# Asymptotic Analysis for One-Name Credit Derivatives

Authors: Yong-Ki Ma; Beom Jin Kim
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We propose approximate solutions to price defaultable zero-coupon bonds as well as the corresponding credit default swaps and bond options. We consider the intensity-based approach of a two-correlated-factor Hull-White model with stochastic volatility of interest rate process. Perturbations from the stochastic volatility are computed by using an asymptotic analysis. We also study the sensitive properties of the defaultable bond prices and the yield curves.


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## Body

## 1. Introduction

It is well known that the methodology for modeling a credit risk can be split into two primary approaches of models that attempt to describe default processes (see Duffie and Singleton [1] and Bielecki and Rutkowski [2] for general references): the structural approach such as those developed by Merton [3], Longstaff and Schwartz [4], Leland and Toft [5], Zhou [6], Duffie and Lando [7], Hilberink and Rogers [8], and Giesecke [9] and the intensity-based approach such as those developed by Jarrow and Turnbull [10], Madan and Unal [11], Lando [12], Duffie and Singleton [13], and Collin-Dufresne and Goldstein [14]. A structural approach assumes that the market has complete information with respect to the underlying firm’s value process and capital structure. In contrast, an intensity-based approach (also known as a reduced-form approach) has been developed where default is determined as the first jump of an exogenously given jump process. Hence, an underlying firm's default time is inaccessible and driven by a default intensity, a function of some latent state variables. Since we are concerned with modeling the default time, we adopt the intensity-based approach with the fractional recovery assumption of Duffie and Singleton [13].Since the initial contribution to the intensity-based approach is given by Jarrow and Turnbull [10], who consider a constant and a deterministic Poisson intensity, there has been many mathematical studies on credit risk. Lando [12] uses that the default process is described by a Cox process. Schönbucher [15] develops the term structure model of defaultable interest rates by using the Heath-Jarrow-Morton model, and Tchuindjo [16] studies the price of a defaultable zero-coupon bond with two-correlated-factor Hull-White model. Duffee [17] and Driessen [18] perform the estimation of an intensity-based model using a Kalman filter approach, and Bayraktar and Yang [19] propose intensity-based unified models with stochastic volatility calibrated to stock options data. Some of other application articles based on an intensity-based framework are as follows. Duffie and Gârleanu [20] extend the single-name intensity setting to a multiname setting by putting firms' default intensities as the sum of an idiosyncratic factor and a common factor that affects the default of all firms. Papageorgiou and Sircar [21] study the pricing for the prices of single-name credit derivatives such as defaultable bonds, options on defaultable bonds, and credit default swaps (CDSs) using a function of two different time-scale intensity models. In particular, Ma and Kim [22] provide a pricing formula for the CDSs based on modeling the intensity as a jump-diffusion process.The main contribution of this paper is as follows. Tchuindjo [16] studies the two-correlated-factor Hull-White model to propose a closed-form solution to price the defaultable bonds, supposing a nonzero correlation between interest rate process and intensity process. However, Tchuindjo's model does not appear hump-shaped yield curve that matches a typical yield curve for the defaultable bonds as in Merton's model [3] (see Figure 2). Cotton et al. [23] also show the bursty nature of stochastic volatility of interest rate process to understand the effect of uncertain and changing volatility on interest rate derivatives. So, we expand the two-correlated-factor Hull-White model by modifying constant volatility of interest rate process and use an asymptotic analysis for prices of the defaultable zero-coupon bonds. Our numerical results indicate that the defaultable zero-coupon bonds with stochastic volatility of interest rate process affects both quantitative and qualitative effect. We also obtain formulas for CDSs and options when the underlying assets are defaultable zero-coupon bonds.This paper is structured as follows. In Section2, we obtain a partial differential equation (PDE) based on stochastic volatility of interest rate process to get the price of the defaultable zero-coupon bonds. In Section 3, an approximate solution to price defaultable zero-coupon bonds is derived by using an asymptotic analysis including some numerical examples. Section 4 applies the results of Section 3 to the CDSs and the bond options. In Section 5, we give the final concluding remarks.


## 2. Defaultable Zero-Coupon Bonds

In this section, we consider an intensity-based approach to price the defaultable zero-coupon bonds with stochastic volatility of the interest rate process.


### 2.1. Constant Volatility of Interest Rate Process

The framework is the intensity-based approach with the fractional recovery assumption of Duffie and Singleton [13] as follows:
(1)𝒵(t,T)=𝔼*[e-∫tT‍{rs+(1-R)λs}ds∣ℱt],
under a risk-neutral probability P*, where t is a current time, T is an expiry time, R is a recovery rate with R∈[0,1], and ℱt is a filtration generated by the joint process of interest rate process rt and intensity process λt. We need to work out the expectation over the possible paths of rt and λt to get the defaultable zero-coupon bond prices.We review the price of the defaultable zero-coupon bonds when the interest rate process and the intensity process are correlated and each of these processes follows a Hull-White model [24]. Under a risk-neutral probability P*, we adopt the Tchuindjo type of stochastic differential equations (SDEs),
(2)drt=(θt-art)dt+σdWt(r)*,dλt=(θ~t-a~λt)dt+σ~dWt(λ)*,dWt(r)*dWt(λ)*=ρdt,
for such models, where a,a~,σ, and σ~ are constants, θt and θ~t are time-varying deterministic functions, and Wt(r)* and Wt(λ)* are correlated Brownian motions. Under the fractional recovery of market value assumption, the PDE for the price of the defaultable zero-coupon bond is derived. Then, this PDE is analytically solved. Refer to Tchuindjo [16].


