minus 119862(119905 119879)120597

120597119904ln119875

00(0 119904)

100381610038161003816100381610038161003816100381610038161003816119904=119905

minus(119861(119905 119879) minus 119862(119905 119879))120597

120597119904lnW(0 119904)

10038161003816100381610038161003816100381610038161003816119904=119905

+1205902

41198863(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862B2 (119905 119879)

+120588119903120582120590120590 (119911)

119886119886 (119886 + 119886)(1 minus 119877)

times (119886 + 119886) (119879 minus 119905) minus (119886119861 (119905 119879) + 1) (119886119862 (119905 119879) + 1) + 1

+2

(119911)

41198863(1 minus 119877) [(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862

1198622

(119905 119879)

minus 119877 2119886 (119879 minus 119905) minus (119886119862 (119905 119879) + 3)

times (119886119862 (119905 119879) minus 1) minus 3 ]

(23)

respectively Here

120590 (119911) = ⟨119891⟩ (119911) =1

radic2120587]2int+infin

minusinfin

119891 (119910 119911) 119890minus(119910minus119898)

22]2

119889119910

2

(119911) = ⟨1198912

⟩ (119911) =1

radic2120587]2int+infin

minusinfin

1198912

(119910 119911) 119890minus(119910minus119898)

22]2

119889119910

(24)

and W(0 119904) is the price of a zero-coupon default-free bondaccording to Hull and White [17]

Proof Multiply (21) by 120598 and then let 120598 rarr 0 This gives thefirst two leading-order terms as

L011987500

= 0

L011987510

+L111987500

= 0(25)

Recall that because the infinitesimal operator L0is the

generator of the OU process 119910119905 the solution 119875

00of (25) must

be a constant with respect to the 119910 variable that is 11987500

=

11987500(119905 119903 120582 119911 119879) Similarly because 119875

00does not rely on the

119910 variable we get L111987500

= 0 and then L011987510

= 0 that is11987510

= 11987510(119905 119903 120582 119911 119879)Thismeans that the first two terms119875

00

and 11987510

do not depend on the current level 119910 of the fast scalevolatility driving the process 119910

119905 In this way we can continue

to eliminate the terms of order 1radic120598 120598 For the order-1term we getL

011987520

+L111987510

+L211987500

= 0This PDE becomes

L011987520

+L211987500

= 0 (26)

Journal of Applied Mathematics 5

because L111987510

= 0 This PDE is a Poisson equation for 11987520

with respect to the infinitesimal operatorL0 It is well-known

that a solution exists only ifL211987500

is centered with respect tothe invariant distribution of the stochastic volatility process119910119905 that is

⟨L211987500⟩ = 0 119905 lt 119879 (27)

with the terminal condition 11987500(119905 119903 120582 119911 119879)|

119905=119879

= 1 Because11987500

does not rely on the 119910 variable the solvability conditionbecomes

⟨L2⟩ 119875

00= 0 (28)

Here ⟨L2⟩ is a partial differential operator given by

⟨L2⟩ =

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582

+1

212059021205972

1205971199032+1

22

(119911)1205972

1205971205822+ 120590120590 (119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

(29)

Finally substituting (22) into (28) we directly obtain theresult of Theorem 1 using the results of Tchuindjo [7]

Note that in the stochastic volatility setting we get theleading order term 119875

00 which is a generalization of the

constant volatility result obtained by Tchuindjo [7] If 119891(119910 119911)is a constant function then our results (22) reduce to those inTchuindjorsquos study

32 First Perturbation Term 119875120598

10 In this subsection we

precisely calculate 119875120598

10by using the result of Theorem 1

Theorem 2 The correction term 11987510

is independent of thevariable 119910 and 119875120598

10= radic120598119875

10(119905 119903 120582 119911 119879) is given by

119875120598

10(119905 119903 120582 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(30)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119866(119879 119879) = 0 where119866(119905 119879) is given by

119866 (119905 119879) = minus int119879

119905

[119880120598

1(1 minus 119877)

2

119861 (119904 119879) 1198622

(119904 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119904 119879) 119862 (119904 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119904 119879)] 119889119904

119880120598

1=

]radic2

radic120572120588119903119910120590⟨120579

1015840

⟩ (119911) +]radic2

radic120572120588120582119910

⟨1198911205951015840

⟩ (119911)

119880120598

2=

]radic2

radic120572120588119903119910120590 ⟨120595

1015840

⟩ (z)

119880120598

3=

]radic2

radic120572120588120582119910

⟨1198911205791015840

⟩ (119911)

(31)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof The order-radic120598 term in (21) leads toL011987530

+ L111987520

+

L211987510

= 0 which is a Poisson equation for 11987530

with respectto the infinitesimal operator L

0whose solvability condition

is given by

⟨L111987520

+L211987510⟩ = 0 (32)

From (26) and (28) we obtain

11987520

= minusLminus1

0(L

2minus ⟨L

2⟩) 119875

00+ 119899 (119905 119903 120582 119911) (33)

for some function 119899(119905 119903 120582 119911) Inserting (33) into (32) wederive a PDE for 119875

10as follows

⟨L2⟩ 119875

10= ⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩ 119875

00 (34)

with the final condition 11987510(119905 119903 120582 119911 119879)|

119905=119879

= 0 Because ourfocus is only the first perturbation to 119875

00 we reset the PDE

(34) with respect to 119875120598

10(119905 119903 120582 119911 119879) = radic120598119875

10(119905 119903 120582 119911 119879) as

follows

⟨L2⟩ 119875

120598

10= A

120598

11987500

A120598

=1

radic120572⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩

(35)

From (15) and (29) we have

L2minus ⟨L

2⟩ =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))1205972

1205971205822

+ 120588119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

1205972

120597119903120597120582

(36)

Then we introduce functions 120579 1198772 rarr 119877 and 120595 1198772 rarr 119877

defined by the solutions of

L0120579 (119910 119911) =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))

L0120595 (119910 119911) = 120588

119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

(37)

respectively and obtain the operatorL120598 denoted by

L120598

= 119880120598

1

1205973

1205971199031205971205822+ 119880

120598

2

1205973

1205971199032120597120582+ 119880

120598

3

1205973

1205971205823 (38)

Hence we obtain (39) in the following form

120597119875120598

10

120597119905+ (120579

119905minus 119886119903)

120597119875120598

10

120597119903+ (120579

119905minus 119886120582)

120597119875120598

10

120597120582+1

212059021205972119875120598

10

1205971199032

+1

22

(119911)1205972119875120598

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120598

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120598

10

= minus [119880120598

1(1 minus 119877)

2

119861 (119905 119879) 1198622

(119905 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119905 119879) 119862 (119905 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119905 119879)] 11987500

(39)

with the terminal condition 119875120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (30) into (39) we obtain the result of Theorem 2by direct computation

6 Journal of Applied Mathematics

33 First Perturbation 119875120575

01 Using similar arguments to those

in Section 32 we will derive the first perturbation term 119875120575

01

We insert 119896 = 1 in (17) as

119875120598

1= 119875

01+ radic120598119875

11+ 120598119875

21+ 120598

32

11987531

+ sdot sdot sdot (40)

Substituting the expansions (20) and (40) into (19) we get

1

120598L

011987501

+1

radic120598(L

011987511

+L111987501)

+ (L011987521

+L111987511

+L211987501)

+ radic120598 (L011987531

+L111987521

+L211987511) + sdot sdot sdot

=1

radic120598M

311987500

+ (M111987500

+M311987510)

+ radic120598 (M111987510

+M311987520) + sdot sdot sdot

(41)

Theorem 3 The correction term 11987501

does not depend on thevariable 119910 and 119875120575

01= radic120575119875

01(119905 119903 120582 119911 119879) is given by

119875120575

01(119905 119903 120582 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(42)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119867(119879 119879) = 0 where119867(119905 119879) is given by

119867(119905 119879) = minusint119879

119905

119881120575

1119861 (119904 119879) + 119881

120575

2(1 minus 119877)119862 (119904 119879) 119889119904

119881120575

1= radic120575120588

119903119911120590ℎ (119911)

119881120575

2= radic120575120588

120582119911ℎ

119911(119911) ⟨119891⟩ (119911) + ℎ (119911) ⟨119891⟩

119911(119911)

(43)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof Multiply (41) by 120598 and let 120598 rarr 0 Then we find thefirst two leading-order terms as follows

L011987501

= 0

L011987511

+L111987501

= M311987500

(44)

Recall that because L0is the infinitesimal generator of the

OU process 119910119905 the solution 119875

01of (44) must be a constant

with respect to the 119910 variable that is 11987501

= 11987501(119905 119903 120582 119911 119879)

The next order also generates L011987511

= 0 Because M3has

a derivative with respect to the 119910 variable and 11987500

does notrely on the 119910 variable we get M

311987500

= 0 and L111987501

= 0respectively as remarked above Then 119875

11does not depend

on the 119910 variable that is 11987511

= 11987511(119905 119903 120582 119911 119879) This means

that the first two terms 11987501

and 11987511

do not depend on thecurrent level 119910 of the fast scale volatility driving the process119910119905 In this way we can continue to eliminate terms of order

1radic120598 120598 For the order-1 term we haveL011987521

+L111987511

+

L211987501

= M111987500

+ M311987510 This PDE becomes L

011987521

+

L211987501

= M111987500 because L

111987511

= 0 and M311987510

= 0This PDE is a Poisson equation for 119875

21with respect to the

infinitesimal operator L0 It is well-known that a solution

exists only if L211987501

= M111987500

is centered with respect to

the invariant distribution of the stochastic volatility process119910119905 Because 119875

00and 119875

01do not depend on the 119910 variable we

obtain

⟨L2⟩ 119875

01= A

120575

11987500 119905 lt 119879 (45)

with the terminal condition 11987501(119905 119903 120582 119911 119879)|

119905=119879

= 0 where

A120575

= 120590ℎ119911(119911) 120588

119903119911

120597

120597119903+ 120588

120582119911(ℎ

119911(119911) ⟨119891⟩ (119911)

+ℎ (119911) ⟨119891⟩119911(119911))

120597

120597120582

⟨119891⟩119911(119911) =

120597

120597119911⟨119891⟩ (119911) ℎ

119911(119911) =

120597

120597119911ℎ (119911)

(46)

respectively Because we concentrate on only the first per-turbation to 119875

00 we reset the PDE (45) with respect to

119875120575

01(119905 119903 120582 119911 119879) = radic120575119875

01(119905 119903 120582 119911 119879) as follows

⟨L2⟩ 119875

120575

01= radic120575A

120575

11987500 119905 lt 119879 (47)

with the final condition 119875120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 We thenobtain the operatorL120575 expressed as

L120575

= 119881120575

1

120597

120597119903+ 119881

120575

2

120597

120597120582 (48)

Hence we obtain (45) as follows

120597119875120575

10

120597119905+ (120579

119905minus 119886119903)

120597119875120575

10

120597119903+ (120579

119905minus 119886120582)

120597119875120575

10

120597120582+1

212059021205972119875120575

10

1205971199032

+1

22

(119911)1205972119875120575

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120575

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120575

10

= minus 119881120575

1119861 (119905 119879) + 119881

120575

2(1 minus 119877)119862 (119905 119879) 119875

00

(49)

with the terminal condition 119875120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (42) into (49) we obtain the result of Theorem 3by direct computation

Note that the group parameters 119880120598

1 119880120598

2 119880120598

3 119881120575

1 and 119881120575

2

which contain model parameters are needed for the pricingof the zero-coupon defaultable bond and simplify the estima-tion procedure See Fouque et al [15] for a general reference

In summary from (16) the asymptotic analysis of thezero-coupon defaultable bond price gives

119875120598120575

(119905 119903 120582 119910 119911 119879)

asymp 11987500

(119905 119903 120582 119911 119879) + 119875120598

10(119905 119903 120582 119911 119879) + 119875

120575

01(119905 119903 120582 119911 119879)

119905 lt 119879

(50)

with the final condition 119875120598120575(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 and

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120598

10(119905 119903 120582 119911 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120575

01(119905 119903 120582 119911 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(51)

Journal of Applied Mathematics 7

Table 1 Effect of the recovery rate

Recovery rate (119877) 01 02 03 05 07 08Fast scale factor 035 009 minus011 minus029 minus027 minus020Slow scale factor minus043 minus026 minus010 024 057 073Multiple scale factors minus010 minus020 minus021 minus003 034 058

where 11987500(119905 119903 120582 119911 119879) 119875120598

10(119905 119903 120582 119911 119879) and 119875120575

01(119905 119903 120582 119911 119879)

are given by (22) (30) and (42) respectively In terms ofaccuracy of the approximation we refer to the error estimateobtained by Papageorgiou and Sircar [8]

34 Numerical Results In this subsection we conclude ourpaper with some sensitivity analyses of themodel parametersThe parameter values used to obtain the results shown inTable 1 are 119886 = 02 119886 = 03119880120598

