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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 567340, 9 pages
Asymptotic Analysis for One-Name Credit Derivatives
1Department of Applied Mathematics, Kongju National University, Chungcheongnam-Do 314-701, Republic of Korea
2Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea
Received 1 March 2013; Accepted 16 May 2013
Academic Editor: Carlos Vazquez
Copyright © 2013 Yong-Ki Ma and Beom Jin Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  and Bielecki and Rutkowski  for general references): the structural approach such as those developed by Merton , Longstaff and Schwartz , Leland and Toft , Zhou , Duffie and Lando , Hilberink and Rogers , and Giesecke  and the intensity-based approach such as those developed by Jarrow and Turnbull , Madan and Unal , Lando , Duffie and Singleton , and Collin-Dufresne and Goldstein . 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 .
Since the initial contribution to the intensity-based approach is given by Jarrow and Turnbull , who consider a constant and a deterministic Poisson intensity, there has been many mathematical studies on credit risk. Lando  uses that the default process is described by a Cox process. Schönbucher  develops the term structure model of defaultable interest rates by using the Heath-Jarrow-Morton model, and Tchuindjo  studies the price of a defaultable zero-coupon bond with two-correlated-factor Hull-White model. Duffee  and Driessen  perform the estimation of an intensity-based model using a Kalman filter approach, and Bayraktar and Yang  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  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  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  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  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  (see Figure 2). Cotton et al.  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 Section 2, 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  as follows: under a risk-neutral probability , where is a current time, is an expiry time, is a recovery rate with , and is a filtration generated by the joint process of interest rate process and intensity process . We need to work out the expectation over the possible paths of and 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 . Under a risk-neutral probability , we adopt the Tchuindjo type of stochastic differential equations (SDEs), for such models, where , and are constants, and are time-varying deterministic functions, and and 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 .
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 constant in the SDE (2) by a stochastic process given by a smooth, bounded below and above, and strictly positive function of an Ornstein-Uhlenbeck (OU) process as follows (see Fouque et al. ): where is a rate of meanreversion of volatility, is a volatility of volatility, and is a Brownian motion in a risk-neutral probability . Note that the OU process is an ergodic process with a unique invariant distribution, which is a Gaussian distribution with mean and variance . For the asymptotic analysis, driving the volatility is large, and we are interested in approximation in the limit with remaining constant. In this OU model, therefore, we set where is a small and strictly positive parameter. So, in the risk-neutral probability the joint process is given by the following SDEs: where the standard Brownian motions , , and are dependent of each other such that the correlation structure is given by
The defaultable zero-coupon bond prices with a fractional recovery rate at time for an interest rate process , an intensity process , and a stochastic volatility process , denoted by , are given by and then, using the three-dimensional Feynman-Kac formula, we have the Kolmogorov PDE: with the terminal condition . Refer to Øksendal .
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 .
The PDE (8) is rewritten as follows: where respectively. In particular, is the infinitesimal generator of the two-correlated-factor Hull-White model at current level of the stochastic volatility process .
3.1. The Leading Order Term
In this subsection, we derive the leading order term .
Theorem 1. One supposes to have the affine representation with . Then, and are given by respectively, where and are given by with the probability density function for . Here, is the price at time of a default-free zero-coupon maturing at time under the Hull-White model .
Proof. Multiplying (12) by and then letting go to zero, we obtain the first two terms as follows: respectively. Since the infinitesimal operator is the generator of the OU process, the solution of (17) must be a constant with respect to variable, namely, . Similarly, from (18) we get since does not rely on variable. So, the solution of (18) must be a constant with respect to variable, namely, . Hence, the two terms and do not depend on the current level of the stochastic volatility process . In this method, we can continue to eliminate the terms of order . For the order- term, we get . This PDE becomes since does not rely on variable. This PDE is a Poisson equation for with respect to infinitesimal operator . It is well known that it has a solution only if is centered with respect to the probability density function of the Gaussian distribution with mean and variance , namely, with the terminal condition , where , called by the solvability condition, denotes the expectation with respect to invariant distribution of . Here, is an infinitesimal operator given by from (10), where and are defined by (15) and (16), respectively. Plugging (21) into (13), we have (14) by using the result of Tchuindjo .