### 2.2. Stochastic Volatility of Interest Rate Process

In this subsection, we enlarge the constant volatility of the interest rate process by incorporating the stochastic volatility of the interest rate process.We replace the constantσ in the SDE (2) by a stochastic process σt given by a smooth, bounded below and above, and strictly positive function f(yt) of an Ornstein-Uhlenbeck (OU) process yt as follows (see Fouque et al. [25]):
(3)dyt=α(m-yt)dt+βdWt(y)*,
where α is a rate of meanreversion of volatility, β is a volatility of volatility, and Wt(y)* is a Brownian motion in a risk-neutral probability P*. Note that the OU process is an ergodic process with a unique invariant distribution, which is a Gaussian distribution with mean m and variance ν2=β2/2α. For the asymptotic analysis, α driving the volatility is large, and we are interested in approximation in the limit α→∞ with ν2 remaining constant. In this OU model, therefore, we set
(4)α=1ϵ,β=𝒪(1ϵ),ν=𝒪(1),
where ϵ is a small and strictly positive parameter. So, in the risk-neutral probability P* the joint process (rt,λt,yt) is given by the following SDEs:
(5)drt=(θt-art)dt+σtdWt(r)*,σt=f(yt),dyt=1ϵ(m-yt)dt+ν2ϵdWt(y)*,dλt=(θ~t-a~λt)dt+σ~dWt(λ)*,
where the standard Brownian motions Wt(r)*, Wt(λ)*, and Wt(y)* are dependent of each other such that the correlation structure is given by
(6)d〈W(r)*,W(λ)*〉t=ρrλdt,d〈W(r)*,W(y)*〉t=ρrydt,d〈W(λ)*,W(y)*〉t=ρλydt.The defaultable zero-coupon bond prices with a fractional recovery rateR at time t for an interest rate process rt=r, an intensity process λt=λ, and a stochastic volatility process yt=y, denoted by P(t,r,λ,y;T), are given by
(7)P(t,r,λ,y;T)=𝔼*[e-∫tT‍{rs+(1-R)λt}ds∣rt=r,λt=λ,yt=y],
and then, using the three-dimensional Feynman-Kac formula, we have the Kolmogorov PDE:
(8)∂P∂t+(θt-ar)∂P∂r+(θ~t-a~λ)∂P∂λ+1ϵ(m-y)∂P∂y+12f2(y)∂2P∂r2+12σ~2∂2P∂λ2+ν2ϵ∂2P∂y2+ρrλσ~f(y)∂2P∂r∂λ+ρryf(y)ν2ϵ∂2P∂r∂y+ρλyσ~ν2ϵ∂2P∂λ∂y-(r+(1-R)λ)P=0,
with the terminal condition P(t,r,λ,y;T)|t=T=1. Refer to Øksendal [26].


## 2.1. Constant Volatility of Interest Rate Process

The framework is the intensity-based approach with the fractional recovery assumption of Duffie and Singleton [13] as follows:
(1)𝒵(t,T)=𝔼*[e-∫tT‍{rs+(1-R)λs}ds∣ℱt],
under a risk-neutral probability P*, where t is a current time, T is an expiry time, R is a recovery rate with R∈[0,1], and ℱt is a filtration generated by the joint process of interest rate process rt and intensity process λt. We need to work out the expectation over the possible paths of rt and λt to get the defaultable zero-coupon bond prices.We review the price of the defaultable zero-coupon bonds when the interest rate process and the intensity process are correlated and each of these processes follows a Hull-White model [24]. Under a risk-neutral probability P*, we adopt the Tchuindjo type of stochastic differential equations (SDEs),
(2)drt=(θt-art)dt+σdWt(r)*,dλt=(θ~t-a~λt)dt+σ~dWt(λ)*,dWt(r)*dWt(λ)*=ρdt,
for such models, where a,a~,σ, and σ~ are constants, θt and θ~t are time-varying deterministic functions, and Wt(r)* and Wt(λ)* are correlated Brownian motions. Under the fractional recovery of market value assumption, the PDE for the price of the defaultable zero-coupon bond is derived. Then, this PDE is analytically solved. Refer to Tchuindjo [16].


## 2.2. Stochastic Volatility of Interest Rate Process

In this subsection, we enlarge the constant volatility of the interest rate process by incorporating the stochastic volatility of the interest rate process.We replace the constantσ in the SDE (2) by a stochastic process σt given by a smooth, bounded below and above, and strictly positive function f(yt) of an Ornstein-Uhlenbeck (OU) process yt as follows (see Fouque et al. [25]):
(3)dyt=α(m-yt)dt+βdWt(y)*,
where α is a rate of meanreversion of volatility, β is a volatility of volatility, and Wt(y)* is a Brownian motion in a risk-neutral probability P*. Note that the OU process is an ergodic process with a unique invariant distribution, which is a Gaussian distribution with mean m and variance ν2=β2/2α. For the asymptotic analysis, α driving the volatility is large, and we are interested in approximation in the limit α→∞ with ν2 remaining constant. In this OU model, therefore, we set
(4)α=1ϵ,β=𝒪(1ϵ),ν=𝒪(1),
where ϵ is a small and strictly positive parameter. So, in the risk-neutral probability P* the joint process (rt,λt,yt) is given by the following SDEs:
(5)drt=(θt-art)dt+σtdWt(r)*,σt=f(yt),dyt=1ϵ(m-yt)dt+ν2ϵdWt(y)*,dλt=(θ~t-a~λt)dt+σ~dWt(λ)*,
where the standard Brownian motions Wt(r)*, Wt(λ)*, and Wt(y)* are dependent of each other such that the correlation structure is given by
(6)d〈W(r)*,W(λ)*〉t=ρrλdt,d〈W(r)*,W(y)*〉t=ρrydt,d〈W(λ)*,W(y)*〉t=ρλydt.The defaultable zero-coupon bond prices with a fractional recovery rateR at time t for an interest rate process rt=r, an intensity process λt=λ, and a stochastic volatility process yt=y, denoted by P(t,r,λ,y;T), are given by
(7)P(t,r,λ,y;T)=𝔼*[e-∫tT‍{rs+(1-R)λt}ds∣rt=r,λt=λ,yt=y],
and then, using the three-dimensional Feynman-Kac formula, we have the Kolmogorov PDE:
(8)∂P∂t+(θt-ar)∂P∂r+(θ~t-a~λ)∂P∂λ+1ϵ(m-y)∂P∂y+12f2(y)∂2P∂r2+12σ~2∂2P∂λ2+ν2ϵ∂2P∂y2+ρrλσ~f(y)∂2P∂r∂λ+ρryf(y)ν2ϵ∂2P∂r∂y+ρλyσ~ν2ϵ∂2P∂λ∂y-(r+(1-R)λ)P=0,
with the terminal condition P(t,r,λ,y;T)|t=T=1. Refer to Øksendal [26].


## 3. Asymptotic Analysis

In this section, we provide the main results of the paper. We present an asymptotic analysis to the solution of the PDE (8) and give an approximate solution of the defaultable zero-coupon bond price for ϵ.The PDE (8) is rewritten as follows:
(9)ℒP(t,r,λ,y;T)=0,t<T,P(t,r,λ,y;T)|t=T=1,
where
(10)ℒ:=1ϵℒ0+1ϵℒ1+ℒ2,ℒ0=(m-y)∂∂y+ν2∂2∂y2,ℒ1=2f(y)νρry∂2∂r∂y+2σ~νρλy∂2∂λ∂y,ℒ2=∂∂t+(θt-ar)∂∂r+(θ~t-a~λ)∂∂λ+12f2(y)∂2∂r2+12σ~2∂2∂λ2+σ~f(y)ρrλ∂2∂r∂λ-(r+(1-R)λ),
respectively. In particular, ℒ2 is the infinitesimal generator of the two-correlated-factor Hull-White model at current level y of the stochastic volatility process yt.Now, we wish to utilize an asymptotic analysis forP(t,r,λ,y;T) in terms of ϵ. We consider an expansion of P in the following form:
(11)P=P0+ϵP1+ϵP2+ϵϵP3+⋯.
Applying (11) to (9) leads to
(12)1ϵℒ0P0+1ϵ(ℒ0P1+ℒ1P0)+(ℒ0P2+ℒ1P1+ℒ2P0)+ϵ(ℒ0P3+ℒ1P2+ℒ2P1)+⋯=0.