1= 001119880120598

2= minus003 119880120598

3= 004

119881120575

1= 002 119881120575

2= minus003 119905 = 0 and 119879 = 1 We calculate

the magnitude of mispricing with respect to the zero-coupondefaultable bond as a percentage of the face value of the bondThat is we show that the zero-coupon defaultable bond tendsto be overpriced or underpriced in terms of the parametersinvolvedThe mispricing as a percentage of the zero-coupondefaultable bond price caused by changes in the value ofthe recovery rate is shown in Table 1 The second row showsresults when only the fast scale factor of the intensity (119881

120575

1=

119881120575

2= 0) is considered and the third row shows results

when only the slow scale factor (119880120598

1= 119880

120598

2= 119880

120598

3= 0) is

considered The fourth row shows the results using multiplescale factors We can see from Table 1 that the mispricing ofthe zero-coupon defaultable bondwith a fast scale factor doesnot behave monotonically as the value of the recovery rateincreases but the mispricing of the zero-coupon defaultablebond with a slow scale factor monotonically increases withthe value of the recovery rate Additionally the mispricing ofthe zero-coupon defaultable bond with multiple scale factorsdoes not behave monotonically as the value of the recoveryrate increases From the above investigation we note that themispricing of the zero-coupon defaultable bond takes variousforms depending on the recovery rate parameters involved

Figures 1 and 2 show results for parameter values of 119886 =

02 119886 = 03 120590 = 01 120590(119911) = 015 2(119911) = 018 120588119903120582

= minus02119880120598

1= minus007 119880120598

2= 001 119880120598

3= minus006 119881120575

1= 001 119881120575

2= minus004

119903 = 015 120582 = 013 119905 = 0 119879 = 10 and 119877 = 04 These figuresillustrate the defaultable bond price and the correspondingyield curve respectively Figures 1 and 2 have two typesof curve Case 1 corresponds to constant volatility in theintensity process whereas Case 2 contains both fast andslow scale factors Figure 1 shows that the defaultable bondprices with stochastic intensity become higher than thosewith constant volatility as the time to maturity increasesIn addition the hump-shaped yield curve which matchesthose in structural models [10] appears in Case 2 in Figure 2That is the multiple time scales have both quantitative andqualitative effects Hence Figures 1 and 2 show the significanteffect of multiple time scales in the stochastic intensity onboth defaultable bond prices and yields

Time to maturity (years)

Def

aulta

ble b

ond

pric

es

0 2 4 6 8 100

02

04

06

08

1

Case 1Case 2

Figure 1 Defaultable bond prices (source KIS Pricing)

Time to maturity (years)0 2 4 6 8 10

00025

0003

00035

0004

00045

0005

00055

0006

Yield

s (

)

Case 1Case 2

Figure 2 Yield curves (source KIS Pricing)

4 Credit Default Swap and Bond Option

In this section we derive a formula for the CDS rate andobtain another formula for options when the underlyingasset is a zero-coupon defaultable bond using the results ofSection 3

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

dynamics of the interest rate process 119903119905 intensity process 120582

119905

and stochastic volatility process 119905are given by the SDEs

119889119903119905= (120579

119905minus 119886119903

119905) 119889119905 + 120590119889119861

(119903)lowast

119905

119889120582119905= (120579

119905minus 119886120582

119905) 119889119905 +

119905119889119861

(120582)lowast

119905

119905= 119891 (119910

119905 119911

119905)

119889119910119905=

1

120598(119898 minus 119910

119905) 119889119905 +

]radic2

radic120598119889119861

(119910)lowast

119905

119889119911119905= 120575119892 (119911

119905) 119889119905 + radic120575ℎ (119911

119905) 119889119861

(119911)lowast

119905

(9)

under the risk-neutral probability 119875lowast where 119886 119886 and 120590 areconstants 120579

119905and 120579

119905are time-varying deterministic functions

and the standard Brownian motions 119861(119903)lowast

119905 119861(120582)lowast

119905 119861(119910)lowast

119905 and

119861(119911)lowast

119905are dependent on each other with the correlation

structure given by

119889⟨119861(119903)lowast

119861(120582)lowast

⟩119905

= 120588119903120582119889119905 119889⟨119861

(119903)lowast

119861(119910)lowast

⟩119905

= 120588119903119910119889119905

119889⟨119861(119903)lowast

119861(119911)lowast

⟩119905

= 120588119903119911119889119905 119889⟨119861

(120582)lowast

119861(119910)lowast

⟩119905

= 120588120582119910119889119905

119889⟨119861(120582)lowast

119861(119911)lowast

⟩119905

= 120588120582119911119889119905 119889⟨119861

(119910)lowast

119861(119911)lowast

⟩119905

= 120588119910119911119889119905

(10)

The zero-coupon defaultable bond price with the frac-tional recovery assumption at time 119905 for an interest rateprocess level 119903

119905= 119903 an intensity process level 120582

119905= 120582 a

fast volatility level 119910119905= 119910 and a slow volatility level 119911

119905= 119911

denoted by 119875(119905 119903 120582 119910 119911 119879) is given by

119875 (119905 119903 120582 119910 119911 119879)

= 119864lowast

[119890minusint

119879

119905(119903119904+(1minus119877)120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(11)

and then using the four-dimensional Feynman-Kac formulawe obtain the Kolmogorov PDE

120597119875

120597119905+ (120579

119905minus 119886119903)

120597119875

120597119903+1

212059021205972119875

1205971199032+ (120579

119905minus 119886120582)

120597119875

120597120582

+1

21198912

(119910 119911)1205972119875

1205971205822+1

120598(119898 minus 119910)

120597119875

120597119910+]2

120598

1205972119875

1205971199102+ 120575119892 (119911)

120597119875

120597119911

+1

2120575ℎ

2

(119911)1205972119875

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119875

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119875

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119875

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119875

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119875

120597119910120597119911

minus 119903 + (1 minus 119877) 120582 119875 = 0

(12)

with the final condition 119875(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 Refer toOslashksendal [16]

3 Asymptotic Analysis of Zero-CouponDefaultable Bonds

In this section we employ an asymptotic analysis for thesolution of the PDE (12) and give an approximate solutionfor the zero-coupon defaultable bond price for the smallindependent parameters 120598 and 120575

The zero-coupon defaultable bond price function119875(119905 119903 120582 119910 119911 119879) consists of the Feynman-Kac formulaproblems

L119875 (119905 119903 120582 119910 119911 119879) = 0 119905 lt 119879

119875(119905 119903 120582 119910 119911 119879)1003816100381610038161003816119905=119879 = 1

(13)

where

L =1

120598L

0+

1

radic120598L

1+L

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

(14)

andL0L

1L

2M

1M

2 andM

3are satisfied by

L0= (119898 minus 119910)

120597

120597119910+ ]2

1205972

1205971199102

L1= radic2120590]120588

119903119910

1205972

120597119903120597119910+ radic2119891 (119910 119911) ]120588

120582119910

1205972

120597120582120597119910

L2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910 119911)1205972

1205971205822+ 120590119891 (119910 119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

M1= 120590ℎ (119911) 120588

119903119911

1205972

120597119903120597119911+ 119891 (119910 119911) ℎ (119911) 120588

120582119911

1205972

120597120582120597119911

M2= 119892 (119911)

120597

120597119911+1

2ℎ2

(119911)1205972

1205971199112

M3= radic2]ℎ (119911) 120588

119910119911

1205972

120597119910120597119911

(15)

Here 120572L0is the infinitesimal generator of the OU process

119910119905 L

1contains the mixed partial derivatives due to the

4 Journal of Applied Mathematics

correlation between 119903119905and 119910

119905and between 120582

119905and 119910

119905 L

2is

the operator of the correlated Hull and White model withconstant volatility at the volatility level 119891(119910 119911) M

1includes

the mixed partial derivatives due to the correlation between119903119905and 119911

119905and between 120582

119905and 119911

119905 M

2is the infinitesimal

generator with respect to 119911119905 and M

3contains the mixed

partial derivative due to the correlation between 119910119905and 119911

119905

Now we use the notation 119875119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 This choiceis somewhat simpler than the reverse ordering Hence weconsider an expansion of 119875120598120575

119875120598120575

= 119875120598

0+ radic120575119875

120598

1+ 120575119875

120598

2+ sdot sdot sdot (16)

119875120598

119896= 119875

0119896+ radic120598119875

1119896+ 120598119875

2119896+ 120598

32

1198753119896

+ sdot sdot sdot (17)

for 119896 = 0 1 2 The expansion (16) leads to the leading-order term 119875120598

0and the next-order term 119875120598

1given by the

solutions of the PDEs

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

0= 0 119905 lt 119879 (18)

with the terminal condition 119875120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 1 and

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

1= (M

1+

1

radic120598M

3)119875

120598

0 119905 lt 119879

(19)

with the terminal condition 119875120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively

31 Leading-Order Term 11987500 We insert 119896 = 0 into (17) as

119875120598

0= 119875

00+ radic120598119875

10+ 120598119875

20+ 120598

32

11987530

+ sdot sdot sdot (20)

Applying the expanded solution (20) to (18) leads to

1

120598L

011987500

+1

radic120598(L

011987510

+L111987500)

+ (L011987520

+L111987510

+L211987500)

+ radic120598 (L011987530

+L111987520

+L211987510) + sdot sdot sdot = 0

(21)

Theorem 1 The leading order term 11987500

of the expansion (17)with 119896 = 0 is independent of the fast scale variable 119910 andfurther it has the affine representation

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(22)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 0 where 119861(119905 119879) 119862(119905 119879)and 119865(119905 119879) are given by

119861 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119862 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119865 (119905 119879)

= ln11987500(0 119879)

11987500(0 119905)

minus 119862(119905 119879)120597

120597119904ln119875

00(0 119904)

100381610038161003816100381610038161003816100381610038161003816119904=119905

minus(119861(119905 119879) minus 119862(119905 119879))120597

120597119904lnW(0 119904)

10038161003816100381610038161003816100381610038161003816119904=119905

+1205902

41198863(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862B2 (119905 119879)

+120588119903120582120590120590 (119911)

119886119886 (119886 + 119886)(1 minus 119877)

times (119886 + 119886) (119879 minus 119905) minus (119886119861 (119905 119879) + 1) (119886119862 (119905 119879) + 1) + 1

+2

(119911)

41198863(1 minus 119877) [(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862

1198622

(119905 119879)

minus 119877 2119886 (119879 minus 119905) minus (119886119862 (119905 119879) + 3)

times (119886119862 (119905 119879) minus 1) minus 3 ]

(23)

respectively Here

120590 (119911) = ⟨119891⟩ (119911) =1

radic2120587]2int+infin

minusinfin

119891 (119910 119911) 119890minus(119910minus119898)

22]2

119889119910

2

(119911) = ⟨1198912

⟩ (119911) =1

radic2120587]2int+infin

minusinfin

1198912

(119910 119911) 119890minus(119910minus119898)

22]2

119889119910

(24)

and W(0 119904) is the price of a zero-coupon default-free bondaccording to Hull and White [17]

Proof Multiply (21) by 120598 and then let 120598 rarr 0 This gives thefirst two leading-order terms as

L011987500

= 0

L011987510

+L111987500

= 0(25)

Recall that because the infinitesimal operator L0is the

generator of the OU process 119910119905 the solution 119875

00of (25) must

be a constant with respect to the 119910 variable that is 11987500

=

11987500(119905 119903 120582 119911 119879) Similarly because 119875

00does not rely on the

119910 variable we get L111987500

= 0 and then L011987510

= 0 that is11987510

= 11987510(119905 119903 120582 119911 119879)Thismeans that the first two terms119875

00

and 11987510

do not depend on the current level 119910 of the fast scalevolatility driving the process 119910

119905 In this way we can continue

to eliminate the terms of order 1radic120598 120598 For the order-1term we getL

011987520

+L111987510

+L211987500

= 0This PDE becomes

L011987520

+L211987500

= 0 (26)

Journal of Applied Mathematics 5

because L111987510

= 0 This PDE is a Poisson equation for 11987520

with respect to the infinitesimal operatorL0 It is well-known

that a solution exists only ifL211987500

is centered with respect tothe invariant distribution of the stochastic volatility process119910119905 that is

⟨L211987500⟩ = 0 119905 lt 119879 (27)

with the terminal condition 11987500(119905 119903 120582 119911 119879)|

119905=119879

= 1 Because11987500

does not rely on the 119910 variable the solvability conditionbecomes

⟨L2⟩ 119875

00= 0 (28)

Here ⟨L2⟩ is a partial differential operator given by

⟨L2⟩ =

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582

+1

212059021205972

1205971199032+1

22

(119911)1205972

1205971205822+ 120590120590 (119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

(29)

Finally substituting (22) into (28) we directly obtain theresult of Theorem 1 using the results of Tchuindjo [7]

Note that in the stochastic volatility setting we get theleading order term 119875

00 which is a generalization of the

constant volatility result obtained by Tchuindjo [7] If 119891(119910 119911)is a constant function then our results (22) reduce to those inTchuindjorsquos study