Note that in the stochastic volatility setting, we get the leading order term , which is a generalization of the constant volatility result obtained by Tchuindjo. If is a constant function, then our results (14) reduce to Tchuindjo's result.
3.2. The First Perturbation Term
In this subsection, we derive the first perturbation term by using the leading order term .
Theorem 2. One supposes to have the affine representation with . Then, , and are given by (14) and respectively.
Proof. The order- terms in (12) lead to which is a Poisson equation for whose solvability condition is given by From (19) and (20), we get for some function . Plugging (25) into (24), we derive a PDE for as follows: with the terminal condition . Since we focus on the first perturbation term to , we reset (26) with respect to as follows: where is given by . From (10) and (21), we have Here, we introduce that the functions and are given by the solutions of respectively, and hence, we get the operator denoted by where , , and are defined by respectively. Hence, we obtain the PDE (27) as follows: with the terminal condition . So, plugging (22) into (32), we obtain the result of Theorem 2 by direct computation.
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. .
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 are , and . Table 1 has three types as follows: the second row only has a stochastic volatility factor , the third row solely has a stochastic volatility factor , and the fourth row uniquely has a stochastic volatility factor . 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 , and . Table 2 shows that the mispricing of the defaultable zero-coupon bond monotonically decreases when the value of the recovery rate increases.
We 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 are , and 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  appears to Case 2 in Figure 2. Hence, the stochastic volatility affects both quantitative and qualitative effect.
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 Section 3.
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  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 time with , for using the results of Section 3.(ii)Let be the time-to-maturity of a forward CDS contract, the premium payment dates, and 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 time . Let the price of the forward CDS rate and the price of the protection buyer paying at time is given by Note that we can easily solve (34) by using (33) when (zero recovery).
On the other hand, let the price of the protection seller at the default time is given by with . Note that the protection seller payment is zero on .
Now, we will calculate (35) by using the asymptotic analysis. For the protection seller payment, we put
Using the three-dimensional Feynman-Kac formula we have the following Kolmogorov PDE: where and are defined by in Section 3, respectively. corresponds to with (zero recovery).
Through similar processes in Section 3, we will calculate the leading order term and the first perturbation term . Namely, we let with the terminal condition and with the terminal condition , where and .
Therefore, synthesizing (45), we derive an asymptotic analysis of the protection seller at time which is given by
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.
Let be the maturity of the option and be the maturity of the defaultable zero-coupon bond with . We assume that the option becomes invalid when a default occurs prior to and that the defaultable zero-coupon bond has the fractional recovery as before. The option price at time for an interest rate process , an intensity process , and a stochastic volatility process , denoted by , is given by under the martingale measure , where the bond price is and is the payoff function of the option at time . It is assumed that the payoff function 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 is not differentiable at the exercise price in the classical European call or put option case. In fact, the smoothness assumption on can be removed as shown in Fouque et al. . We take, however, the smoothness assumption for the simplicity of our argument here.
The three-dimensional Feynman-Kac formula corresponding to the price function is the same as the one for the defaultable zero-coupon bond in (9) but with a terminal condition, namely, where is defined in (9). We take the expansions where and are given by Theorems 1 and 2, respectively.
The order- terms give a Poisson equation in from which the solvability condition is satisfied. From (49) the corresponding terminal condition is given by . Then, as calculated in Section 3.1, we have where is given by Theorem 1 at time .
The order- terms lead to a Poisson equation in with the solvability condition . If we put , then this solvability condition leads to the following PDE: with the terminal condition given by from (50), where the operator and 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:
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. , 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 .
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