### 3.1. The Leading Order Term

In this subsection, we derive the leading order termP0.Theorem 1.
One supposesP0 to have the affine representation
(13)P0(t,r,λ;T)=eA(t,T)-B(t,T)r-(1-R)C(t,T)λ,
with A(T,T)=B(T,T)=C(T,T)=0. Then, A(t,T),B(t,T) and C(t,T) are given by
(14)A(t,T)=lnP0(0,T)P0(0,t)-C(t,T)∂∂slnP0(0,s)|s=t-(B(t,T)-C(t,T))∂∂sln𝒵~(0,s)|s=t+σ^24a3{(e-aT-e-at)2-a2B2(t,T)}+ρrλσ~σ-aa~(a+a~)(1-R)×{(a+a~)(T-t)-(a~B(t,T)+1)(aC(t,T)+1)+1}+σ~24a~3(1-R)[(e-a~T-e-a~t)2-a~2C2(t,T)-R{2a~(T-t)-(a~C(t,T)+3)×(a~C(t,T)-1)-3}(e-a~T-e-a~t)2],B(t,T)=1a{1-e-a(T-t)},C(t,T)=1a~{1-e-a~(T-t)},
respectively, where σ- and σ^2 are given by
(15)σ-:=〈f〉=∫ℝ‍f(y)ϕ(y)dy,(16)σ^2:=〈f2〉=∫ℝ‍f2(y)ϕ(y)dy
with the probability density function ϕ(y)=(1/2πν2)exp{-(y-m)2/2ν2} for y∈ℝ. Here, 𝒵~(0,t) is the price at time 0 of a default-free zero-coupon maturing at time t under the Hull-White model [24].Proof.
Multiplying (12) by ϵ and then letting ϵ go to zero, we obtain the first two terms as follows:
(17)ℒ0P0=0,(18)ℒ0P1+ℒ1P0=0,
respectively. Since the infinitesimal operator ℒ0 is the generator of the OU process, the solution P0 of (17) must be a constant with respect to y variable, namely, P0=P0(t,r,λ;T). Similarly, from (18) we get ℒ1P0=0 since P0 does not rely on y variable. So, the solution P1 of (18) must be a constant with respect to y variable, namely, P1=P1(t,r,λ;T). Hence, the two terms P0 and P1 do not depend on the current level y of the stochastic volatility process yt. In this method, we can continue to eliminate the terms of order 1,ϵ,ϵ,…. For the order-1 term, we get ℒ0P2+ℒ1P1+ℒ2P0=0. This PDE becomes
(19)ℒ0P2+ℒ2P0=0,
since P1 does not rely on y variable. This PDE is a Poisson equation for P2 with respect to infinitesimal operator ℒ0. It is well known that it has a solution only if ℒ2P0 is centered with respect to the probability density function ϕ(y) of the Gaussian distribution with mean m and variance ν2, namely,
(20)〈ℒ2P0〉=〈ℒ2〉P0=0,
with the terminal condition P0(t,r,λ;T)|t=T=1, where 〈·〉, called by the solvability condition, denotes the expectation with respect to invariant distribution of yt. Here, 〈ℒ2〉 is an infinitesimal operator given by
(21)〈ℒ2〉=∂∂t+(θt-ar)∂∂r+(θ~t-a~λ)∂∂λ+12σ^2∂2∂r2+12σ~2∂2∂λ2+σ~σ-ρrλ∂2∂r∂λ-(r+(1-R)λ),
from (10), where σ- and σ^2 are defined by (15) and (16), respectively. Plugging (21) into (13), we have (14) by using the result of Tchuindjo [16].Note that in the stochastic volatility setting, we get the leading order termP0, which is a generalization of the constant volatility result obtained by Tchuindjo. If f(y) is a constant function, then our results (14) reduce to Tchuindjo's result.


### 3.2. The First Perturbation TermP~1

In this subsection, we derive the first perturbation termP1 by using the leading order term P0.Theorem 2.
One supposesP~1 to have the affine representation
(22)P~1(t,r,λ;T)=D(t,T)eA(t,T)-B(t,T)r-(1-R)C(t,T)λ,
with A(T,T)=B(T,T)=C(T,T)=D(T,T)=0. Then, A(t,T),B(t,T),C(t,T), and D(t,T) are given by (14) and
(23)D(t,T)=-∫tT‍[K1ϵ{(1-R)2B(s,T)C2(s,T)}+K2ϵ{(1-R)B2(s,T)C(s,T)}+K3ϵB3(s,T)]ds,
respectively.Proof.
The order-ϵ terms in (12) lead to ℒ0P3+ℒ1P2+ℒ2P1=0 which is a Poisson equation for P3 whose solvability condition is given by
(24)〈ℒ1P2+ℒ2P1〉=0.
From (19) and (20), we get
(25)P2=-ℒ0-1(ℒ2-〈ℒ2〉)P0+c(t,r,λ),
for some function c(t,r,λ). Plugging (25) into (24), we derive a PDE for P1 as follows:
(26)〈ℒ2〉P1=〈ℒ1ℒ0-1(ℒ2-〈ℒ2〉)〉P0,
with the terminal condition P1(t,r,λ;T)|t=T=0. Since we focus on the first perturbation term to P0, we reset (26) with respect to P~1:=ϵP1 as follows:
(27)〈ℒ2〉P~1=𝒜P0,
where 𝒜 is given by (1/α)〈ℒ1ℒ0-1(ℒ2-〈ℒ2〉)〉. From (10) and (21), we have
(28)ℒ2-〈ℒ2〉=12(f2(y)-〈f2〉)∂2∂r2+ρrλσ~(f(y)-〈f〉)∂2∂r∂λ.
Here, we introduce that the functions θ:ℝ→ℝ and ψ:ℝ→ℝ are given by the solutions of
(29)ℒ0θ=f2(y)-〈f2〉,ℒ0ψ=f(y)-〈f〉,
respectively, and hence, we get the operator 𝒜 denoted by
(30)𝒜=K1ϵ∂3∂r∂λ2+K2ϵ∂3∂r2∂λ+K3ϵ∂3∂r3,
where K1ϵ, K2ϵ, and K3ϵ are defined by
(31)K1ϵ=2νασ~2ρrλ〈ψ′〉,K2ϵ=ν2ασ~ρλy〈θ′〉+2νασ~ρrλρry〈fψ′〉,K3ϵ=ν2αρry〈fθ′〉,
respectively. Hence, we obtain the PDE (27) as follows:
(32)∂P~1∂t+(θt-ar)∂P~1∂r+(θ~t-a~λ)∂P~1∂λ+12σ^2∂2P~1∂λ2+12σ~2∂2P~1∂r2+σ-σ˘ρrλ∂2P~1∂r∂λ-(r+(1-R)λ)P~1=-[K1ϵ{B(t,T)(1-R)2C2(t,T)}+K2ϵ{B2(t,T)(1-R)C(t,T)}+K3ϵB3(t,T)]P0,
with the terminal condition P~1(t,r,λ;T)|t=T=0. So, plugging (22) into (32), we obtain the result of Theorem 2 by direct computation.Therefore, synthesizing Theorems1 and 2, we derive an asymptotic analysis of the defaultable zero-coupon bond prices at time t which is given by
(33)P(t,r,λ,y;T)≈(1+D(t,T))eA(t,T)-B(t,T)r-(1-R)C(t,T)λ,
where A(t,T),B(t,T),C(t,T), and D(t,T) are given by (14), and (23), respectively.


