Credit Risky Securities Valuation under a Contagion Model with Interacting Intensities
We study a three-firm contagion model with counterparty risk and apply this model to price defaultable bonds and credit default swap (CDS). This model assumes that default intensities are driven by external common factors as well as other defaults in the system. Using the “total hazard” approach, default times can be generated and the joint density function is obtained. We represent the pricing method of defaultable bonds and obtain the closed-form pricing formulas. By the approach of “change of measure,” analytical solutions of CDS swap rate (swap premuim) are derived in the continuous time framework and the discrete time framework, respectively.
The corporate bonds and their credit derivatives are typical financial tools in the markets which undertake and avoid the credit risk of the companies. There are two basic approaches to modeling the pricing of defaultable securities: the value-of-the-firm (or structural) approach and the intensity-based approach. The structural model is based on the work of Merton , Black and Cox , and Geske : the default occurs when the firm assets are insufficient to meet payments on debt or the value of the firm asset falls below a prespecified level.
Nevertheless, the value of the firm assets is not observable, which brings difficulties to the pricing of credit derivatives. Reduced-form approach for credit risks avoids the disadvantage of structural approach which models the firm's value directly. They use risk-neutral pricing principle of contingent claims and take the time of default or other credit events as an exogenous random variable.
Reduced-form models are developed by Artzner and Delbaen , Duffie et al. , Jarrow and Turnbull , and Madan and Unal . Duffie and Lando  show that a reduced-form model can be obtained from a structural model with incomplete accounting information. The simplest type of reduced-form model is that the default time or the credit migration is the first jump of an exogenously given jump process with an intensity. In Jarrow et al. , the intensity for credit migration is constant; see also Litterman and Iben  for a Markov chain model of credit migration. In the papers by Duffie et al. , Duffie and Singleton , and Lando , the intensity of default is a random process. The common feature of the reduced-form models is that default cannot be predicted and can occur at any time. Therefore, reduced-form models have been used to price a wide variety of instruments. In recent years, some papers on estimating the parameters of these models are Collin-Dufresne and Solnik  and Duffee . Jarrow and Yu  set up a reduced-form model in which estimation can be based on bond prices as well as credit default swap prices. A systematic development of mathematical tools for reduced-form models has been given by Elliott et al. , and Jamshidian  develops change of numeraire methodology for reduced-form models.
In this paper, we mainly discuss the pricing of the defaultable zero-coupon bonds and CDS based on the intensity model with correlated default. The structure of this paper is organized as follows: in Section 2, we give the basic setup and the three-firm contagion model with an interaction term, comparing it with the model in Leung and Kwok . In Section 3, we give the general pricing formulas in various cases. In this general framework, a pricing formula of defaultable bonds is provided for three-firm model. In Section 4, using the approach of “total hazard” and “change of measure,” we present the construction of default time, derive the joint density function, and obtain the closed forms of CDS swap rate (swap premium) in the continuous time and the discrete time framework, respectively. We conclude this paper with Section 5.
2. Basic Setup and Three-Firm Contagion Model
2.1. Basic Setup and Construction of Default Time
We consider an uncertain economy with a time horizon of described by a filtered probability space (in this paper we follow the symbols and notations of Jarrow and Yu ) satisfying the usual conditions of right continuity and completeness with respect to -null sets, where and is an equivalent martingale measure under which discounted bond prices are martingale. We assume the existence and uniqueness of , so that bond markets are complete and no arbitrage, as shown in discrete time by Harrison and Kreps  and in continuous time by Harrison and Pliska . Subsequent specifications of the model are all under the equivalent martingale measure .
On this probability space, there is an -valued process , which presents -dimensional economy-wide state variables. There are also point processes, , initialized at 0. These represent the default processes of the firms in the economy such that the default of the th firm occurs when jumps from 0 to 1.
According to the information contained in the state variables and the default processes, the filtration is where are the filtrations generated by and , respectively.
Let where . We know that contains complete information on the state variables and the default processes of all firms other than that of the th, all the way up to time .
According to the filtration , it is possible to select a nonnegative, -measurable process , satisfying . for all , so that we can define an inhomogeneous Poisson process , using the process as its intensity function.
Let denote the default time of firm , namely, be the first jump time of , in a typical reduced-form model, which can be defined as where is independent of .