32 First Perturbation Term 119875120598

10 In this subsection we

precisely calculate 119875120598

10by using the result of Theorem 1

Theorem 2 The correction term 11987510

is independent of thevariable 119910 and 119875120598

10= radic120598119875

10(119905 119903 120582 119911 119879) is given by

119875120598

10(119905 119903 120582 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(30)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119866(119879 119879) = 0 where119866(119905 119879) is given by

119866 (119905 119879) = minus int119879

119905

[119880120598

1(1 minus 119877)

2

119861 (119904 119879) 1198622

(119904 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119904 119879) 119862 (119904 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119904 119879)] 119889119904

119880120598

1=

]radic2

radic120572120588119903119910120590⟨120579

1015840

⟩ (119911) +]radic2

radic120572120588120582119910

⟨1198911205951015840

⟩ (119911)

119880120598

2=

]radic2

radic120572120588119903119910120590 ⟨120595

1015840

⟩ (z)

119880120598

3=

]radic2

radic120572120588120582119910

⟨1198911205791015840

⟩ (119911)

(31)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof The order-radic120598 term in (21) leads toL011987530

+ L111987520

+

L211987510

= 0 which is a Poisson equation for 11987530

with respectto the infinitesimal operator L

0whose solvability condition

is given by

⟨L111987520

+L211987510⟩ = 0 (32)

From (26) and (28) we obtain

11987520

= minusLminus1

0(L

2minus ⟨L

2⟩) 119875

00+ 119899 (119905 119903 120582 119911) (33)

for some function 119899(119905 119903 120582 119911) Inserting (33) into (32) wederive a PDE for 119875

10as follows

⟨L2⟩ 119875

10= ⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩ 119875

00 (34)

with the final condition 11987510(119905 119903 120582 119911 119879)|

119905=119879

= 0 Because ourfocus is only the first perturbation to 119875

00 we reset the PDE

(34) with respect to 119875120598

10(119905 119903 120582 119911 119879) = radic120598119875

10(119905 119903 120582 119911 119879) as

follows

⟨L2⟩ 119875

120598

10= A

120598

11987500

A120598

=1

radic120572⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩

(35)

From (15) and (29) we have

L2minus ⟨L

2⟩ =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))1205972

1205971205822

+ 120588119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

1205972

120597119903120597120582

(36)

Then we introduce functions 120579 1198772 rarr 119877 and 120595 1198772 rarr 119877

defined by the solutions of

L0120579 (119910 119911) =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))

L0120595 (119910 119911) = 120588

119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

(37)

respectively and obtain the operatorL120598 denoted by

L120598

= 119880120598

1

1205973

1205971199031205971205822+ 119880

120598

2

1205973

1205971199032120597120582+ 119880

120598

3

1205973

1205971205823 (38)

Hence we obtain (39) in the following form

120597119875120598

10

120597119905+ (120579

119905minus 119886119903)

120597119875120598

10

120597119903+ (120579

119905minus 119886120582)

120597119875120598

10

120597120582+1

212059021205972119875120598

10

1205971199032

+1

22

(119911)1205972119875120598

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120598

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120598

10

= minus [119880120598

1(1 minus 119877)

2

119861 (119905 119879) 1198622

(119905 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119905 119879) 119862 (119905 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119905 119879)] 11987500

(39)

with the terminal condition 119875120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (30) into (39) we obtain the result of Theorem 2by direct computation

6 Journal of Applied Mathematics

33 First Perturbation 119875120575

01 Using similar arguments to those

in Section 32 we will derive the first perturbation term 119875120575

01

We insert 119896 = 1 in (17) as

119875120598

1= 119875

01+ radic120598119875

11+ 120598119875

21+ 120598

32

11987531

+ sdot sdot sdot (40)

Substituting the expansions (20) and (40) into (19) we get

1

120598L

011987501

+1

radic120598(L

011987511

+L111987501)

+ (L011987521

+L111987511

+L211987501)

+ radic120598 (L011987531

+L111987521

+L211987511) + sdot sdot sdot

=1

radic120598M

311987500

+ (M111987500

+M311987510)

+ radic120598 (M111987510

+M311987520) + sdot sdot sdot

(41)

Theorem 3 The correction term 11987501

does not depend on thevariable 119910 and 119875120575

01= radic120575119875

01(119905 119903 120582 119911 119879) is given by

119875120575

01(119905 119903 120582 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(42)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119867(119879 119879) = 0 where119867(119905 119879) is given by

119867(119905 119879) = minusint119879

119905

119881120575

1119861 (119904 119879) + 119881

120575

2(1 minus 119877)119862 (119904 119879) 119889119904

119881120575

1= radic120575120588

119903119911120590ℎ (119911)

119881120575

2= radic120575120588

120582119911ℎ

119911(119911) ⟨119891⟩ (119911) + ℎ (119911) ⟨119891⟩

119911(119911)

(43)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof Multiply (41) by 120598 and let 120598 rarr 0 Then we find thefirst two leading-order terms as follows

L011987501

= 0

L011987511

+L111987501

= M311987500

(44)

Recall that because L0is the infinitesimal generator of the

OU process 119910119905 the solution 119875

01of (44) must be a constant

with respect to the 119910 variable that is 11987501

= 11987501(119905 119903 120582 119911 119879)

The next order also generates L011987511

= 0 Because M3has

a derivative with respect to the 119910 variable and 11987500

does notrely on the 119910 variable we get M

311987500

= 0 and L111987501

= 0respectively as remarked above Then 119875

11does not depend

on the 119910 variable that is 11987511

= 11987511(119905 119903 120582 119911 119879) This means

that the first two terms 11987501

and 11987511

do not depend on thecurrent level 119910 of the fast scale volatility driving the process119910119905 In this way we can continue to eliminate terms of order

1radic120598 120598 For the order-1 term we haveL011987521

+L111987511

+

L211987501

= M111987500

+ M311987510 This PDE becomes L

011987521

+

L211987501

= M111987500 because L

111987511

= 0 and M311987510

= 0This PDE is a Poisson equation for 119875

21with respect to the

infinitesimal operator L0 It is well-known that a solution

exists only if L211987501

= M111987500

is centered with respect to

the invariant distribution of the stochastic volatility process119910119905 Because 119875

00and 119875

01do not depend on the 119910 variable we

obtain

⟨L2⟩ 119875

01= A

120575

11987500 119905 lt 119879 (45)

with the terminal condition 11987501(119905 119903 120582 119911 119879)|

119905=119879

= 0 where

A120575

= 120590ℎ119911(119911) 120588

119903119911

120597

120597119903+ 120588

120582119911(ℎ

119911(119911) ⟨119891⟩ (119911)

+ℎ (119911) ⟨119891⟩119911(119911))

120597

120597120582

⟨119891⟩119911(119911) =

120597

120597119911⟨119891⟩ (119911) ℎ

119911(119911) =

120597

120597119911ℎ (119911)

(46)

respectively Because we concentrate on only the first per-turbation to 119875

00 we reset the PDE (45) with respect to

119875120575

01(119905 119903 120582 119911 119879) = radic120575119875

01(119905 119903 120582 119911 119879) as follows

⟨L2⟩ 119875

120575

01= radic120575A

120575

11987500 119905 lt 119879 (47)

with the final condition 119875120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 We thenobtain the operatorL120575 expressed as

L120575

= 119881120575

1

120597

120597119903+ 119881

120575

2

120597

120597120582 (48)

Hence we obtain (45) as follows

120597119875120575

10

120597119905+ (120579

119905minus 119886119903)

120597119875120575

10

120597119903+ (120579

119905minus 119886120582)

120597119875120575

10

120597120582+1

212059021205972119875120575

10

1205971199032

+1

22

(119911)1205972119875120575

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120575

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120575

10

= minus 119881120575

1119861 (119905 119879) + 119881

120575

2(1 minus 119877)119862 (119905 119879) 119875

00

(49)

with the terminal condition 119875120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (42) into (49) we obtain the result of Theorem 3by direct computation

Note that the group parameters 119880120598

1 119880120598

2 119880120598

3 119881120575

1 and 119881120575

2

which contain model parameters are needed for the pricingof the zero-coupon defaultable bond and simplify the estima-tion procedure See Fouque et al [15] for a general reference

In summary from (16) the asymptotic analysis of thezero-coupon defaultable bond price gives

119875120598120575

(119905 119903 120582 119910 119911 119879)

asymp 11987500

(119905 119903 120582 119911 119879) + 119875120598

10(119905 119903 120582 119911 119879) + 119875

120575

01(119905 119903 120582 119911 119879)

119905 lt 119879

(50)

with the final condition 119875120598120575(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 and

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120598

10(119905 119903 120582 119911 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120575

01(119905 119903 120582 119911 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(51)

Journal of Applied Mathematics 7

Table 1 Effect of the recovery rate

Recovery rate (119877) 01 02 03 05 07 08Fast scale factor 035 009 minus011 minus029 minus027 minus020Slow scale factor minus043 minus026 minus010 024 057 073Multiple scale factors minus010 minus020 minus021 minus003 034 058

where 11987500(119905 119903 120582 119911 119879) 119875120598

10(119905 119903 120582 119911 119879) and 119875120575

01(119905 119903 120582 119911 119879)

are given by (22) (30) and (42) respectively In terms ofaccuracy of the approximation we refer to the error estimateobtained by Papageorgiou and Sircar [8]

34 Numerical Results In this subsection we conclude ourpaper with some sensitivity analyses of themodel parametersThe parameter values used to obtain the results shown inTable 1 are 119886 = 02 119886 = 03119880120598

1= 001119880120598

2= minus003 119880120598

3= 004

119881120575

1= 002 119881120575

2= minus003 119905 = 0 and 119879 = 1 We calculate

the magnitude of mispricing with respect to the zero-coupondefaultable bond as a percentage of the face value of the bondThat is we show that the zero-coupon defaultable bond tendsto be overpriced or underpriced in terms of the parametersinvolvedThe mispricing as a percentage of the zero-coupondefaultable bond price caused by changes in the value ofthe recovery rate is shown in Table 1 The second row showsresults when only the fast scale factor of the intensity (119881

120575

1=

119881120575

2= 0) is considered and the third row shows results

when only the slow scale factor (119880120598

1= 119880

120598

2= 119880

120598

3= 0) is

considered The fourth row shows the results using multiplescale factors We can see from Table 1 that the mispricing ofthe zero-coupon defaultable bondwith a fast scale factor doesnot behave monotonically as the value of the recovery rateincreases but the mispricing of the zero-coupon defaultablebond with a slow scale factor monotonically increases withthe value of the recovery rate Additionally the mispricing ofthe zero-coupon defaultable bond with multiple scale factorsdoes not behave monotonically as the value of the recoveryrate increases From the above investigation we note that themispricing of the zero-coupon defaultable bond takes variousforms depending on the recovery rate parameters involved

Figures 1 and 2 show results for parameter values of 119886 =

02 119886 = 03 120590 = 01 120590(119911) = 015 2(119911) = 018 120588119903120582

= minus02119880120598

1= minus007 119880120598

2= 001 119880120598

3= minus006 119881120575

1= 001 119881120575

2= minus004

119903 = 015 120582 = 013 119905 = 0 119879 = 10 and 119877 = 04 These figuresillustrate the defaultable bond price and the correspondingyield curve respectively Figures 1 and 2 have two typesof curve Case 1 corresponds to constant volatility in theintensity process whereas Case 2 contains both fast andslow scale factors Figure 1 shows that the defaultable bondprices with stochastic intensity become higher than thosewith constant volatility as the time to maturity increasesIn addition the hump-shaped yield curve which matchesthose in structural models [10] appears in Case 2 in Figure 2That is the multiple time scales have both quantitative andqualitative effects Hence Figures 1 and 2 show the significanteffect of multiple time scales in the stochastic intensity onboth defaultable bond prices and yields

Time to maturity (years)

Def

aulta

ble b

ond

pric

es

0 2 4 6 8 100

02

04

06

08

1

Case 1Case 2

Figure 1 Defaultable bond prices (source KIS Pricing)

Time to maturity (years)0 2 4 6 8 10

00025

0003

00035

0004

00045

0005

00055

0006

Yield

s (

)

Case 1Case 2

Figure 2 Yield curves (source KIS Pricing)

4 Credit Default Swap and Bond Option

In this section we derive a formula for the CDS rate andobtain another formula for options when the underlyingasset is a zero-coupon defaultable bond using the results ofSection 3

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

correlation between 119903119905and 119910

119905and between 120582

119905and 119910

119905 L

2is

the operator of the correlated Hull and White model withconstant volatility at the volatility level 119891(119910 119911) M

1includes

the mixed partial derivatives due to the correlation between119903119905and 119911

119905and between 120582

119905and 119911

119905 M

2is the infinitesimal

generator with respect to 119911119905 and M

3contains the mixed

partial derivative due to the correlation between 119910119905and 119911

119905

Now we use the notation 119875119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 This choiceis somewhat simpler than the reverse ordering Hence weconsider an expansion of 119875120598120575