### 3.3. Numerical Results

In this subsection, we show effects of the first perturbation term with some sensitive analyses of the model parameters. Note that the main numerical implication of the asymptotic analysis is reduction of the parametric dependence of the price formula. Refer to Cotton et al. [23].We calculate the magnitude of mispricing with respect to defaultable zero-coupon bonds as a percentage of the face value of bond. Namely, we show that the defaultable zero-coupon bonds with stochastic volatility tend to be overpriced or underpriced in terms of parameters involved. The parameter values used to calculate area=0.2,a~=0.3,R=0.8,t=0, and T=1. Table 1 has three types as follows: the second row only has a stochastic volatility K1ϵ factor (K2ϵ=K3ϵ=0), the third row solely has a stochastic volatility K2ϵ factor (K1ϵ=K3ϵ=0), and the fourth row uniquely has a stochastic volatility K3ϵ factor (K1ϵ=K2ϵ=0). We can see from Table 1 that the mispricing of the defaultable zero-coupon bond moves in the same direction from −0.07 to 0.07 with respect to each of the stochastic volatility terms. We also study the mispricing of the defaultable zero-coupon bond with respect to recovery rate. The same coefficients are used as for Table 1 except for K1ϵ=0.01,K2ϵ=-0.03, and K3ϵ=-0.02. Table 2 shows that the mispricing of the defaultable zero-coupon bond monotonically decreases when the value of the recovery rate increases.Table 1
Effect of stochastic volatility terms.
Stochastic volatility value
−0.07
−0.05
−0.01
0
0.01
0.05
0.07
Mispricing as percentage of defaultablezero-coupon bond price with respect to K1ϵ
0.051
0.037
0.007
0
−0.007
−0.037
−0.051
Mispricing as percentage of defaultablezero-coupon bond price with respect to K2ϵ
0.265
0.190
0.038
0
−0.038
−0.191
−0.267
Mispricing as percentage of defaultablezero-coupon bond price with respect to K3ϵ
1.364
0.978
0.197
0
−0.002
−0.010
−1.402Table 2
Effect of recovery rate.
Recovery rate (R)
0
0.2
0.4
0.6
0.8
1
Mispricing as percentage ofdefaultable zero-coupon bond price
0.78
0.72
0.67
0.59
0.49
0.39We report the defaultable bond price and the yield curve caused by changes in the value of time to maturity (source: KIS Pricing’s). The parameter values used to calculate area=0.1,a~=0.2,σ=0.1,σ-(z)=0.16,σ˘2(z)=0.19,ρrλ=-0.1,K1ϵ=-0.06,K2ϵ=0.02,K3ϵ=-0.04,r=0.17,λ=0.15, and R=0.3 in Figures 1 and 2, respectively. Figures 1 and 2 have two cases of curve. Case 1 corresponds to the constant volatility of the interest rate process. Case 2 contains the stochastic volatility of the interest rate process. Figure 1 shows that the defaultable bond prices with the stochastic intensity becomes higher than the defaultable bond prices with the constant volatility as the time to maturity increase. Also, the hump-shaped yield curve that matches a yield curve for structural model as in the Merton [3] appears to Case 2 in Figure 2. Hence, the stochastic volatility affects both quantitative and qualitative effect.Figure 1
Defaultable zero-coupon bond prices.Figure 2
Yield curves.


## 3.1. The Leading Order Term

In this subsection, we derive the leading order termP0.Theorem 1.
One supposesP0 to have the affine representation
(13)P0(t,r,λ;T)=eA(t,T)-B(t,T)r-(1-R)C(t,T)λ,
with A(T,T)=B(T,T)=C(T,T)=0. Then, A(t,T),B(t,T) and C(t,T) are given by
(14)A(t,T)=lnP0(0,T)P0(0,t)-C(t,T)∂∂slnP0(0,s)|s=t-(B(t,T)-C(t,T))∂∂sln𝒵~(0,s)|s=t+σ^24a3{(e-aT-e-at)2-a2B2(t,T)}+ρrλσ~σ-aa~(a+a~)(1-R)×{(a+a~)(T-t)-(a~B(t,T)+1)(aC(t,T)+1)+1}+σ~24a~3(1-R)[(e-a~T-e-a~t)2-a~2C2(t,T)-R{2a~(T-t)-(a~C(t,T)+3)×(a~C(t,T)-1)-3}(e-a~T-e-a~t)2],B(t,T)=1a{1-e-a(T-t)},C(t,T)=1a~{1-e-a~(T-t)},
respectively, where σ- and σ^2 are given by
(15)σ-:=〈f〉=∫ℝ‍f(y)ϕ(y)dy,(16)σ^2:=〈f2〉=∫ℝ‍f2(y)ϕ(y)dy
with the probability density function ϕ(y)=(1/2πν2)exp{-(y-m)2/2ν2} for y∈ℝ. Here, 𝒵~(0,t) is the price at time 0 of a default-free zero-coupon maturing at time t under the Hull-White model [24].Proof.
Multiplying (12) by ϵ and then letting ϵ go to zero, we obtain the first two terms as follows:
(17)ℒ0P0=0,(18)ℒ0P1+ℒ1P0=0,
respectively. Since the infinitesimal operator ℒ0 is the generator of the OU process, the solution P0 of (17) must be a constant with respect to y variable, namely, P0=P0(t,r,λ;T). Similarly, from (18) we get ℒ1P0=0 since P0 does not rely on y variable. So, the solution P1 of (18) must be a constant with respect to y variable, namely, P1=P1(t,r,λ;T). Hence, the two terms P0 and P1 do not depend on the current level y of the stochastic volatility process yt. In this method, we can continue to eliminate the terms of order 1,ϵ,ϵ,…. For the order-1 term, we get ℒ0P2+ℒ1P1+ℒ2P0=0. This PDE becomes
(19)ℒ0P2+ℒ2P0=0,
since P1 does not rely on y variable. This PDE is a Poisson equation for P2 with respect to infinitesimal operator ℒ0. It is well known that it has a solution only if ℒ2P0 is centered with respect to the probability density function ϕ(y) of the Gaussian distribution with mean m and variance ν2, namely,
(20)〈ℒ2P0〉=〈ℒ2〉P0=0,
with the terminal condition P0(t,r,λ;T)|t=T=1, where 〈·〉, called by the solvability condition, denotes the expectation with respect to invariant distribution of yt. Here, 〈ℒ2〉 is an infinitesimal operator given by
(21)〈ℒ2〉=∂∂t+(θt-ar)∂∂r+(θ~t-a~λ)∂∂λ+12σ^2∂2∂r2+12σ~2∂2∂λ2+σ~σ-ρrλ∂2∂r∂λ-(r+(1-R)λ),
from (10), where σ- and σ^2 are defined by (15) and (16), respectively. Plugging (21) into (13), we have (14) by using the result of Tchuindjo [16].Note that in the stochastic volatility setting, we get the leading order termP0, which is a generalization of the constant volatility result obtained by Tchuindjo. If f(y) is a constant function, then our results (14) reduce to Tchuindjo's result.