According to the Doob-Meyer decomposition, we have that is a -martingale.
Under the above characterization, the conditional survival probability of firm is given by
The unconditional survival probability of firm is given by
Now, we give the recursive construction of default time as Yu in his paper . Specifically, we start with the case of no state variable. Let the notation denote the intensity for firm given the observed default times of other firms, , where .
The total hazard accumulated by firm by time given observed defaults is defined as where is the total hazard accumulated by firm for a period of length following the th default. At the same time, we assume that there is no default between and .
Define the inverse functions for . We can use the following recursive procedure to construct a collection of random variables.
Step 1. Let be the i.i.d. unit exponentials and and define
Given that Step 1 up to step have resulted in , define the set and as the set of firms excluding . Let and let
2.2. Three-Firms Contagion Model
In this subsection, we explore the three-firms contagion model with an interaction term. Consider the case where the default intensity of one firm is affected by the default of other two firms, so that when one firm defaults the default probabilities of other two firms will jump. In the three-firms contagion model, the interdependent structure between firm , firm , and firm is characterized by the correlated default intensities.
Recall Leung and Kwok's three-firms model:where , , and satisfying , , . Similarly, , , and reflect the effect of macroeconomic factor and itself on firms , , , respectively.
Nevertheless, Leung and Kwok have not allow the effect of two parties' simultaneous default on the third party, namely, there is not an interaction term in their model. Thus, if three firms are copartners, then the default risk of each firm may be overestimated and the asset value may be underestimated because there exists the case in which the default events might overlap. If they are competitors, then the case is contrary.
For the above reason, we allow the following three-firms contagion model:where , , and satisfying , , . , , and reflect the effect of macroeconomic factor and itself on firms , , , respectively.
3. Bond Pricing under Three-Firm Model
3.1. The General Pricing Formulas
Definition 3.1. A defaultable claim maturing at is the quadruple , where is an -measurable random variable, is an -adapted, continuous process of finite variation with , is an -predictable process, and is a random time.
Definition 3.2. The dividend process of the above defaultable claim maturing at equals, for every , where is the promised payoff, represents the process of promised dividends, and the process is the recovery process.
Definition 3.3. The exdividend price process of a defaultable claim equals, for every , where is the money market account, is a constant risk-free spot rate, and represents the conditional expectation on under the equivalent martingale measure .
Lemma 3.4. The exdividend price of the defaultable claim equals, for , where .
From Lemma 3.4, We can explore the following special cases. For the default-free zero-coupon bond which pays one dollar, the dividend process is . Let be the time- price, then is given by If the dividend process is , then the value of the bond is always 1.For the defaultable zero-coupon bond which pays one dollar if not default and pays times the price of a default-free bond at maturity, where is introduced by Jarrow and Turnbull  and Jarrow et al.  as “recovery of Treasury,” let denote the time- price, issued by firm , is the recovery rate of the firm , then is given by If the dividend process , using the Doob-Meyer decomposition of , then the value of the defaultable bond is
3.2. Bond Pricing under Three-Firms Model
We assume that there are three firms , , and . Now, we consider the case that each firm holds the other two firms' defaultable bonds, so that when one party defaults, the other two parties' default probability will jump. The default intensities are described as (2.14a)–(2.14c).
We adopt the change of measure introduced by Collin-Dufresne et al.  to define a firm-specific probability measure which puts zero probability on the paths where default occurs prior to the maturity . Specifically, the change of measure is defined by where is a firm-specific (firm ) probability measure which is absolutely continuous with respect to on the stochastic interval . To proceed the calculations under the measure , we enlarge the filtration to as the completion of by the null sets of the probability measure .
Applying the result of Jarrow and Yu , we know that the defaultable bond price of firm is given by or
Because of the symmetry of default intensities, we need only to compute one firm's value of the three firms. In the remainder of this subsection, we will derive the closed-form pricing formula of firm .
For firm , the time- value of the defaultable bond maturity at satisfies where Conditional on , , namely, neither firm nor firm has defaulted by time , the default intensities and under the measure and are given by According to the result of Leung and Kwok , the conditional joint density function of is The integration region of (3.11) is then appropriately divided into five pieces: : , ; : , ; : , ; : , ; : , : where Conditional on , , namely, firm has defaulted and firm has not defaulted by time , the default intensities are given by and the density function of is so Conditional on , , namely, firm has defaulted and firm has not defaulted by time , similar to the computation in (II), we have Conditional on , , namely, firm and firm have defaulted by time , we have From (3.10), (3.11), and the discussions in (I)–(IV), we have the following theorem.