119875120598120575

= 119875120598

0+ radic120575119875

120598

1+ 120575119875

120598

2+ sdot sdot sdot (16)

119875120598

119896= 119875

0119896+ radic120598119875

1119896+ 120598119875

2119896+ 120598

32

1198753119896

+ sdot sdot sdot (17)

for 119896 = 0 1 2 The expansion (16) leads to the leading-order term 119875120598

0and the next-order term 119875120598

1given by the

solutions of the PDEs

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

0= 0 119905 lt 119879 (18)

with the terminal condition 119875120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 1 and

(1

120598L

0+

1

radic120598L

1+L

2)119875

120598

1= (M

1+

1

radic120598M

3)119875

120598

0 119905 lt 119879

(19)

with the terminal condition 119875120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively

31 Leading-Order Term 11987500 We insert 119896 = 0 into (17) as

119875120598

0= 119875

00+ radic120598119875

10+ 120598119875

20+ 120598

32

11987530

+ sdot sdot sdot (20)

Applying the expanded solution (20) to (18) leads to

1

120598L

011987500

+1

radic120598(L

011987510

+L111987500)

+ (L011987520

+L111987510

+L211987500)

+ radic120598 (L011987530

+L111987520

+L211987510) + sdot sdot sdot = 0

(21)

Theorem 1 The leading order term 11987500

of the expansion (17)with 119896 = 0 is independent of the fast scale variable 119910 andfurther it has the affine representation

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(22)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 0 where 119861(119905 119879) 119862(119905 119879)and 119865(119905 119879) are given by

119861 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119862 (119905 119879) =1

1198861 minus 119890

minus119886(119879minus119905)

119865 (119905 119879)

= ln11987500(0 119879)

11987500(0 119905)

minus 119862(119905 119879)120597

120597119904ln119875

00(0 119904)

100381610038161003816100381610038161003816100381610038161003816119904=119905

minus(119861(119905 119879) minus 119862(119905 119879))120597

120597119904lnW(0 119904)

10038161003816100381610038161003816100381610038161003816119904=119905

+1205902

41198863(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862B2 (119905 119879)

+120588119903120582120590120590 (119911)

119886119886 (119886 + 119886)(1 minus 119877)

times (119886 + 119886) (119879 minus 119905) minus (119886119861 (119905 119879) + 1) (119886119862 (119905 119879) + 1) + 1

+2

(119911)

41198863(1 minus 119877) [(119890

minus119886119879

minus 119890minus119886119905

)2

minus 1198862

1198622

(119905 119879)

minus 119877 2119886 (119879 minus 119905) minus (119886119862 (119905 119879) + 3)

times (119886119862 (119905 119879) minus 1) minus 3 ]

(23)

respectively Here

120590 (119911) = ⟨119891⟩ (119911) =1

radic2120587]2int+infin

minusinfin

119891 (119910 119911) 119890minus(119910minus119898)

22]2

119889119910

2

(119911) = ⟨1198912

⟩ (119911) =1

radic2120587]2int+infin

minusinfin

1198912

(119910 119911) 119890minus(119910minus119898)

22]2

119889119910

(24)

and W(0 119904) is the price of a zero-coupon default-free bondaccording to Hull and White [17]

Proof Multiply (21) by 120598 and then let 120598 rarr 0 This gives thefirst two leading-order terms as

L011987500

= 0

L011987510

+L111987500

= 0(25)

Recall that because the infinitesimal operator L0is the

generator of the OU process 119910119905 the solution 119875

00of (25) must

be a constant with respect to the 119910 variable that is 11987500

=

11987500(119905 119903 120582 119911 119879) Similarly because 119875

00does not rely on the

119910 variable we get L111987500

= 0 and then L011987510

= 0 that is11987510

= 11987510(119905 119903 120582 119911 119879)Thismeans that the first two terms119875

00

and 11987510

do not depend on the current level 119910 of the fast scalevolatility driving the process 119910

119905 In this way we can continue

to eliminate the terms of order 1radic120598 120598 For the order-1term we getL

011987520

+L111987510

+L211987500

= 0This PDE becomes

L011987520

+L211987500

= 0 (26)

Journal of Applied Mathematics 5

because L111987510

= 0 This PDE is a Poisson equation for 11987520

with respect to the infinitesimal operatorL0 It is well-known

that a solution exists only ifL211987500

is centered with respect tothe invariant distribution of the stochastic volatility process119910119905 that is

⟨L211987500⟩ = 0 119905 lt 119879 (27)

with the terminal condition 11987500(119905 119903 120582 119911 119879)|

119905=119879

= 1 Because11987500

does not rely on the 119910 variable the solvability conditionbecomes

⟨L2⟩ 119875

00= 0 (28)

Here ⟨L2⟩ is a partial differential operator given by

⟨L2⟩ =

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582

+1

212059021205972

1205971199032+1

22

(119911)1205972

1205971205822+ 120590120590 (119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

(29)

Finally substituting (22) into (28) we directly obtain theresult of Theorem 1 using the results of Tchuindjo [7]

Note that in the stochastic volatility setting we get theleading order term 119875

00 which is a generalization of the

constant volatility result obtained by Tchuindjo [7] If 119891(119910 119911)is a constant function then our results (22) reduce to those inTchuindjorsquos study

32 First Perturbation Term 119875120598

10 In this subsection we

precisely calculate 119875120598

10by using the result of Theorem 1

Theorem 2 The correction term 11987510

is independent of thevariable 119910 and 119875120598

10= radic120598119875

10(119905 119903 120582 119911 119879) is given by

119875120598

10(119905 119903 120582 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(30)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119866(119879 119879) = 0 where119866(119905 119879) is given by

119866 (119905 119879) = minus int119879

119905

[119880120598

1(1 minus 119877)

2

119861 (119904 119879) 1198622

(119904 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119904 119879) 119862 (119904 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119904 119879)] 119889119904

119880120598

1=

]radic2

radic120572120588119903119910120590⟨120579

1015840

⟩ (119911) +]radic2

radic120572120588120582119910

⟨1198911205951015840

⟩ (119911)

119880120598

2=

]radic2

radic120572120588119903119910120590 ⟨120595

1015840

⟩ (z)

119880120598

3=

]radic2

radic120572120588120582119910

⟨1198911205791015840

⟩ (119911)

(31)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof The order-radic120598 term in (21) leads toL011987530

+ L111987520

+

L211987510

= 0 which is a Poisson equation for 11987530

with respectto the infinitesimal operator L

0whose solvability condition

is given by

⟨L111987520

+L211987510⟩ = 0 (32)

From (26) and (28) we obtain

11987520

= minusLminus1

0(L

2minus ⟨L

2⟩) 119875

00+ 119899 (119905 119903 120582 119911) (33)

for some function 119899(119905 119903 120582 119911) Inserting (33) into (32) wederive a PDE for 119875

10as follows

⟨L2⟩ 119875

10= ⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩ 119875

00 (34)

with the final condition 11987510(119905 119903 120582 119911 119879)|

119905=119879

= 0 Because ourfocus is only the first perturbation to 119875

00 we reset the PDE

(34) with respect to 119875120598

10(119905 119903 120582 119911 119879) = radic120598119875

10(119905 119903 120582 119911 119879) as

follows

⟨L2⟩ 119875

120598

10= A

120598

11987500

A120598

=1

radic120572⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩

(35)

From (15) and (29) we have

L2minus ⟨L

2⟩ =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))1205972

1205971205822

+ 120588119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

1205972

120597119903120597120582

(36)

Then we introduce functions 120579 1198772 rarr 119877 and 120595 1198772 rarr 119877

defined by the solutions of

L0120579 (119910 119911) =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))

L0120595 (119910 119911) = 120588

119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

(37)

respectively and obtain the operatorL120598 denoted by

L120598

= 119880120598

1

1205973

1205971199031205971205822+ 119880

120598

2

1205973

1205971199032120597120582+ 119880

120598

3

1205973

1205971205823 (38)

Hence we obtain (39) in the following form

120597119875120598

10

120597119905+ (120579

119905minus 119886119903)

120597119875120598

10

120597119903+ (120579

119905minus 119886120582)

120597119875120598

10

120597120582+1

212059021205972119875120598

10

1205971199032

+1

22

(119911)1205972119875120598

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120598

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120598

10

= minus [119880120598

1(1 minus 119877)

2

119861 (119905 119879) 1198622

(119905 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119905 119879) 119862 (119905 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119905 119879)] 11987500

(39)

with the terminal condition 119875120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (30) into (39) we obtain the result of Theorem 2by direct computation

6 Journal of Applied Mathematics

33 First Perturbation 119875120575

01 Using similar arguments to those

in Section 32 we will derive the first perturbation term 119875120575

01

We insert 119896 = 1 in (17) as

119875120598

1= 119875

01+ radic120598119875

11+ 120598119875

21+ 120598

32

11987531

+ sdot sdot sdot (40)

Substituting the expansions (20) and (40) into (19) we get

1

120598L

011987501

+1

radic120598(L

011987511

+L111987501)

+ (L011987521

+L111987511

+L211987501)

+ radic120598 (L011987531

+L111987521

+L211987511) + sdot sdot sdot

=1

radic120598M

311987500

+ (M111987500

+M311987510)

+ radic120598 (M111987510

+M311987520) + sdot sdot sdot

(41)

Theorem 3 The correction term 11987501

does not depend on thevariable 119910 and 119875120575

01= radic120575119875

01(119905 119903 120582 119911 119879) is given by

119875120575

01(119905 119903 120582 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(42)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119867(119879 119879) = 0 where119867(119905 119879) is given by

119867(119905 119879) = minusint119879

119905

119881120575

1119861 (119904 119879) + 119881

120575

2(1 minus 119877)119862 (119904 119879) 119889119904

119881120575

1= radic120575120588

119903119911120590ℎ (119911)

119881120575

2= radic120575120588

120582119911ℎ

119911(119911) ⟨119891⟩ (119911) + ℎ (119911) ⟨119891⟩

119911(119911)

(43)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof Multiply (41) by 120598 and let 120598 rarr 0 Then we find thefirst two leading-order terms as follows

L011987501

= 0

L011987511

+L111987501

= M311987500

(44)

Recall that because L0is the infinitesimal generator of the

OU process 119910119905 the solution 119875

01of (44) must be a constant

with respect to the 119910 variable that is 11987501

= 11987501(119905 119903 120582 119911 119879)

The next order also generates L011987511

= 0 Because M3has

a derivative with respect to the 119910 variable and 11987500

does notrely on the 119910 variable we get M

311987500

= 0 and L111987501

= 0respectively as remarked above Then 119875

11does not depend

on the 119910 variable that is 11987511

= 11987511(119905 119903 120582 119911 119879) This means

that the first two terms 11987501

and 11987511

do not depend on thecurrent level 119910 of the fast scale volatility driving the process119910119905 In this way we can continue to eliminate terms of order

1radic120598 120598 For the order-1 term we haveL011987521

+L111987511

+

L211987501

= M111987500

+ M311987510 This PDE becomes L

011987521

+

L211987501

= M111987500 because L

111987511

= 0 and M311987510

= 0This PDE is a Poisson equation for 119875

21with respect to the

infinitesimal operator L0 It is well-known that a solution

exists only if L211987501

= M111987500

is centered with respect to

the invariant distribution of the stochastic volatility process119910119905 Because 119875

00and 119875

01do not depend on the 119910 variable we

obtain

⟨L2⟩ 119875

01= A

120575

11987500 119905 lt 119879 (45)

with the terminal condition 11987501(119905 119903 120582 119911 119879)|

119905=119879

= 0 where

A120575

= 120590ℎ119911(119911) 120588

119903119911

120597

120597119903+ 120588

120582119911(ℎ

119911(119911) ⟨119891⟩ (119911)

+ℎ (119911) ⟨119891⟩119911(119911))

120597

120597120582

⟨119891⟩119911(119911) =

120597

120597119911⟨119891⟩ (119911) ℎ

119911(119911) =

120597

120597119911ℎ (119911)

(46)

respectively Because we concentrate on only the first per-turbation to 119875

00 we reset the PDE (45) with respect to

119875120575

01(119905 119903 120582 119911 119879) = radic120575119875

01(119905 119903 120582 119911 119879) as follows

⟨L2⟩ 119875

120575

01= radic120575A

120575

11987500 119905 lt 119879 (47)

with the final condition 119875120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 We thenobtain the operatorL120575 expressed as

L120575

= 119881120575

1

120597

120597119903+ 119881

120575

2

120597

120597120582 (48)

Hence we obtain (45) as follows

120597119875120575

10

120597119905+ (120579

119905minus 119886119903)

120597119875120575

10

120597119903+ (120579

119905minus 119886120582)