## 3.2. The First Perturbation TermP~1

In this subsection, we derive the first perturbation termP1 by using the leading order term P0.Theorem 2.
One supposesP~1 to have the affine representation
(22)P~1(t,r,λ;T)=D(t,T)eA(t,T)-B(t,T)r-(1-R)C(t,T)λ,
with A(T,T)=B(T,T)=C(T,T)=D(T,T)=0. Then, A(t,T),B(t,T),C(t,T), and D(t,T) are given by (14) and
(23)D(t,T)=-∫tT‍[K1ϵ{(1-R)2B(s,T)C2(s,T)}+K2ϵ{(1-R)B2(s,T)C(s,T)}+K3ϵB3(s,T)]ds,
respectively.Proof.
The order-ϵ terms in (12) lead to ℒ0P3+ℒ1P2+ℒ2P1=0 which is a Poisson equation for P3 whose solvability condition is given by
(24)〈ℒ1P2+ℒ2P1〉=0.
From (19) and (20), we get
(25)P2=-ℒ0-1(ℒ2-〈ℒ2〉)P0+c(t,r,λ),
for some function c(t,r,λ). Plugging (25) into (24), we derive a PDE for P1 as follows:
(26)〈ℒ2〉P1=〈ℒ1ℒ0-1(ℒ2-〈ℒ2〉)〉P0,
with the terminal condition P1(t,r,λ;T)|t=T=0. Since we focus on the first perturbation term to P0, we reset (26) with respect to P~1:=ϵP1 as follows:
(27)〈ℒ2〉P~1=𝒜P0,
where 𝒜 is given by (1/α)〈ℒ1ℒ0-1(ℒ2-〈ℒ2〉)〉. From (10) and (21), we have
(28)ℒ2-〈ℒ2〉=12(f2(y)-〈f2〉)∂2∂r2+ρrλσ~(f(y)-〈f〉)∂2∂r∂λ.
Here, we introduce that the functions θ:ℝ→ℝ and ψ:ℝ→ℝ are given by the solutions of
(29)ℒ0θ=f2(y)-〈f2〉,ℒ0ψ=f(y)-〈f〉,
respectively, and hence, we get the operator 𝒜 denoted by
(30)𝒜=K1ϵ∂3∂r∂λ2+K2ϵ∂3∂r2∂λ+K3ϵ∂3∂r3,
where K1ϵ, K2ϵ, and K3ϵ are defined by
(31)K1ϵ=2νασ~2ρrλ〈ψ′〉,K2ϵ=ν2ασ~ρλy〈θ′〉+2νασ~ρrλρry〈fψ′〉,K3ϵ=ν2αρry〈fθ′〉,
respectively. Hence, we obtain the PDE (27) as follows:
(32)∂P~1∂t+(θt-ar)∂P~1∂r+(θ~t-a~λ)∂P~1∂λ+12σ^2∂2P~1∂λ2+12σ~2∂2P~1∂r2+σ-σ˘ρrλ∂2P~1∂r∂λ-(r+(1-R)λ)P~1=-[K1ϵ{B(t,T)(1-R)2C2(t,T)}+K2ϵ{B2(t,T)(1-R)C(t,T)}+K3ϵB3(t,T)]P0,
with the terminal condition P~1(t,r,λ;T)|t=T=0. So, plugging (22) into (32), we obtain the result of Theorem 2 by direct computation.Therefore, synthesizing Theorems1 and 2, we derive an asymptotic analysis of the defaultable zero-coupon bond prices at time t which is given by
(33)P(t,r,λ,y;T)≈(1+D(t,T))eA(t,T)-B(t,T)r-(1-R)C(t,T)λ,
where A(t,T),B(t,T),C(t,T), and D(t,T) are given by (14), and (23), respectively.


## 3.3. Numerical Results

In this subsection, we show effects of the first perturbation term with some sensitive analyses of the model parameters. Note that the main numerical implication of the asymptotic analysis is reduction of the parametric dependence of the price formula. Refer to Cotton et al. [23].We calculate the magnitude of mispricing with respect to defaultable zero-coupon bonds as a percentage of the face value of bond. Namely, we show that the defaultable zero-coupon bonds with stochastic volatility tend to be overpriced or underpriced in terms of parameters involved. The parameter values used to calculate area=0.2,a~=0.3,R=0.8,t=0, and T=1. Table 1 has three types as follows: the second row only has a stochastic volatility K1ϵ factor (K2ϵ=K3ϵ=0), the third row solely has a stochastic volatility K2ϵ factor (K1ϵ=K3ϵ=0), and the fourth row uniquely has a stochastic volatility K3ϵ factor (K1ϵ=K2ϵ=0). We can see from Table 1 that the mispricing of the defaultable zero-coupon bond moves in the same direction from −0.07 to 0.07 with respect to each of the stochastic volatility terms. We also study the mispricing of the defaultable zero-coupon bond with respect to recovery rate. The same coefficients are used as for Table 1 except for K1ϵ=0.01,K2ϵ=-0.03, and K3ϵ=-0.02. Table 2 shows that the mispricing of the defaultable zero-coupon bond monotonically decreases when the value of the recovery rate increases.Table 1
Effect of stochastic volatility terms.
Stochastic volatility value
−0.07
−0.05
−0.01
0
0.01
0.05
0.07
Mispricing as percentage of defaultablezero-coupon bond price with respect to K1ϵ
0.051
0.037
0.007
0
−0.007
−0.037
−0.051
Mispricing as percentage of defaultablezero-coupon bond price with respect to K2ϵ
0.265
0.190
0.038
0
−0.038
−0.191
−0.267
Mispricing as percentage of defaultablezero-coupon bond price with respect to K3ϵ
1.364
0.978
0.197
0
−0.002
−0.010
−1.402Table 2
Effect of recovery rate.
Recovery rate (R)
0
0.2
0.4
0.6
0.8
1
Mispricing as percentage ofdefaultable zero-coupon bond price
0.78
0.72
0.67
0.59
0.49
0.39We report the defaultable bond price and the yield curve caused by changes in the value of time to maturity (source: KIS Pricing’s). The parameter values used to calculate area=0.1,a~=0.2,σ=0.1,σ-(z)=0.16,σ˘2(z)=0.19,ρrλ=-0.1,K1ϵ=-0.06,K2ϵ=0.02,K3ϵ=-0.04,r=0.17,λ=0.15, and R=0.3 in Figures 1 and 2, respectively. Figures 1 and 2 have two cases of curve. Case 1 corresponds to the constant volatility of the interest rate process. Case 2 contains the stochastic volatility of the interest rate process. Figure 1 shows that the defaultable bond prices with the stochastic intensity becomes higher than the defaultable bond prices with the constant volatility as the time to maturity increase. Also, the hump-shaped yield curve that matches a yield curve for structural model as in the Merton [3] appears to Case 2 in Figure 2. Hence, the stochastic volatility affects both quantitative and qualitative effect.Figure 1
Defaultable zero-coupon bond prices.Figure 2
Yield curves.