4. CDS Valuation under Three-Firms Model
4.1. The Basics
As one of the important credit derivatives, CDS is a contract agreement which allows the transfer of credit risk of a risky asset (basket of risky assets) from one party to the other. A financial institution may use a CDS to transfer credit risk of a risky asset while continues to retain the legal ownership of the asset. To determine a fair swap rate of a CDS in the presence of counterparty risks, the interdependent default risk structures between these parties must be considered simultaneously.
On CDS valuation, there have been numerous works in recent years. Based on the reduced-form approach with correlated market and credit risks, the closed-form valuation formula for the swap rate of a CDS is obtained by Jarrow and Yildirim . They assume that the default intensity is “almost” linear in the short interest rate. Recently, considering the impact of counterparty risk on the pricing of a CDS, Jarrow and Yu in  assume an interdependent default structure that avoids “looping default” by involving primary-secondary framework and simplifies the payoff structure where the protection seller's compensation is made only at the maturity of the swap contract. They discover that the default risk of the protection seller and reference entity is ignored. Hull and White  apply the credit index model for valuing CDS with counterparty risk. M. A. Kim and T. S. Kim conclude that if the default correlation between the counterparty and reference bond is ignored, then the pricing error in a CDS can be quite substantial. Chen and Filpovic in their paper  develop a generalized affine model to price credit default swaps under default correlations and counterparty risk. Yu  uses the “total hazard” approach to construct the default process from independent and identically distributed exponential random variables and obtains an analytic expression of the joint distribution of default times in his two-firms and three-firms contagion models. Leung and Kwok  use the “change of measure” approach introduced by Collin-Dufresne et al.  to price the CDS in two-firms model and three-firms contagion model, respectively, and obtain the closed-form formulas.
We assume that party (CDS protection buyer) holds a corporate bond of party C (reference asset) and party is subject to default. Party faces the credit risk arising from default of party . To seek protection against such default risk, party enters a CDS contract in which he agrees to make premium payments, known as the swap premium to party (CDS protection seller). In exchange, party promises to compensate for its loss in the event of default of the bond (reference asset).
We employ the three-firms model specified by (2.14a)–(2.14c) to price the CDS and study the effect of the default of each party on the swap rate. Suppose that party (a corporate bond investing firm) holds a corporate bond (reference asset) issued by party (a corporate bond issuer) (refer to 1 in Figure 1) and firm is subject to default. At bond maturity, if firm does not default, then it will pay the bond principle and interest to firm (see 1). Otherwise, it has no payments (refer to 1). On the other hand, to hedge the default risk of firm , firm , and firm (the protection seller, such as a monoline insurer), enter into a CDS contract. Firm and are also subject to default. If firm and have no default, then firm makes fixed premium payments, known as the swap premium to firm (see 1). Either firm or firm defaults, there is no premium payments to firm (refer to 1). In exchange, firm promises to compensate (if does not default) for its loss in the event of default of the bond (reference asset) as long as does not default (refer to 1). If the protection seller defaults prior to the default of either the reference asset or the protection buyer , then the protection seller can simply walk away from the contract and has no obligation to pay the compensation to the protection buyer (see 1).
In this section, we will analyze the effect of correlated risks between three parties in a CDS using a similar contagion model as in Leung and Kwok's model . Differently from their model on CDS valuation with counterparty risk, we allow an interaction term in the default intensity model for three firms, namely, we discuss how the simultaneous default of two parties impacts on the third one.
4.2. The Joint Density Function for Three-Firms Model
To price CDS swap rate , we firstly need to provide the joint density function of three firms , , and . We adopt the “total hazard” approach by Yu  and Zheng and Jiang  as description in Section 2.1; the result is the following Lemma.
Proof. With the total hazard method (2.10) and (2.12) introduced in Section 2, we can express default time in terms of standard exponential variables , and vice versa. If , then we have
The Jacobi determinant of with respect to is given by The density of is therefore Substituting into , we get
The density function in other regions can be expressed similarly with permutation. Thus, we complete the proof of the lemma.