120597119875120575

10

120597120582+1

212059021205972119875120575

10

1205971199032

+1

22

(119911)1205972119875120575

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120575

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120575

10

= minus 119881120575

1119861 (119905 119879) + 119881

120575

2(1 minus 119877)119862 (119905 119879) 119875

00

(49)

with the terminal condition 119875120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (42) into (49) we obtain the result of Theorem 3by direct computation

Note that the group parameters 119880120598

1 119880120598

2 119880120598

3 119881120575

1 and 119881120575

2

which contain model parameters are needed for the pricingof the zero-coupon defaultable bond and simplify the estima-tion procedure See Fouque et al [15] for a general reference

In summary from (16) the asymptotic analysis of thezero-coupon defaultable bond price gives

119875120598120575

(119905 119903 120582 119910 119911 119879)

asymp 11987500

(119905 119903 120582 119911 119879) + 119875120598

10(119905 119903 120582 119911 119879) + 119875

120575

01(119905 119903 120582 119911 119879)

119905 lt 119879

(50)

with the final condition 119875120598120575(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 and

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120598

10(119905 119903 120582 119911 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120575

01(119905 119903 120582 119911 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(51)

Journal of Applied Mathematics 7

Table 1 Effect of the recovery rate

Recovery rate (119877) 01 02 03 05 07 08Fast scale factor 035 009 minus011 minus029 minus027 minus020Slow scale factor minus043 minus026 minus010 024 057 073Multiple scale factors minus010 minus020 minus021 minus003 034 058

where 11987500(119905 119903 120582 119911 119879) 119875120598

10(119905 119903 120582 119911 119879) and 119875120575

01(119905 119903 120582 119911 119879)

are given by (22) (30) and (42) respectively In terms ofaccuracy of the approximation we refer to the error estimateobtained by Papageorgiou and Sircar [8]

34 Numerical Results In this subsection we conclude ourpaper with some sensitivity analyses of themodel parametersThe parameter values used to obtain the results shown inTable 1 are 119886 = 02 119886 = 03119880120598

1= 001119880120598

2= minus003 119880120598

3= 004

119881120575

1= 002 119881120575

2= minus003 119905 = 0 and 119879 = 1 We calculate

the magnitude of mispricing with respect to the zero-coupondefaultable bond as a percentage of the face value of the bondThat is we show that the zero-coupon defaultable bond tendsto be overpriced or underpriced in terms of the parametersinvolvedThe mispricing as a percentage of the zero-coupondefaultable bond price caused by changes in the value ofthe recovery rate is shown in Table 1 The second row showsresults when only the fast scale factor of the intensity (119881

120575

1=

119881120575

2= 0) is considered and the third row shows results

when only the slow scale factor (119880120598

1= 119880

120598

2= 119880

120598

3= 0) is

considered The fourth row shows the results using multiplescale factors We can see from Table 1 that the mispricing ofthe zero-coupon defaultable bondwith a fast scale factor doesnot behave monotonically as the value of the recovery rateincreases but the mispricing of the zero-coupon defaultablebond with a slow scale factor monotonically increases withthe value of the recovery rate Additionally the mispricing ofthe zero-coupon defaultable bond with multiple scale factorsdoes not behave monotonically as the value of the recoveryrate increases From the above investigation we note that themispricing of the zero-coupon defaultable bond takes variousforms depending on the recovery rate parameters involved

Figures 1 and 2 show results for parameter values of 119886 =

02 119886 = 03 120590 = 01 120590(119911) = 015 2(119911) = 018 120588119903120582

= minus02119880120598

1= minus007 119880120598

2= 001 119880120598

3= minus006 119881120575

1= 001 119881120575

2= minus004

119903 = 015 120582 = 013 119905 = 0 119879 = 10 and 119877 = 04 These figuresillustrate the defaultable bond price and the correspondingyield curve respectively Figures 1 and 2 have two typesof curve Case 1 corresponds to constant volatility in theintensity process whereas Case 2 contains both fast andslow scale factors Figure 1 shows that the defaultable bondprices with stochastic intensity become higher than thosewith constant volatility as the time to maturity increasesIn addition the hump-shaped yield curve which matchesthose in structural models [10] appears in Case 2 in Figure 2That is the multiple time scales have both quantitative andqualitative effects Hence Figures 1 and 2 show the significanteffect of multiple time scales in the stochastic intensity onboth defaultable bond prices and yields

Time to maturity (years)

Def

aulta

ble b

ond

pric

es

0 2 4 6 8 100

02

04

06

08

1

Case 1Case 2

Figure 1 Defaultable bond prices (source KIS Pricing)

Time to maturity (years)0 2 4 6 8 10

00025

0003

00035

0004

00045

0005

00055

0006

Yield

s (

)

Case 1Case 2

Figure 2 Yield curves (source KIS Pricing)

4 Credit Default Swap and Bond Option

In this section we derive a formula for the CDS rate andobtain another formula for options when the underlyingasset is a zero-coupon defaultable bond using the results ofSection 3

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

because L111987510

= 0 This PDE is a Poisson equation for 11987520

with respect to the infinitesimal operatorL0 It is well-known

that a solution exists only ifL211987500

is centered with respect tothe invariant distribution of the stochastic volatility process119910119905 that is

⟨L211987500⟩ = 0 119905 lt 119879 (27)

with the terminal condition 11987500(119905 119903 120582 119911 119879)|

119905=119879

= 1 Because11987500

does not rely on the 119910 variable the solvability conditionbecomes

⟨L2⟩ 119875

00= 0 (28)

Here ⟨L2⟩ is a partial differential operator given by

⟨L2⟩ =

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582

+1

212059021205972

1205971199032+1

22

(119911)1205972

1205971205822+ 120590120590 (119911) 120588

119903120582

1205972

120597119903120597120582

minus 119903 + (1 minus 119877) 120582

(29)

Finally substituting (22) into (28) we directly obtain theresult of Theorem 1 using the results of Tchuindjo [7]

Note that in the stochastic volatility setting we get theleading order term 119875

00 which is a generalization of the

constant volatility result obtained by Tchuindjo [7] If 119891(119910 119911)is a constant function then our results (22) reduce to those inTchuindjorsquos study

32 First Perturbation Term 119875120598

10 In this subsection we

precisely calculate 119875120598

10by using the result of Theorem 1

Theorem 2 The correction term 11987510

is independent of thevariable 119910 and 119875120598

10= radic120598119875

10(119905 119903 120582 119911 119879) is given by

119875120598

10(119905 119903 120582 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(30)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119866(119879 119879) = 0 where119866(119905 119879) is given by

119866 (119905 119879) = minus int119879

119905

[119880120598

1(1 minus 119877)

2

119861 (119904 119879) 1198622

(119904 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119904 119879) 119862 (119904 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119904 119879)] 119889119904

119880120598

1=

]radic2

radic120572120588119903119910120590⟨120579

1015840

⟩ (119911) +]radic2

radic120572120588120582119910

⟨1198911205951015840

⟩ (119911)

119880120598

2=

]radic2

radic120572120588119903119910120590 ⟨120595

1015840

⟩ (z)

119880120598

3=

]radic2

radic120572120588120582119910

⟨1198911205791015840

⟩ (119911)

(31)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof The order-radic120598 term in (21) leads toL011987530

+ L111987520

+

L211987510

= 0 which is a Poisson equation for 11987530

with respectto the infinitesimal operator L

0whose solvability condition

is given by

⟨L111987520

+L211987510⟩ = 0 (32)

From (26) and (28) we obtain

11987520

= minusLminus1

0(L

2minus ⟨L

2⟩) 119875

00+ 119899 (119905 119903 120582 119911) (33)

for some function 119899(119905 119903 120582 119911) Inserting (33) into (32) wederive a PDE for 119875

10as follows

⟨L2⟩ 119875

10= ⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩ 119875

00 (34)

with the final condition 11987510(119905 119903 120582 119911 119879)|

119905=119879

= 0 Because ourfocus is only the first perturbation to 119875

00 we reset the PDE

(34) with respect to 119875120598

10(119905 119903 120582 119911 119879) = radic120598119875

10(119905 119903 120582 119911 119879) as

follows

⟨L2⟩ 119875

120598

10= A

120598

11987500

A120598

=1

radic120572⟨L

1L

minus1

0(L

2minus ⟨L

2⟩)⟩

(35)

From (15) and (29) we have

L2minus ⟨L

2⟩ =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))1205972

1205971205822

+ 120588119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

1205972

120597119903120597120582

(36)

Then we introduce functions 120579 1198772 rarr 119877 and 120595 1198772 rarr 119877

defined by the solutions of

L0120579 (119910 119911) =

1

2(119891

2

(119910 119911) minus ⟨1198912

⟩ (119911))

L0120595 (119910 119911) = 120588

119903120582120590 (119891 (119910 119911) minus ⟨119891⟩ (119911))

(37)

respectively and obtain the operatorL120598 denoted by

L120598

= 119880120598

1

1205973

1205971199031205971205822+ 119880

120598

2

1205973

1205971199032120597120582+ 119880

120598

3

1205973

1205971205823 (38)

Hence we obtain (39) in the following form

120597119875120598

10

120597119905+ (120579

119905minus 119886119903)

120597119875120598

10

120597119903+ (120579

119905minus 119886120582)

120597119875120598

10

120597120582+1

212059021205972119875120598

10

1205971199032

+1

22

(119911)1205972119875120598

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120598

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120598

10

= minus [119880120598

1(1 minus 119877)

2

119861 (119905 119879) 1198622

(119905 119879)

+ 119880120598

2(1 minus 119877) 119861

2

(119905 119879) 119862 (119905 119879)

+119880120598

3(1 minus 119877)

3

1198623

(119905 119879)] 11987500

(39)

with the terminal condition 119875120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (30) into (39) we obtain the result of Theorem 2by direct computation

6 Journal of Applied Mathematics

33 First Perturbation 119875120575

01 Using similar arguments to those

in Section 32 we will derive the first perturbation term 119875120575

01

We insert 119896 = 1 in (17) as

119875120598

1= 119875

01+ radic120598119875

11+ 120598119875

21+ 120598

32

11987531

+ sdot sdot sdot (40)

Substituting the expansions (20) and (40) into (19) we get

1

120598L

011987501

+1

radic120598(L

011987511

+L111987501)

+ (L011987521

+L111987511

+L211987501)

+ radic120598 (L011987531

+L111987521

+L211987511) + sdot sdot sdot

=1

radic120598M

311987500

+ (M111987500

+M311987510)

+ radic120598 (M111987510

+M311987520) + sdot sdot sdot

(41)

Theorem 3 The correction term 11987501

does not depend on thevariable 119910 and 119875120575

01= radic120575119875

01(119905 119903 120582 119911 119879) is given by

119875120575

01(119905 119903 120582 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(42)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119867(119879 119879) = 0 where119867(119905 119879) is given by

119867(119905 119879) = minusint119879

119905

119881120575

1119861 (119904 119879) + 119881

120575

2(1 minus 119877)119862 (119904 119879) 119889119904

119881120575

1= radic120575120588

119903119911120590ℎ (119911)

119881120575

2= radic120575120588

120582119911ℎ

119911(119911) ⟨119891⟩ (119911) + ℎ (119911) ⟨119891⟩

119911(119911)

(43)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof Multiply (41) by 120598 and let 120598 rarr 0 Then we find thefirst two leading-order terms as follows

L011987501

= 0

L011987511

+L111987501

= M311987500

(44)

Recall that because L0is the infinitesimal generator of the

OU process 119910119905 the solution 119875

01of (44) must be a constant

with respect to the 119910 variable that is 11987501

= 11987501(119905 119903 120582 119911 119879)

The next order also generates L011987511

= 0 Because M3has

a derivative with respect to the 119910 variable and 11987500

does notrely on the 119910 variable we get M

311987500

= 0 and L111987501

= 0respectively as remarked above Then 119875

11does not depend

on the 119910 variable that is 11987511

= 11987511(119905 119903 120582 119911 119879) This means

that the first two terms 11987501

and 11987511

do not depend on thecurrent level 119910 of the fast scale volatility driving the process119910119905 In this way we can continue to eliminate terms of order

1radic120598 120598 For the order-1 term we haveL011987521

+L111987511

+

L211987501

= M111987500

+ M311987510 This PDE becomes L

011987521

+

L211987501

= M111987500 because L

111987511

= 0 and M311987510

= 0This PDE is a Poisson equation for 119875

21with respect to the

infinitesimal operator L0 It is well-known that a solution

exists only if L211987501

= M111987500

is centered with respect to

the invariant distribution of the stochastic volatility process119910119905 Because 119875

00and 119875

01do not depend on the 119910 variable we

obtain

⟨L2⟩ 119875

01= A

120575

11987500 119905 lt 119879 (45)

with the terminal condition 11987501(119905 119903 120582 119911 119879)|

119905=119879

= 0 where

A120575

= 120590ℎ119911(119911) 120588

119903119911

120597

120597119903+ 120588

120582119911(ℎ

119911(119911) ⟨119891⟩ (119911)

+ℎ (119911) ⟨119891⟩119911(119911))