## 4. Pricing Credit Default Swaps and Bond Options

In this section, we get formulas for the CDS and the bond option by using the results of Section3.


### 4.1. Pricing Credit Default Swap

A CDS is a bilateral contract in which one party (the protection buyer) pays a periodic, fixed premium to another (the protection seller) for protection related to credit events on an underlying reference entity. If a credit event occurs, the protection seller is obliged to make a payment to the protection buyer in order to compensate him for any losses that he might otherwise incur. Thus, the credit risk of the reference entity is transferred from the protection buyer to the protection seller. In particular, Ma and Kim [22] first study the problem of default correlation when the reference entity and the protection seller can occur default at the same time.For simplicity, we suppose the following.(i)
We consider a forward CDS rate, valuable after some initial timet0 with 0≤t0<t1, for using the results of Section 3.(ii)
LetM be the time-to-maturity of a forward CDS contract, t1<⋯<tN=M the premium payment dates, and 𝒯 the payment tenor.(iii)
The bond coupon dates match the payment dates of the CDS.(iv)
If a credit event happens, then the settlement takes place at coupon date following default, but we do not consider the accrued premium payment.The premium leg is the series of payments of the forward CDS rate until maturity or until the first default timeτ. Let the price C(t,t0;𝒯) of the forward CDS rate and the price Cpb(t,t0;𝒯) of the protection buyer paying 1(τ>t) at time t is given by
(34)Cpb(t,t0;𝒯)=C(t,t0;𝒯)𝔼*∑n=1N‍[e-∫ttn‍(rs+λs)ds∣rt=r,λt=λ,yt=y].
Note that we can easily solve (34) by using (33) when R=0 (zero recovery).On the other hand, let the priceCps(t,t0;𝒯) of the protection seller at the default time τ is given by
(35)Cps(t,t0;𝒯)=(1-R)𝔼*[∫t0tN‍e-∫tu‍(rs+λs)dsλudu∣rt=r,λt=λ,yt=y],
with t<τ. Note that the protection seller payment is zero on {t≥τ}.Finally, plugging (34) and (35), we obtain the forward CDS rate C(t,t0;𝒯) as follows:
(36)C(t,t0;𝒯)=(1-R)×(𝔼*[∫t0tN‍e-∫tu‍(rs+λs)dsλudu∣rt=r,λt=λ,yt=y])×(∑n=1N‍𝔼*[e-∫ttn‍(rs+λs)ds∣rt=r,λt=λ,yt=y])-1,
with t<τ. Note that the spread of a CDS rate is given by the spread of a forward CDS when t0=t. Refer to Ma and Kim [22].Now, we will calculate (35) by using the asymptotic analysis. For the protection seller payment, we put
(37)C˘(t,r,λ,y;T)=𝔼*[∫tT‍e-∫tu‍(rs+λs)dsλudu∣rt=r,λt=λ,yt=y],Cps(t,t0;𝒯)=C˘(t,r,λ,y;tN)-C˘(t,r,λ,y;t0).Using the three-dimensional Feynman-Kac formula we have the following Kolmogorov PDE:(38)ℒ˘C˘(t,r,λ,y;T)=-λ,t<T,C˘(t,r,λ,y;T)|t=T=0,
where
(39)ℒ˘:=1ϵℒ0+1ϵℒ1+ℒ˘2,ℒ˘2=∂∂t+(θt-ar)∂∂r+(θ~t-a~λ)∂∂λ+12σ2∂2∂r2+12f2(y)∂2∂λ2+σf(y)ρrλ∂2∂r∂λ-(r+λ),ℒ0 and ℒ1 are defined by in Section 3, respectively. ℒ˘2 corresponds to ℒ2 with R=0 (zero recovery).We consider an expansion ofC˘ in the following form:
(40)C˘=C˘0+ϵC˘1+ϵC˘2+ϵϵC˘3+⋯.
Applying (40) to (38) leads to
(41)1ϵℒ0C˘0+1ϵ(ℒ0C˘1+ℒ1C˘0)+(ℒ0C˘2+ℒ1C˘1+ℒ˘2C˘0)+ϵ(ℒ0C˘3+ℒ1C˘2+ℒ˘2C˘1)+⋯=0.Through similar processes in Section3, we will calculate the leading order term C˘0 and the first perturbation term C˘1. Namely, we let
(42)〈ℒ˘2〉C˘0=-λ,
with the terminal condition C˘0(t,r,λ;T)|t=T=0 and
(43)〈ℒ˘2〉C-1=𝒜˘C˘0,
with the terminal condition C-1(t,r,λ;T)|t=T=0, where C-1:=ϵC˘1 and 𝒜˘:=(1/α)〈ℒ1ℒ0-1(ℒ˘2-〈ℒ˘2〉)〉.LetP0(t,r,λ;T) be equal to the leading order term P0(t,r,λ;T) in (13) of the approximate solution of a defaultable zero-coupon bond for zero recovery, that is,
(44)P0(t,r,λ;T)=eA(t,T)-B(t,T)r-C(t,T)λ,
with A(T,T)=B(T,T)=C(T,T)=0, where A(t,T),B(t,T) and C(t,T) are given by (14).Hence, the solutions of (42) and (43) are
(45)C˘0(t,r,λ;T)=λ∫tT‍P0(s,r,λ;T)ds,C-1(t,r,λ;T)=λ∫tT‍∫sT‍[K1ϵB(s,h)C2(s,h)+K2ϵB2(s,h)C(s,h)+K3ϵC3(s,h)]P0(s,r,λ;h)dhds,
respectively.Therefore, synthesizing (45), we derive an asymptotic analysis of the protection seller at time t which is given by
(46)C˘(t,r,λ,y;T)≈C˘0(t,r,λ;T)+C-1(t,r,λ;T).


