Valuation of Credit Default Swap with Counterparty Default Risk by Structural Model *

This paper provides a methodology for valuing a credit default swap (CDS) with considering a counterparty default risk. Using a structural framework, we study the correlation of the reference entity and the counterparty through the joint distribution of them. The default event discussed in our model is associated to whether the minimum value of the companies in stochastic processes has reached their thresholds (default barriers). The joint probability of minimums of correlated Brownian motions solves the backward Kolmogorov equation, which is a two dimensional partial differential equation. A closed pricing formula is obtained. Numerical methodology, parameter analysis and calculation examples are implemented.


Introduction
A vanilla credit default swap (CDS) is a kind of insurance against credit risk.The buyer of the CDS is the buyer of protection who pays a fixed fee or premium to the seller of protection for a period of time.If a certain pre-specified "credit event" occurs, the seller pays compensation to the buyer.The "credit event" can be a bankruptcy of a company, called the "reference entity", or a default of a bond or other debt issued by the reference entity.In this paper, the "credit event" also includes the default of the protection seller.If there is no credit event occurs during the term of the swap, the buyer continues to pay the premium until the CDS maturity.
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.As the rapid growth of the credit default swap market, credit default swaps on reference entity are more actively traded than bonds issued by the reference entities.
There are two primary types of models of default risk in the literature: structural models and reduced form (or intensity) models.A structural model uses the evolution of a firm's structural variables, such as an asset and debt values, to determine the time of a default.Merton's model [1] is considered as the first structural model.In Merton's model, it is assumed that a company has a very simple capital structure where its debt has a face value of D and maturity of time T, provides a zero coupon.Merton shows that the company's equity can be regarded as a European call option on its asset with a strike price of D and maturity of T. A default occurs at T if the option is not exercised.The second approach, within the structural framework, was introduced by Black-Cox [2] and Longstaff-Schwartz [3].In this approach a default occurs as soon as the firm's asset value falls below a certain level.In contrast to the Merton approach, the default can occur at any time.Zhou [4,5] produces an analytic result for the default correlation between two firms by this model.Using this model, credit spread with jump is considered by Zhou [6].
Reduced form models do not care the relation between default and firm value in an explicit manner.In contrast to structural models, the time of default in intensity models is not determined via the value of the firm, but the first jump of an exogenously given jump process.The parameters governing the default hazard rate are inferred from market data.These models can incorporate correlations between defaults by allowing hazard rates to be stochastic and correlated with macroeconomic variables.Duffie-Singleton [7,8] and Lando [9] provide examples of research following this approach.
There have been many works on the pricing of credit default swaps.Hull -White [10] first considered the valuation of a vanilla credit default swap when there is no counterparty default risk.Their methodology is a twostage procedure.The first stage is to calculate the default probabilities at different future times from the yields on bonds issued by the reference entity.The second stage is to calculate the present value of both the expected future payoff and expected future payments on the CDS.They extended their study to the situation where there is possibility of counterparty default risk and obtained a pricing formula with Monte Carlo simulation [11].They argued that if the default correlation between the protection seller and the reference entity is positive, the default of the counterparty will result in a positive replacement cost for the protection buyer.
Affection of the correlation on a CDS pricing remains interesting.The valuation of the credit default swap is based on computing the joint default probability of the reference entity and the counterparty (protection seller).Technically it is difficult because correlation between the entities involved in the contract is hard to deal with.Jarrow and Yildirim [12] obtained a closed form valuation formula for a CDS based on reduced form approach with correlated credit risk.In their model, the default intensity is assumed to be linear in the short interest rate.Jarrow and Yu [13] also assumed an inter-dependent default structure that avoided looping default and simplified the payoff structure where the seller's compensation was only made at the maturity of the swap.They discovered that a CDS may be significantly overpriced if the counterparty default probability was ignored.Yu [14] constructed the default processes from independent and identically distributed exponential random variables using the "total hazard" approach.He obtained an analytic expression of the joint distribution of default times when there were two or three firms in his model.Leung and Kwok [15] considered the valuation of a CDS with counterparty risk using a contagion model.In their model, if one firm defaults, the default intensity of another party will increase.They considered a more realistic scenario in which the compensation payment upon default of the reference entity was made at the end of the settlement period after default.They also extended their model to the three-firm situation.
In this paper we develop a partial differential equation (PDE) procedure for valuing a credit default swap with counterparty default risk.In our model, a default event is supposed to occur at most one time, which means either reference entity or counterparty may default once.Our work is based on the structural framework, where the default event is associated to whether the minimum value of stochastic processes (value of the companies) have reached their thresholds (default barriers).Usually we choose the companies' liability as the thresholds [1,24].We show that the joint probability of minimums of correlated Brownian motions solves the backward Kolmogorov equation, which is a two dimensional PDE with cross derivative term.This equation can be solved as a summation of Bessel and Sturm-Liouville eigenfunctions.The defaultable CDS studied in this paper, same as Hull-White's, is a special case.More complicated features of that kind of CDS are not considered.
The paper is organized as follows.In Section 2, we present a CDS spread expression.In Section 3, we establish a partial differential equation model which solves the joint probability distribution of two correlated companies used in Section 2 under the some assumptions.We obtain an explicit solution for this PDE.The main result of the closed form of the pricing the CDS then follows and shows in this section.Numerical calculation, example tests and parameter analysis for our model are collected in Section 4. We conclude the paper in Section 5.