120597

120597120582

⟨119891⟩119911(119911) =

120597

120597119911⟨119891⟩ (119911) ℎ

119911(119911) =

120597

120597119911ℎ (119911)

(46)

respectively Because we concentrate on only the first per-turbation to 119875

00 we reset the PDE (45) with respect to

119875120575

01(119905 119903 120582 119911 119879) = radic120575119875

01(119905 119903 120582 119911 119879) as follows

⟨L2⟩ 119875

120575

01= radic120575A

120575

11987500 119905 lt 119879 (47)

with the final condition 119875120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 We thenobtain the operatorL120575 expressed as

L120575

= 119881120575

1

120597

120597119903+ 119881

120575

2

120597

120597120582 (48)

Hence we obtain (45) as follows

120597119875120575

10

120597119905+ (120579

119905minus 119886119903)

120597119875120575

10

120597119903+ (120579

119905minus 119886120582)

120597119875120575

10

120597120582+1

212059021205972119875120575

10

1205971199032

+1

22

(119911)1205972119875120575

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120575

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120575

10

= minus 119881120575

1119861 (119905 119879) + 119881

120575

2(1 minus 119877)119862 (119905 119879) 119875

00

(49)

with the terminal condition 119875120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (42) into (49) we obtain the result of Theorem 3by direct computation

Note that the group parameters 119880120598

1 119880120598

2 119880120598

3 119881120575

1 and 119881120575

2

which contain model parameters are needed for the pricingof the zero-coupon defaultable bond and simplify the estima-tion procedure See Fouque et al [15] for a general reference

In summary from (16) the asymptotic analysis of thezero-coupon defaultable bond price gives

119875120598120575

(119905 119903 120582 119910 119911 119879)

asymp 11987500

(119905 119903 120582 119911 119879) + 119875120598

10(119905 119903 120582 119911 119879) + 119875

120575

01(119905 119903 120582 119911 119879)

119905 lt 119879

(50)

with the final condition 119875120598120575(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 and

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120598

10(119905 119903 120582 119911 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120575

01(119905 119903 120582 119911 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(51)

Journal of Applied Mathematics 7

Table 1 Effect of the recovery rate

Recovery rate (119877) 01 02 03 05 07 08Fast scale factor 035 009 minus011 minus029 minus027 minus020Slow scale factor minus043 minus026 minus010 024 057 073Multiple scale factors minus010 minus020 minus021 minus003 034 058

where 11987500(119905 119903 120582 119911 119879) 119875120598

10(119905 119903 120582 119911 119879) and 119875120575

01(119905 119903 120582 119911 119879)

are given by (22) (30) and (42) respectively In terms ofaccuracy of the approximation we refer to the error estimateobtained by Papageorgiou and Sircar [8]

34 Numerical Results In this subsection we conclude ourpaper with some sensitivity analyses of themodel parametersThe parameter values used to obtain the results shown inTable 1 are 119886 = 02 119886 = 03119880120598

1= 001119880120598

2= minus003 119880120598

3= 004

119881120575

1= 002 119881120575

2= minus003 119905 = 0 and 119879 = 1 We calculate

the magnitude of mispricing with respect to the zero-coupondefaultable bond as a percentage of the face value of the bondThat is we show that the zero-coupon defaultable bond tendsto be overpriced or underpriced in terms of the parametersinvolvedThe mispricing as a percentage of the zero-coupondefaultable bond price caused by changes in the value ofthe recovery rate is shown in Table 1 The second row showsresults when only the fast scale factor of the intensity (119881

120575

1=

119881120575

2= 0) is considered and the third row shows results

when only the slow scale factor (119880120598

1= 119880

120598

2= 119880

120598

3= 0) is

considered The fourth row shows the results using multiplescale factors We can see from Table 1 that the mispricing ofthe zero-coupon defaultable bondwith a fast scale factor doesnot behave monotonically as the value of the recovery rateincreases but the mispricing of the zero-coupon defaultablebond with a slow scale factor monotonically increases withthe value of the recovery rate Additionally the mispricing ofthe zero-coupon defaultable bond with multiple scale factorsdoes not behave monotonically as the value of the recoveryrate increases From the above investigation we note that themispricing of the zero-coupon defaultable bond takes variousforms depending on the recovery rate parameters involved

Figures 1 and 2 show results for parameter values of 119886 =

02 119886 = 03 120590 = 01 120590(119911) = 015 2(119911) = 018 120588119903120582

= minus02119880120598

1= minus007 119880120598

2= 001 119880120598

3= minus006 119881120575

1= 001 119881120575

2= minus004

119903 = 015 120582 = 013 119905 = 0 119879 = 10 and 119877 = 04 These figuresillustrate the defaultable bond price and the correspondingyield curve respectively Figures 1 and 2 have two typesof curve Case 1 corresponds to constant volatility in theintensity process whereas Case 2 contains both fast andslow scale factors Figure 1 shows that the defaultable bondprices with stochastic intensity become higher than thosewith constant volatility as the time to maturity increasesIn addition the hump-shaped yield curve which matchesthose in structural models [10] appears in Case 2 in Figure 2That is the multiple time scales have both quantitative andqualitative effects Hence Figures 1 and 2 show the significanteffect of multiple time scales in the stochastic intensity onboth defaultable bond prices and yields

Time to maturity (years)

Def

aulta

ble b

ond

pric

es

0 2 4 6 8 100

02

04

06

08

1

Case 1Case 2

Figure 1 Defaultable bond prices (source KIS Pricing)

Time to maturity (years)0 2 4 6 8 10

00025

0003

00035

0004

00045

0005

00055

0006

Yield

s (

)

Case 1Case 2

Figure 2 Yield curves (source KIS Pricing)

4 Credit Default Swap and Bond Option

In this section we derive a formula for the CDS rate andobtain another formula for options when the underlyingasset is a zero-coupon defaultable bond using the results ofSection 3

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

33 First Perturbation 119875120575

01 Using similar arguments to those

in Section 32 we will derive the first perturbation term 119875120575

01

We insert 119896 = 1 in (17) as

119875120598

1= 119875

01+ radic120598119875

11+ 120598119875

21+ 120598

32

11987531

+ sdot sdot sdot (40)

Substituting the expansions (20) and (40) into (19) we get

1

120598L

011987501

+1

radic120598(L

011987511

+L111987501)

+ (L011987521

+L111987511

+L211987501)

+ radic120598 (L011987531

+L111987521

+L211987511) + sdot sdot sdot

=1

radic120598M

311987500

+ (M111987500

+M311987510)

+ radic120598 (M111987510

+M311987520) + sdot sdot sdot

(41)

Theorem 3 The correction term 11987501

does not depend on thevariable 119910 and 119875120575

01= radic120575119875

01(119905 119903 120582 119911 119879) is given by

119875120575

01(119905 119903 120582 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(42)

with 119861(119879 119879) = 119862(119879 119879) = 119865(119879 119879) = 119867(119879 119879) = 0 where119867(119905 119879) is given by

119867(119905 119879) = minusint119879

119905

119881120575

1119861 (119904 119879) + 119881

120575

2(1 minus 119877)119862 (119904 119879) 119889119904

119881120575

1= radic120575120588

119903119911120590ℎ (119911)

119881120575

2= radic120575120588

120582119911ℎ

119911(119911) ⟨119891⟩ (119911) + ℎ (119911) ⟨119891⟩

119911(119911)

(43)

Here 119861(119905 119879) 119862(119905 119879) and 119865(119905 119879) are given by (23)

Proof Multiply (41) by 120598 and let 120598 rarr 0 Then we find thefirst two leading-order terms as follows

L011987501

= 0

L011987511

+L111987501

= M311987500

(44)

Recall that because L0is the infinitesimal generator of the

OU process 119910119905 the solution 119875

01of (44) must be a constant

with respect to the 119910 variable that is 11987501

= 11987501(119905 119903 120582 119911 119879)

The next order also generates L011987511

= 0 Because M3has

a derivative with respect to the 119910 variable and 11987500

does notrely on the 119910 variable we get M

311987500

= 0 and L111987501

= 0respectively as remarked above Then 119875

11does not depend

on the 119910 variable that is 11987511

= 11987511(119905 119903 120582 119911 119879) This means

that the first two terms 11987501

and 11987511

do not depend on thecurrent level 119910 of the fast scale volatility driving the process119910119905 In this way we can continue to eliminate terms of order

1radic120598 120598 For the order-1 term we haveL011987521

+L111987511

+

L211987501

= M111987500

+ M311987510 This PDE becomes L

011987521

+

L211987501

= M111987500 because L

111987511

= 0 and M311987510

= 0This PDE is a Poisson equation for 119875

21with respect to the

infinitesimal operator L0 It is well-known that a solution

exists only if L211987501

= M111987500

is centered with respect to

the invariant distribution of the stochastic volatility process119910119905 Because 119875

00and 119875

01do not depend on the 119910 variable we

obtain

⟨L2⟩ 119875

01= A

120575

11987500 119905 lt 119879 (45)

with the terminal condition 11987501(119905 119903 120582 119911 119879)|

119905=119879

= 0 where

A120575

= 120590ℎ119911(119911) 120588

119903119911

120597

120597119903+ 120588

120582119911(ℎ

119911(119911) ⟨119891⟩ (119911)

+ℎ (119911) ⟨119891⟩119911(119911))

120597

120597120582

⟨119891⟩119911(119911) =

120597

120597119911⟨119891⟩ (119911) ℎ

119911(119911) =

120597

120597119911ℎ (119911)

(46)

respectively Because we concentrate on only the first per-turbation to 119875

00 we reset the PDE (45) with respect to

119875120575

01(119905 119903 120582 119911 119879) = radic120575119875

01(119905 119903 120582 119911 119879) as follows

⟨L2⟩ 119875

120575

01= radic120575A

120575

11987500 119905 lt 119879 (47)

with the final condition 119875120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 We thenobtain the operatorL120575 expressed as

L120575

= 119881120575

1

120597

120597119903+ 119881

120575

2

120597

120597120582 (48)

Hence we obtain (45) as follows

120597119875120575

10

120597119905+ (120579

119905minus 119886119903)

120597119875120575

10

120597119903+ (120579

119905minus 119886120582)

120597119875120575

10

120597120582+1

212059021205972119875120575

10

1205971199032

+1

22

(119911)1205972119875120575

10

1205971205822+ 120588

119903120582120590120590 (119911)

1205972119875120575

10

120597119903120597120582minus (119903 + (1 minus 119877) 120582) 119875

120575

10

= minus 119881120575

1119861 (119905 119879) + 119881

120575

2(1 minus 119877)119862 (119905 119879) 119875

00

(49)

with the terminal condition 119875120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 Finallysubstituting (42) into (49) we obtain the result of Theorem 3by direct computation

Note that the group parameters 119880120598

1 119880120598

2 119880120598

3 119881120575

1 and 119881120575

2

which contain model parameters are needed for the pricingof the zero-coupon defaultable bond and simplify the estima-tion procedure See Fouque et al [15] for a general reference

In summary from (16) the asymptotic analysis of thezero-coupon defaultable bond price gives

119875120598120575

(119905 119903 120582 119910 119911 119879)

asymp 11987500

(119905 119903 120582 119911 119879) + 119875120598

10(119905 119903 120582 119911 119879) + 119875

120575

01(119905 119903 120582 119911 119879)

119905 lt 119879

(50)

with the final condition 119875120598120575(119905 119903 120582 119910 119911 119879)|119905=119879

= 1 and

11987500

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120598

10(119905 119903 120582 119911 119879) = 119866 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

119875120575

01(119905 119903 120582 119911 119879) = 119867 (119905 119879) 119890

119865(119905119879)minus119861(119905119879)119903minus(1minus119877)119862(119905119879)120582

(51)

Journal of Applied Mathematics 7

Table 1 Effect of the recovery rate

Recovery rate (119877) 01 02 03 05 07 08Fast scale factor 035 009 minus011 minus029 minus027 minus020Slow scale factor minus043 minus026 minus010 024 057 073Multiple scale factors minus010 minus020 minus021 minus003 034 058

where 11987500(119905 119903 120582 119911 119879) 119875120598

10(119905 119903 120582 119911 119879) and 119875120575

01(119905 119903 120582 119911 119879)

are given by (22) (30) and (42) respectively In terms ofaccuracy of the approximation we refer to the error estimateobtained by Papageorgiou and Sircar [8]

34 Numerical Results In this subsection we conclude ourpaper with some sensitivity analyses of themodel parametersThe parameter values used to obtain the results shown inTable 1 are 119886 = 02 119886 = 03119880120598

1= 001119880120598

2= minus003 119880120598

3= 004

119881120575

1= 002 119881120575

2= minus003 119905 = 0 and 119879 = 1 We calculate

the magnitude of mispricing with respect to the zero-coupondefaultable bond as a percentage of the face value of the bondThat is we show that the zero-coupon defaultable bond tendsto be overpriced or underpriced in terms of the parametersinvolvedThe mispricing as a percentage of the zero-coupondefaultable bond price caused by changes in the value ofthe recovery rate is shown in Table 1 The second row showsresults when only the fast scale factor of the intensity (119881