### 4.2. Bond Option Pricing

In this subsection, we obtain an asymptotic option pricing formula when the underlying asset itself is a defaultable zero-coupon bond.LetT0 be the maturity of the option and T be the maturity of the defaultable zero-coupon bond with T0<T. We assume that the option becomes invalid when a default occurs prior to T0 and that the defaultable zero-coupon bond has the fractional recovery R as before. The option price at time t for an interest rate process rt=r, an intensity process λt=λ, and a stochastic volatility process yt=y, denoted by U(t,r,λ,y;T0,T), is given by
(47)U(t,r,λ,y;T0,T)=E*[e-∫tT0‍rsdsl(ℬ(T0,T))∣rt=r,λt=λ,yt=y],
under the martingale measure P*, where the bond price ℬ(T0,T) is
(48)ℬ(T0,T):=P(T0,rT0,λT0,yT0;T)=E*[e-∫T0T‍(rs+(1-R)λs)ds∣rT0,λT0,yT0],
and l(ℬ(T0,T)) is the payoff function of the option at time T0. It is assumed that the payoff function h is at the best linearly growing at infinity and is a smooth function. This smoothness assumption may be too severe in practical point of view as h is not differentiable at the exercise price in the classical European call or put option case. In fact, the smoothness assumption on h can be removed as shown in Fouque et al. [27]. We take, however, the smoothness assumption for the simplicity of our argument here.The three-dimensional Feynman-Kac formula corresponding to the price functionU is the same as the one for the defaultable zero-coupon bond in (9) but with a terminal condition, namely,
(49)ℒU(t,r,λ,y;T0)=0,t<T0,U(T0,rT0,λT0,yT0;T)=l(P(T0,rT0,λT0,yT0;T)),
where ℒ is defined in (9). We take the expansions
(50)U=U0+ϵU1+ϵU2+⋯,U(T0,rT0,λT0,yT0;T)=l(P0(T0,rT0,λT0;T))+P~1(T0,rT0,λT0;T)l′(P0(T0,rT0,λT0;T))+⋯,
where P0 and P~1 are given by Theorems 1 and 2, respectively.Using the similar argument as in Section3, the terms of order 1/ϵ and 1/ϵ in the asymptotic PDE of (9) provide the y-independence of U0 and U1.The order-1 terms give a Poisson equation in U2 from which the solvability condition 〈ℒ2〉U0=0 is satisfied. From (49) the corresponding terminal condition is given by U0|t=T0=l(P0(T0,rT0,λT0;T)). Then, as calculated in Section 3.1, we have
(51)U0(t,r,λ;T0,T)=E*[e-∫tT0‍rsdsl(P0(T0,rT0,λT0;T))∣rt=r,λt=λ],
where P0(T0,rT0,λT0;T) is given by Theorem 1 at time t=T0.The order-ϵ terms lead to a Poisson equation in U3 with the solvability condition 〈ℒ1U2+ℒ2U1〉=0. If we put U~1:=ϵU1, then this solvability condition leads to the following PDE:
(52)〈ℒ2〉U~1(t,r,λ;T0,T)=𝒜U0(t,r,λ;T0,T),
with the terminal condition given by
(53)U~1(T0,rT0,λT0;T0,T)=P~1(T0,rT0,λT0;T)l′(P0(T0,rT0,λT0;T)),
from (50), where the operator 𝒜 and P~1 are given by (30) and Theorem 2, respectively. Then, by the Feynman-Kac formula applied to (52) and (53), we obtain the following probabilistic representation:(54)U~1(t,r,λ;T0,T)=E*[e-∫tT0‍rsdsP~1(T0,rT0,λT0;T)l′(P0(T0,rT0,λT0;T))-∫tT0‍e-∫tu‍rsds𝒜U0(u,r,λ;T0,T)du∣rt=r,λt=λ].Synthesizing the above results, we obtain an asymptotic expansion of the defaultable zero-coupon bond option price as follows:(55)U(t,r,λ,y;T0,T)≈U0(t,r,λ;T0,T)+U~1(t,r,λ;T0,T),
where U0 and U~1 are given by (51) and (54), respectively.


## 4.1. Pricing Credit Default Swap

A CDS is a bilateral contract in which one party (the protection buyer) pays a periodic, fixed premium to another (the protection seller) for protection related to credit events on an underlying reference entity. If a credit event occurs, the protection seller is obliged to make a payment to the protection buyer in order to compensate him for any losses that he might otherwise incur. Thus, the credit risk of the reference entity is transferred from the protection buyer to the protection seller. In particular, Ma and Kim [22] first study the problem of default correlation when the reference entity and the protection seller can occur default at the same time.For simplicity, we suppose the following.(i)
We consider a forward CDS rate, valuable after some initial timet0 with 0≤t0<t1, for using the results of Section 3.(ii)
LetM be the time-to-maturity of a forward CDS contract, t1<⋯<tN=M the premium payment dates, and 𝒯 the payment tenor.(iii)
The bond coupon dates match the payment dates of the CDS.(iv)
If a credit event happens, then the settlement takes place at coupon date following default, but we do not consider the accrued premium payment.The premium leg is the series of payments of the forward CDS rate until maturity or until the first default timeτ. Let the price C(t,t0;𝒯) of the forward CDS rate and the price Cpb(t,t0;𝒯) of the protection buyer paying 1(τ>t) at time t is given by
(34)Cpb(t,t0;𝒯)=C(t,t0;𝒯)𝔼*∑n=1N‍[e-∫ttn‍(rs+λs)ds∣rt=r,λt=λ,yt=y].
Note that we can easily solve (34) by using (33) when R=0 (zero recovery).On the other hand, let the priceCps(t,t0;𝒯) of the protection seller at the default time τ is given by
(35)Cps(t,t0;𝒯)=(1-R)𝔼*[∫t0tN‍e-∫tu‍(rs+λs)dsλudu∣rt=r,λt=λ,yt=y],
with t<τ. Note that the protection seller payment is zero on {t≥τ}.Finally, plugging (34) and (35), we obtain the forward CDS rate C(t,t0;𝒯) as follows:
(36)C(t,t0;𝒯)=(1-R)×(𝔼*[∫t0tN‍e-∫tu‍(rs+λs)dsλudu∣rt=r,λt=λ,yt=y])×(∑n=1N‍𝔼*[e-∫ttn‍(rs+λs)ds∣rt=r,λt=λ,yt=y])-1,
with t<τ. Note that the spread of a CDS rate is given by the spread of a forward CDS when t0=t. Refer to Ma and Kim [22].Now, we will calculate (35) by using the asymptotic analysis. For the protection seller payment, we put
(37)C˘(t,r,λ,y;T)=𝔼*[∫tT‍e-∫tu‍(rs+λs)dsλudu∣rt=r,λt=λ,yt=y],Cps(t,t0;𝒯)=C˘(t,r,λ,y;tN)-C˘(t,r,λ,y;t0).Using the three-dimensional Feynman-Kac formula we have the following Kolmogorov PDE:(38)ℒ˘C˘(t,r,λ,y;T)=-λ,t<T,C˘(t,r,λ,y;T)|t=T=0,
where
(39)ℒ˘:=1ϵℒ0+1ϵℒ1+ℒ˘2,ℒ˘2=∂∂t+(θt-ar)∂∂r+(θ~t-a~λ)∂∂λ+12σ2∂2∂r2+12f2(y)∂2∂λ2+σf(y)ρrλ∂2∂r∂λ-(r+λ),ℒ0 and ℒ1 are defined by in Section 3, respectively. ℒ˘2 corresponds to ℒ2 with R=0 (zero recovery).We consider an expansion ofC˘ in the following form:
(40)C˘=C˘0+ϵC˘1+ϵC˘2+ϵϵC˘3+⋯.
Applying (40) to (38) leads to
(41)1ϵℒ0C˘0+1ϵ(ℒ0C˘1+ℒ1C˘0)+(ℒ0C˘2+ℒ1C˘1+ℒ˘2C˘0)+ϵ(ℒ0C˘3+ℒ1C˘2+ℒ˘2C˘1)+⋯=0.Through similar processes in Section3, we will calculate the leading order term C˘0 and the first perturbation term C˘1. Namely, we let
(42)〈ℒ˘2〉C˘0=-λ,
with the terminal condition C˘0(t,r,λ;T)|t=T=0 and
(43)〈ℒ˘2〉C-1=𝒜˘C˘0,
with the terminal condition C-1(t,r,λ;T)|t=T=0, where C-1:=ϵC˘1 and 𝒜˘:=(1/α)〈ℒ1ℒ0-1(ℒ˘2-〈ℒ˘2〉)〉.LetP0(t,r,λ;T) be equal to the leading order term P0(t,r,λ;T) in (13) of the approximate solution of a defaultable zero-coupon bond for zero recovery, that is,
(44)P0(t,r,λ;T)=eA(t,T)-B(t,T)r-C(t,T)λ,
with A(T,T)=B(T,T)=C(T,T)=0, where A(t,T),B(t,T) and C(t,T) are given by (14).Hence, the solutions of (42) and (43) are
(45)C˘0(t,r,λ;T)=λ∫tT‍P0(s,r,λ;T)ds,C-1(t,r,λ;T)=λ∫tT‍∫sT‍[K1ϵB(s,h)C2(s,h)+K2ϵB2(s,h)C(s,h)+K3ϵC3(s,h)]P0(s,r,λ;h)dhds,
respectively.Therefore, synthesizing (45), we derive an asymptotic analysis of the protection seller at time t which is given by
(46)C˘(t,r,λ,y;T)≈C˘0(t,r,λ;T)+C-1(t,r,λ;T).