4.3. CDS Valuation
In this subsection, we employ the three-firms model specified by (2.14a)–(2.14c) to price the CDS swap rate (or swap premium) in continuous time framework and discrete time framework, respectively. We assume that the recovery rate is zero and the risk-free spot rate is a constant.
4.3.1. In Continuous Time Framework
In this framework, the value of the contingent leg at time 0 is equal to and the value of the fee leg at time 0 is equal to where is the expiration, is the length of the settlement period, and represents the settlement date at the end of the settlement period.
We can derive by computing the expectation of and ; the result is the following theorem.
Proof. According to the arbitrage-free principle, we set the present value of protection buyer's payment equal to the present value of the compensation payment made at , conditional on default of prior to , no default of prior to , and no default of prior to .
Since it takes no cost to enter a CDS, the value of under this three-firms model is determined by where represents the settlement date at the end of the settlement period.
Recall that the change of measure is defined by
Thus, by (4.1), (4.11) and the Fubini Theorem, we can derive the left side of (4.10): where the fourth equation is according to the two-firms model of Leung and Kwok in ,
The right side of (4.10) equals where
By (4.10) and (4.12)–(4.16), we obtain expression (4.9) of .
Remark 4.3. From (4.9), we can see that the swap rate is not dependent on the expiration date . The default of the buyer impacts on the swap rate , so it is not strict if assuming that the buyer has no default throughout the process though the default risk of the protection buyer has little impact on the swap rate. The reference asset's default risk proxied by gives the most significant impact on the swap rate, and an increasing higher value of gives rise to a higher swap rate. The contagion effect of the protection buyer and the protection seller on the reference asset has no effect on the swap rate (there are no , , and terms). This shows that when valuating CDS in “loop-default” models, without loss of generality, we can assume that the reference asset is the primary firm and the protection buyer and the seller are secondary firms.
Remark 4.4. From (4.9), if the settlement period is zero, the swap rate is , which is the default intensity of reference asset determined by macroeconomic factor and itself, and has nothing to do with the credit risk of the protection buyer and seller.
4.4. Valuation of CDS in the Discrete Time Framework
In the discrete time framework, let , , ,, be the swap payment dates, where . We assume that the payment dates are uniformly distributed; that is, for and .
The value of the contingent leg at time 0 is given by
The value of the fee leg at time 0 is given by According to the arbitrage-free pricing principle, we have the following theorem.
Proof. Similar to the discussion in the continuous time, since it takes no cost to enter a CDS, the value of the swap rate under this three-firms model is determined by
where is still the length of the settlement period. The first term in (4.20) gives the present value of the sum of periodic swap payments (terminated when either , , or defaults or at maturity), and is the present value of the accrued swap premium for the fraction of period between and the last payment date. The right term represents the present value which the protection seller () pays if the reference asset defaults prior to the maturity:
We can obtain The right-hand side of (4.20) is From (4.22)–(4.26), we can obtain the expression of from the following equation:
Remark 4.6. As analyzed in the continuous time, the expression for the swap premium in (4.27) shows no dependence on , , , , , . In the financial sense, prior to the default of the underlying asset, the default event of the protection buyer or the protection seller will terminate the contract. This is why , , , , , have no influence on the swap premium. Moreover, we discover that the swap premium is also insensitive to maturity.
In this paper, we present a three-firms contagion model with an interaction term which is an improvement in the model of Leung and Kwok . Under this model, we analyze the pricing of defaultable bonds and obtain the closed forms. We also discuss the CDS valuation in continuous time and discrete time framework, respectively. The analytical solutions of CDS swap rate (swap premium) are obtained by the approaches of “total hazard construction” and “change of measure.” Besides, we analyze the effect of the default of the protection buyer, the protection seller, and the reference asset on the swap rate.
Our model has its actual background. For example, before and during the global financial crisis, as default risk of the reference asset issuer increased, the protection seller collected higher CDS swap premiums. Thus, default risk of the protection buyer increased since more CDS swap premiums were payed. On the other hand, the protection seller compensated more and more for the loss of reference asset (if it defaulted). When the protection seller (such as a monoline insurer) had no ability to compensate for the loss of reference asset, it went bankrupt. All of these could be important reasons for the financial crisis. So our model is of some significance.
This paper was supported by the National Basic Research Program of China (973 program 2007CB814903).
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