120575

1=

119881120575

2= 0) is considered and the third row shows results

when only the slow scale factor (119880120598

1= 119880

120598

2= 119880

120598

3= 0) is

considered The fourth row shows the results using multiplescale factors We can see from Table 1 that the mispricing ofthe zero-coupon defaultable bondwith a fast scale factor doesnot behave monotonically as the value of the recovery rateincreases but the mispricing of the zero-coupon defaultablebond with a slow scale factor monotonically increases withthe value of the recovery rate Additionally the mispricing ofthe zero-coupon defaultable bond with multiple scale factorsdoes not behave monotonically as the value of the recoveryrate increases From the above investigation we note that themispricing of the zero-coupon defaultable bond takes variousforms depending on the recovery rate parameters involved

Figures 1 and 2 show results for parameter values of 119886 =

02 119886 = 03 120590 = 01 120590(119911) = 015 2(119911) = 018 120588119903120582

= minus02119880120598

1= minus007 119880120598

2= 001 119880120598

3= minus006 119881120575

1= 001 119881120575

2= minus004

119903 = 015 120582 = 013 119905 = 0 119879 = 10 and 119877 = 04 These figuresillustrate the defaultable bond price and the correspondingyield curve respectively Figures 1 and 2 have two typesof curve Case 1 corresponds to constant volatility in theintensity process whereas Case 2 contains both fast andslow scale factors Figure 1 shows that the defaultable bondprices with stochastic intensity become higher than thosewith constant volatility as the time to maturity increasesIn addition the hump-shaped yield curve which matchesthose in structural models [10] appears in Case 2 in Figure 2That is the multiple time scales have both quantitative andqualitative effects Hence Figures 1 and 2 show the significanteffect of multiple time scales in the stochastic intensity onboth defaultable bond prices and yields

Time to maturity (years)

Def

aulta

ble b

ond

pric

es

0 2 4 6 8 100

02

04

06

08

1

Case 1Case 2

Figure 1 Defaultable bond prices (source KIS Pricing)

Time to maturity (years)0 2 4 6 8 10

00025

0003

00035

0004

00045

0005

00055

0006

Yield

s (

)

Case 1Case 2

Figure 2 Yield curves (source KIS Pricing)

4 Credit Default Swap and Bond Option

In this section we derive a formula for the CDS rate andobtain another formula for options when the underlyingasset is a zero-coupon defaultable bond using the results ofSection 3

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Table 1 Effect of the recovery rate

Recovery rate (119877) 01 02 03 05 07 08Fast scale factor 035 009 minus011 minus029 minus027 minus020Slow scale factor minus043 minus026 minus010 024 057 073Multiple scale factors minus010 minus020 minus021 minus003 034 058

where 11987500(119905 119903 120582 119911 119879) 119875120598

10(119905 119903 120582 119911 119879) and 119875120575

01(119905 119903 120582 119911 119879)

are given by (22) (30) and (42) respectively In terms ofaccuracy of the approximation we refer to the error estimateobtained by Papageorgiou and Sircar [8]

34 Numerical Results In this subsection we conclude ourpaper with some sensitivity analyses of themodel parametersThe parameter values used to obtain the results shown inTable 1 are 119886 = 02 119886 = 03119880120598

1= 001119880120598

2= minus003 119880120598

3= 004

119881120575

1= 002 119881120575

2= minus003 119905 = 0 and 119879 = 1 We calculate

the magnitude of mispricing with respect to the zero-coupondefaultable bond as a percentage of the face value of the bondThat is we show that the zero-coupon defaultable bond tendsto be overpriced or underpriced in terms of the parametersinvolvedThe mispricing as a percentage of the zero-coupondefaultable bond price caused by changes in the value ofthe recovery rate is shown in Table 1 The second row showsresults when only the fast scale factor of the intensity (119881

120575

1=

119881120575

2= 0) is considered and the third row shows results

when only the slow scale factor (119880120598

1= 119880

120598

2= 119880

120598

3= 0) is

considered The fourth row shows the results using multiplescale factors We can see from Table 1 that the mispricing ofthe zero-coupon defaultable bondwith a fast scale factor doesnot behave monotonically as the value of the recovery rateincreases but the mispricing of the zero-coupon defaultablebond with a slow scale factor monotonically increases withthe value of the recovery rate Additionally the mispricing ofthe zero-coupon defaultable bond with multiple scale factorsdoes not behave monotonically as the value of the recoveryrate increases From the above investigation we note that themispricing of the zero-coupon defaultable bond takes variousforms depending on the recovery rate parameters involved

Figures 1 and 2 show results for parameter values of 119886 =

02 119886 = 03 120590 = 01 120590(119911) = 015 2(119911) = 018 120588119903120582

= minus02119880120598

1= minus007 119880120598

2= 001 119880120598

3= minus006 119881120575

1= 001 119881120575

2= minus004

119903 = 015 120582 = 013 119905 = 0 119879 = 10 and 119877 = 04 These figuresillustrate the defaultable bond price and the correspondingyield curve respectively Figures 1 and 2 have two typesof curve Case 1 corresponds to constant volatility in theintensity process whereas Case 2 contains both fast andslow scale factors Figure 1 shows that the defaultable bondprices with stochastic intensity become higher than thosewith constant volatility as the time to maturity increasesIn addition the hump-shaped yield curve which matchesthose in structural models [10] appears in Case 2 in Figure 2That is the multiple time scales have both quantitative andqualitative effects Hence Figures 1 and 2 show the significanteffect of multiple time scales in the stochastic intensity onboth defaultable bond prices and yields

Time to maturity (years)

Def

aulta

ble b

ond

pric

es

0 2 4 6 8 100

02

04

06

08

1

Case 1Case 2

Figure 1 Defaultable bond prices (source KIS Pricing)

Time to maturity (years)0 2 4 6 8 10

00025

0003

00035

0004

00045

0005

00055

0006

Yield

s (

)

Case 1Case 2

Figure 2 Yield curves (source KIS Pricing)

4 Credit Default Swap and Bond Option

In this section we derive a formula for the CDS rate andobtain another formula for options when the underlyingasset is a zero-coupon defaultable bond using the results ofSection 3

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

41 Credit Default Swap

411 Preliminaries A CDS is a bilateral contract in whichone party (the protection buyer) pays a periodic fixedpremium to another (the protection seller) for protectionrelated to credit events on a reference entity If a credit eventoccurs the protection seller is obliged to make a paymentto the protection buyer to compensate him for any lossesthat he might otherwise incur Then the credit risk of thereference entity is transferred from the protection buyer tothe protection seller In particular Ma and Kim [6] studiedthe problem of default correlation when the reference entityand the protection seller can default simultaneously Weassume the following

(i) We consider a forward CDS rate valuable after someinitial time 119905

0with 0 le 119905

0lt 119905

1 to use the results of

Section 3(ii) Let 119879 be the time-to-maturity of a forward CDS

contract 1199051lt sdot sdot sdot lt 119905

119873= 119879 the premium payment

dates andT = (1199051 119905

119873) the payment tenor

(iii) We assume that in a credit event the bond recoversa proportion 119877 of its face value and the protectionseller provides the remaining proportion 1 minus 119877 to theprotection buyer

(iv) If the settlement takes place at a coupon date fol-lowing the credit event occurring to the referenceentity then we do not take into account this accruedpremium payment

For a detailed explanation refer to OrsquoKane and Turnbull [18]Let us denote by 119862(119905 119905

0T) the price of the forward CDS

rate Note that the spread of a CDS rate is given by the spreadof a forward CDS when 119905

0= 119905 Let us first consider the

case of the protection buyer The premium leg is the seriesof payments of the forward CDS rate until maturity or untilthe first credit event 120591 Let us denote by119862119901119887(119905 119905

0T) the price

of the protection buyer paying 1(120591gt119905)

at time 119905This is given by

119862119901119887

(119905 1199050T)

= 119862 (119905 1199050T)

times

119873

sum119899=1

119864lowast

[119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(52)

Note that the payment made by the protection buyer is zeroif 120591 le 119905 Here we can easily solve (52) using the results ofSection 3 when 119877 = 0 (zero recovery) On the other hand theprice 119862

119901119904(119905 1199050T) demanded by the protection seller at the

credit event time 120591 is given by

119862119901119904

(119905 1199050T)

= (1 minus 119877) 119864lowast

times [int119905119873

1199050

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(53)

with 119905 lt 120591 Note that the payment demanded by theprotection seller is zero if 120591 le 119905 Finally from the no-arbitragecondition based on the pricing of each counterpartyrsquos posi-tion by substituting (52) and (53) we obtain the forward CDSrate 119862(119905 119905

0T) as follows

119862 (119905 1199050T)

= (1 minus 119877)

times119864lowast [int

119905119873

1199050

119890minusint119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905=119903 120582

119905=120582 119910

119905=119910 119911

119905= 119911]

sum119873

119899=1119864lowast [119890minusint

119905119899

119905(119903119904+120582119904)119889119904 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(54)

Now we will calculate (53) using the asymptotic analysis

412 Asymptotic Analysis of the CDS Rate For the protectionseller payment we put

(119905 119903 120582 119910 119879)

= 119864lowast

[int119879

119905

119890minusint

119906

119905(119903119904+120582119904)119889119904120582

119906119889119906 | 119903

119905= 119903 120582

119905= 120582 119910

119905= 119910 119911

119905= 119911]

(55)

and then

119862119901119904

(119905 1199050T) = (119905 119903 120582 119910 119879) minus (119905 119903 120582 119910 119905

0) (56)

Using the four-dimensional Feynman-Kac formula we havethe Kolmogorov PDE

L119901119904

(119905 119903 120582 119910 119911 119879) + 120582 = 0 119905 lt 119879

(119905 119903 120582 119910 119911 119879)10038161003816100381610038161003816119905=119879

= 0(57)

where

L119901119904

=1

120598L

0+

1

radic120598L

1+L

119901119904

2+ radic120575M

1+ 120575M

2+ radic

120575

120598M

3

L119901119904

2=

120597

120597119905+ (120579

119905minus 119886119903)

120597

120597119903+ (120579

119905minus 119886120582)

120597

120597120582+1

212059021205972

1205971199032

+1

21198912

(119910)1205972

1205971205822+ 120590119891 (119910) 120588

119903120582

1205972

120597119903120597120582minus (119903 + 120582)

(58)

where L0 L

1 M

1 M

2 and M

3are defined in Section 3

Here the operatorL119901119904

2is equal to the operatorL

2when 119877 =

0Now we use the notation

119894119895for the 12059811989421205751198952-order term

for 119894 = 0 1 2 and 119895 = 0 1 2 We first expand withrespect to half-powers of 120575 and then for each of these termswe expand with respect to half-powers of 120598 The expansion of120598120575 is

120598120575

= 120598

0+ radic120575

120598

1+ 120575

120598

2+ sdot sdot sdot

120598

119896=

0119896+ radic120598

1119896+ 120598

2119896+ 120598

32

3119896

+ sdot sdot sdot

(59)

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

for 119896 = 0 1 2 The expansion (59) leads to the leading-order term 120598

0and the next-order term 120598

1is defined by

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

0= 0 119905 lt 119879 (60)

with the terminal condition 120598

0(119905 119903 120582 119910 119911 119879)|

119905=119879= 0 and

(1

120598L

0+

1

radic120598L

1+L

119901119904

2)

120598

1= (M

1+

1

radic120598M

3)

120598

0 119905 lt 119879

(61)

with the terminal condition 120598

1(119905 119903 120582 119910 119911 119879)|

119905=119879= 0

respectively We insert 119896 = 0 in (59) as

120598

0=

00+ radic120598

10+ 120598

20+ 120598

32

30

+ sdot sdot sdot (62)

Applying the expanded solution (62) to (60) leads to

1

120598L

000

+1

radic120598(L

010

+L100)

+ (L020

+L110

+L119901119904

200)

+ radic120598 (L030

+L120

+L119901119904

210) + sdot sdot sdot = 0

(63)

Using similar processes as in Section 3 we will calculate theleading-order term

00and the first perturbation term

10

That is we let

⟨L119901119904

2⟩

00+ 120582 = 0 (64)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and

⟨L119901119904

2⟩

120598

10= A

119901119904

00

120598

10= radic120598

10

A119901119904

=1

radic120572⟨L

1L

minus1

0(L

119901119904

2minus ⟨L

119901119904

2⟩)⟩

(65)

with the terminal condition 120598

10(119905 119903 120582 119911 119879)|

119905=119879

= 0Let

00(119905 119903 120582 119911 119879) be equal to the leading-order term

11987500(119905 119903 120582 119911 119879) in (22) with zero recovery that is

00

(119905 119903 120582 119911 119879) = 119890119865(119905119879)minus119861(119905119879)119903minus119862(119905119879)120582