## 4.2. Bond Option Pricing

In this subsection, we obtain an asymptotic option pricing formula when the underlying asset itself is a defaultable zero-coupon bond.LetT0 be the maturity of the option and T be the maturity of the defaultable zero-coupon bond with T0<T. We assume that the option becomes invalid when a default occurs prior to T0 and that the defaultable zero-coupon bond has the fractional recovery R as before. The option price at time t for an interest rate process rt=r, an intensity process λt=λ, and a stochastic volatility process yt=y, denoted by U(t,r,λ,y;T0,T), is given by
(47)U(t,r,λ,y;T0,T)=E*[e-∫tT0‍rsdsl(ℬ(T0,T))∣rt=r,λt=λ,yt=y],
under the martingale measure P*, where the bond price ℬ(T0,T) is
(48)ℬ(T0,T):=P(T0,rT0,λT0,yT0;T)=E*[e-∫T0T‍(rs+(1-R)λs)ds∣rT0,λT0,yT0],
and l(ℬ(T0,T)) is the payoff function of the option at time T0. It is assumed that the payoff function h is at the best linearly growing at infinity and is a smooth function. This smoothness assumption may be too severe in practical point of view as h is not differentiable at the exercise price in the classical European call or put option case. In fact, the smoothness assumption on h can be removed as shown in Fouque et al. [27]. We take, however, the smoothness assumption for the simplicity of our argument here.The three-dimensional Feynman-Kac formula corresponding to the price functionU is the same as the one for the defaultable zero-coupon bond in (9) but with a terminal condition, namely,
(49)ℒU(t,r,λ,y;T0)=0,t<T0,U(T0,rT0,λT0,yT0;T)=l(P(T0,rT0,λT0,yT0;T)),
where ℒ is defined in (9). We take the expansions
(50)U=U0+ϵU1+ϵU2+⋯,U(T0,rT0,λT0,yT0;T)=l(P0(T0,rT0,λT0;T))+P~1(T0,rT0,λT0;T)l′(P0(T0,rT0,λT0;T))+⋯,
where P0 and P~1 are given by Theorems 1 and 2, respectively.Using the similar argument as in Section3, the terms of order 1/ϵ and 1/ϵ in the asymptotic PDE of (9) provide the y-independence of U0 and U1.The order-1 terms give a Poisson equation in U2 from which the solvability condition 〈ℒ2〉U0=0 is satisfied. From (49) the corresponding terminal condition is given by U0|t=T0=l(P0(T0,rT0,λT0;T)). Then, as calculated in Section 3.1, we have
(51)U0(t,r,λ;T0,T)=E*[e-∫tT0‍rsdsl(P0(T0,rT0,λT0;T))∣rt=r,λt=λ],
where P0(T0,rT0,λT0;T) is given by Theorem 1 at time t=T0.The order-ϵ terms lead to a Poisson equation in U3 with the solvability condition 〈ℒ1U2+ℒ2U1〉=0. If we put U~1:=ϵU1, then this solvability condition leads to the following PDE:
(52)〈ℒ2〉U~1(t,r,λ;T0,T)=𝒜U0(t,r,λ;T0,T),
with the terminal condition given by
(53)U~1(T0,rT0,λT0;T0,T)=P~1(T0,rT0,λT0;T)l′(P0(T0,rT0,λT0;T)),
from (50), where the operator 𝒜 and P~1 are given by (30) and Theorem 2, respectively. Then, by the Feynman-Kac formula applied to (52) and (53), we obtain the following probabilistic representation:(54)U~1(t,r,λ;T0,T)=E*[e-∫tT0‍rsdsP~1(T0,rT0,λT0;T)l′(P0(T0,rT0,λT0;T))-∫tT0‍e-∫tu‍rsds𝒜U0(u,r,λ;T0,T)du∣rt=r,λt=λ].Synthesizing the above results, we obtain an asymptotic expansion of the defaultable zero-coupon bond option price as follows:(55)U(t,r,λ,y;T0,T)≈U0(t,r,λ;T0,T)+U~1(t,r,λ;T0,T),
where U0 and U~1 are given by (51) and (54), respectively.


## 5. Final Remarks

In this paper, we consider the intensity-based defaultable bonds of two-correlated-factor Hull-White model with stochastic volatility of interest rate process. Using the asymptotic analysis, developed by Fouque et al. [25], we get approximate solutions to price the defaultable zero-coupon bonds as well as the credit default swaps and the bond options. It shows how these small perturbations can affect the shape of the yield curve of the defaultable zero-coupon bond. For further study, one can apply the framework of this paper to the multiname intensity models studied by Ma and Kim [28].



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