(66)

where 119861(119905 119879)119862(119905 119879) and 119865(119905 119879) are given by (23)Then thesolution of the inhomogeneous PDE (64) is given by

00

(119905 119903 120582 119911 119879) = 120582int119879

119905

11987500

(119904 119903 120582 119911 119879) 119889119904 (67)

with the terminal condition 00(119905 119903 120582 119911 119879)|

119905=119879

= 0 and thesolution of the PDE (65) is given by

120598

10(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

[119880120598

1119861 (119904 ℎ) 119862

2

(119904 ℎ) + 119880120598

2119861

2

(119904 ℎ) 119862 (119904 ℎ)

+119880120598

3119862

3

(119904 ℎ)] 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(68)

where 119880120598

1 119880120598

2 and 119880120598

3are given by (30) Also we insert 119896 = 1

in (59) as

120598

1=

01+ radic120598

11+ 120598

21+ 120598

32

31

+ sdot sdot sdot (69)

Substituting the expansions (62) and (69) into (61) we get

1

120598L

001

+1

radic120598(L

011

+L101)

+ (L021

+L111

+L119901119904

201)

+ radic120598 (L011987531

+L121

+L119901119904

211) + sdot sdot sdot

=1

radic120598M

300

+ (M100

+M310)

+ radic120598 (M110

+M320) + sdot sdot sdot

(70)

Using similar processes as in Section 3 we will calculate thefirst perturbation term 120575

01 That is we let

⟨L119901119904

2⟩

120575

01= radic120575M

100

120575

01= radic120575

01 (71)

with the terminal condition 120575

01(119905 119903 120582 119911 119879)|

119905=119879

= 0 Thenthe solution of the PDE (71) is given by

120575

01(119905 119903 120582 119879)

= 120582int119879

119905

int119879

119904

(119881120575

1119861 (119904 ℎ) + 119881

120575

2119862 (119904 ℎ))

times 00

(119904 119903 120582 119911 ℎ) 119889ℎ 119889119904

(72)

with the terminal condition 120575

10(119905 119903 120582 119911 119879)|

119905=119879

= 0 where119881120575

1and 119881120575

2are given by (43) Therefore combining (67)

(68) and (72) we derive an asymptotic expression for theprotection seller which is given by

119901119904

(119905 119903 120582 119910 119879)

asymp 00

(119905 119903 120582 119879) + 120598

10(119905 119903 120582 119879) +

120575

01(119905 119903 120582 119879)

(73)

42 Bond Option Pricing We use 119879 and 1198790 119905 le 119879

0lt 119879

to denote the maturity of the zero-coupon defaultable bondand the maturity of the option written on that zero-coupondefaultable bond respectively We assume that the optionbecomes invalid when a credit event occurs before 119879

0 Then

the option price with the fractional recovery assumptiondenoted by119883(119905 119903 120582 119910 119911 119879

0 119879) is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119910

119905= 119910 119911

119905= 119911]

(74)

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

under risk-neutral probability 119875lowast where the bond price119875(119879

0 1199031198790 120582

11987901199101198790 119911

1198790 119879) is

119875 (1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

= 119864lowast

[119890minusint

119879

1198790

(119903119904+(1minus119877)120582119904)119889119904 | 1199031198790 120582

1198790 119910

1198790 119911

1198790]

(75)

and 119897(119875(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)) is the payoff function of the

option at time 1198790 Here for simplicity we assume that the

payoff function 119897 is at best linearly growing at infinity and issmooth In fact the nonsmoothness assumption on 119897 can betreated by a nontrivial regularization argument as presentedin Fouque et al [19]

From the four-dimensional Feynman-Kac formula weobtain 119883(119905 119903 120582 119910 119911 119879

0 119879) as a solution of the Kolmogorov

PDE

120597119883

120597119905+ (120579

119905minus 119886119903)

120597119883

120597119903+1

212059021205972

119883

1205971199032+ (120579

119905minus 119886120582)

120597119883

120597120582

+1

21198912

(119910 119911)1205972

119883

1205971205822+1

120598(119898 minus 119910)

120597119883

120597119910+]2

120598

1205972

119883

1205971199102+ 120575119892 (119911)

120597119883

120597119911

+1

2120575ℎ

2

(119911)1205972119883

1205971199112+ 120588

119903120582120590119891 (119910 119911)

1205972119883

120597119903120597120582+ 120588

119903119910120590]radic2

radic120598

1205972119883

120597119903120597119910

+ 120588119903119911120590radic120575ℎ (119911)

1205972119883

120597119903120597119911+ 120588

120582119910119891 (119910 119911)

]radic2

radic120598

1205972119875

120597120582120597119910

+ 120588120582119911119891 (119910 119911)radic120575ℎ (119911)

1205972119883

120597120582120597119911+ 120588

119910119911

]radic2

radic120598radic120575ℎ (119911)

1205972119883

120597119910120597119911

minus 119903 + (1 minus 119877) 120582119883 = 0

(76)

with the final condition 119883(119905 119903 120582 119910 119911 1198790 119879)|

119905=1198790

= 119897(119875(1198790

1199031198790120582

1198790 119910

1198790 119911

1198790 119879)) Hence keeping the notation used in the

pricing of zero-coupon defaultable bonds with fractionalrecovery in Section 3 but with a different terminal conditionthe option price 119883(119905 119903 120582 119910 119911 119879

0 119879) contains the Feynman-

Kac formula problems

L119883(119905 119903 120582 119910 119911 1198790 119879) = 0 119905 lt 119879

0

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790= 119897 (119875 (119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(77)

with

119883 = 119883120598

0+ radic120575119883

120598

1+ 120575119883

120598

2+ sdot sdot sdot

119883120598

119896= 119883

0119896+ radic120598119883

1119896+ 120598119883

2119896+ 120598

32

1198833119896

+ sdot sdot sdot

(78)

for 119896 = 0 1 2 Using the assumed smoothness of the payofffunction 119897 the terminal condition (77) can be expanded asfollows

119883(119905 119903 120582 119910 119911 1198790 119879)

1003816100381610038161003816119905=1198790

= 119897 (11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(79)

+ 119875120598

10(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(80)

+ 119875120575

01(119879

0 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119910

1198790 119911

1198790 119879))

(81)

where 11987500 119875120598

10 and 119875

120575

01are given by (22) (30) and (42)

respectively Using a similar argument as in Section 3 theterms of order 1120598 and 1radic120598 have 119910-independence in 119883

00

11988310 and 119883

01 The order-1 terms give a Poisson equation in

11988320 with which the solvability condition ⟨L

2⟩119883

00= 0 is

satisfied From the solution calculated in Section 31 we have

11988300

(119905 119903 120582 119911 1198790 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904ℎ (119875

00(119879

0 1199031198790 120582

1198790 119911

1198790 119879)) | 119903

119905= 119903

120582119905= 120582 119911

119905= 119911]

(82)

with the terminal condition (79) where11987500(119879

0 1199031198790 120582

1198790 119911

1198790 119879) is given by (22) at time 119905 = 119879

0

The order-radic120598 terms give a Poisson equation in 11988330 with

which the solvability condition ⟨L111988320

+ L211988310⟩ = 0 If

we put119883120598

10= radic120598119883

10 then this solvability condition leads to

the PDE

⟨L2⟩119883

120598

10(119905 119903 120582 119911 119879

0 119879) = radic120598A

120598

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(83)

with the terminal condition (80) where the infinitesimaloperator A120598 is given by (35) Hence by applying theFeynman-Kac formula to (82) and (83) we get the followingprobabilistic representation of the first perturbation term

119883120598

10(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120598

10(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(11987500

(1198790 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120598

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905=119903 120582

119905=120582 119885

119905= 119911]

(84)

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

The order-radic120575 terms give a Poisson equation11988321 with which

the solvability condition ⟨L211988301⟩ = ⟨A120575119883

00⟩ If we put

119883120575

01= radic120575119883

01 then this solvability condition leads to the

PDE

⟨L2⟩119883

120575

01(119905 119903 120582 119911 119879

0 119879) = radic120575A

120575

11988300

(119905 119903 120582 119911 1198790 119879)

119905 lt 1198790

(85)

where the operator A120575 is obtained by (46) Hence byapplying the Feynman-Kac formula to (82) and (85) wehave the following probabilistic representation of the firstperturbation term

119883120575

01(119905 119903 120582 119911 119879

0 119879)

= 119864lowast

[119890minusint

1198790

119905119903119904119889119904119875

120575

01(119879

0 1199031198790 120582

1198790 119911

1198790 119879)

times 1198971015840

(1198750(119879

0 1199031198790 120582

1198790 119911

1198790 119879))

minus int1198790

119905

119890minusint

119906

119905119903119904119889119904A

120575

11988300

times (119906 119903119906 120582

119906 119911

119906 119879

0 119879) 119889119906 | 119903

119905= 119903 120582

119905= 120582 119911

119905= 119911]

(86)

In summary we have an asymptotic expression for thezero-coupon defaultable bond option price with fractionalrecovery which is given by

119883(119905 119903 120582 119910 119911 1198790 119879)

asymp 11988300

(119905 119903 120582 119911 1198790 119879) + 119883

120598

10(119905 119903 120582 119911 119879

0 119879)

+ 119883120575

01(119905 119903 120582 119911 119879

0 119879)

(87)

where 11988300(119905 119903 120582 119911 119879

0 119879) 119883120598

10(119905 119903 120582 119911 119879

0 119879) and 119883120575

01(119905 119903

120582 119911 1198790 119879) are given by (82) (84) and (86) respectively

5 Final Remarks

In this paper we have studied the effect of applying stochasticvolatility to the default intensity in zero-coupon defaultablebonds To model this we considered the correlated Hull andWhitemodel developed by Tchuindjo [7] withmultiple timescales in the stochastic volatility of the intensity processUsing asymptotic analysis we obtained approximate solu-tions to price zero-coupon defaultable bonds credit defaultswap rates and bond options when 120598 and 120575 are indepen-dently small parameters To understand multiple time scaleswe provided numerical examples to price the zero-coupondefaultable bonds as well as the yield curve This showedhow these multiple time scales can have both quantitativeand qualitative effects In future work we will provide anefficient tool for calibrating the zero-coupon defaultable bondintensity models from market yield spreads

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Korea Institute of Science andTechnology Information (KISTI) under Project no K-14-L06-C14-S01

References

[1] D Duffie and K J Singleton ldquoModeling term structures ofdefaultable bondsrdquo Review of Financial Studies vol 12 no 4pp 687ndash720 1999

[2] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash85 1995

[3] D Lando ldquoOn cox processes and credit risky securitiesrdquo Reviewof Derivatives Research vol 2 no 2-3 pp 99ndash120 1998

[4] P J Schonbucher ldquoTerm structure of defaultable bond pricesrdquoReview of Derivatives Research vol 2 pp 161ndash192 1998

[5] G R Duffee ldquoEstimating the price of default riskrdquo Review ofFinancial Studies vol 12 no 1 pp 197ndash226 1999

[6] Y-K Ma and J-H Kim ldquoPricing the credit default swap ratefor jump diffusion default intensity processesrdquo QuantitativeFinance vol 10 no 8 pp 809ndash817 2010

[7] L Tchuindjo ldquoPricing of multi-defaultable bonds with a two-correlated-factor Hull-White modelrdquo Applied MathematicalFinance vol 14 no 1 pp 19ndash39 2007

[8] E Papageorgiou and R Sircar ldquoMultiscale intensity models forsingle name credit derivativesrdquo Applied Mathematical Financevol 15 no 1-2 pp 73ndash105 2008

[9] J-P Fouque G Papanicolaou R Sircar andK SoslashlnaMultiscaleStochastic Volatility for Equity Interest Rate and Credit Deriva-tives Cambridge University Press 2011

[10] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Journal of Finance vol 29 pp 449ndash470 1974

[11] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoMulti-scale stochastic volatility asymptoticsrdquoA SIAM InterdisciplinaryJournal vol 2 no 1 pp 22ndash42 2003

[12] J-P Fouque R Sircar and K Soslashlna ldquoStochastic volatility effectsondefaultable bondsrdquoAppliedMathematical Finance vol 13 no3 pp 215ndash244 2006

[13] S Alizadeh M W Brandt and F X Diebold ldquoRange-basedestimation of stochastic volatility modelsrdquo Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[14] J Masoliver and J Perello ldquoMultiple time scales and theexponential Ornstein-Uhlenbeck stochastic volatility modelrdquoQuantitative Finance vol 6 no 5 pp 423ndash433 2006

[15] J-P Fouque G Papanicolaou and R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[16] B Oslashksendal Stochastic Differential Equations Springer 2003[17] J Hull andAWhite ldquoPricing interest-rate derivative securitiesrdquo

Review of Financial Studies vol 3 no 4 pp 573ndash592 1990

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

[18] K OKane and S Turnbull Valuation of Credit Default SwapsFixed Income Quantitative Credit Research Lehman Brother2003

[19] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of