CDS Spread with Counterparty Default Risk
In A vanilla CDS contract usually specifies two potential cash flow streams -a fixed premium leg and a contingent leg.On the premium leg side, the buyer of protection makes a series of fixed, periodic payments of the CDS premium until maturity or until a credit event occurs.On the contingent leg side, the protection seller makes a single payment in the case of the credit event.The value of the CDS contract to the protection buyer at any given point of time is the difference between the expected present value of the contingent leg, which is the protection buyer expects to receive, and that of the fixed leg, which he expects to pay, or Similar to the vanilla CDS, we assume that the payments are made at dates 1 On the contingent leg side, if the reference entity defaults at time  , the present value of the payoff form the CDS is given by where  is the liquidation period.The expected payoff is According to (1), the value of the CDS at time t is The value of the swap at origination must be equal to zero.The CDS spread s is the value of w which makes the value of the CDS equal to zero.Thus The variable s is referred as the credit default swap spread or the CDS spread.It is the total of the payments per year, as a percentage of the notional principal.
In expression (2), the joint probability densities are still unknown.We will focus on how to obtain these probability densities in following sections.

Modelling and Solution
In this section, we present several mathematical theorems which are necessary for the valuation of the CDS with the counterparty default risk.In order to describe the correlation between the reference entity and the counterparty, we study the joint probability distribution functions of the minimum values of two correlated Brownian motions.
The following are Basic Assumptions for our model.
 with  being a constant; 3) Firm i defaults as soon as its asset value ( ) i V t reaches the default barrier i D .In this paper, we use the Black-Cox type default barrier, which is where i F and i  are given constants respectively (see [2]); 4) The credit event occurs at most once.

Default Probability
and a credit event occurs when i X reaches i m .
Define the running minimum of In order to get the probability density needed in (2), define , , u x x t is the probability of the event that 1 X defaults (i.e.
Next, we eliminate the cross-partial derivative and normalize the Brownian motions by a suitable transformation of coordinates, this idea was introduced by He etc. ( [26].Define new coordinates 1 z and 2 z as the following Then where Because the boundary conditions are more conveniently expressed in polar coordinates, we introduce thus ,0, = 0, , , = 0, , ,0 = , ,0 . In order to solve PDE (15), we consider the Green's function Proof.We try to find separable solutions to this equation in the form of Plugging ( 18) into ( 16), we find that Divide the previous equation by Since the left side of ( 19) is a function of  and the right side is a function of r and  , so it must be a constant.Denote this constant by This is a Sturm-Liouville problem.We try to find separable solutions in the form of Plugging this into (16) we get with boundary conditions It is easy to see that   , we get the standard form of Bes- sel's equation The well known fundamental solutions of this Bessel's equation is to be bounded, the solution

 
k n Y y is not permitted.Hence the general radial part of the solution is Integral the previous equation over  , we obtain the general solution to PDE (16) for , , = sin .
Then solution of PDE ( 13) is Returning to the original coordinates and variables where 1 z , 2 z , r ,  are defined in ( 9), ( 10) and (12).
That is, we have Theorem 1 The solution of the initial boundary problem (4) has a closed form solution (29) associated by (26), ( 14) and (17).
By now, we have already obtained the probability of that company 1 defaults and company 2 does not defaults between time t and T. Change t into 0, T into t, here comes the probability between time 0 and t.
In order to obtain the probability   1 2 , , v x x t of that company 2 defaults and company 1 does not default, we only need to change the positions of the parameters of the two companies such as   , , , 0 , , u x x t .Now apply our result to the spread Formula (2) when = 0  .In the valuation formula of ( 2), what we need are the default probability density functions of     and     while we only have the probability functions.
Therefore, we need to modify (2).In fact, notice that   a  and   e  are piecewise continuous functions and on every piece, Integrate the numerator and denominator of (2) by parts, and we have for u and v are solved in this subsection.

Survival Probability 1 2 ( ) π x , x ,t
To calculate the credit default swap spread s, we still need to study the joint survival probability of   In the previous section, we get the solutions to these two PDEs in the domain  , , =1, , , = 0, , , = 0, where 1 2 , 0. m m  Compare the boundary and final conditions of PDE (31), (32), (33), the solution to (31) can be written as the linear combination of the other two , , =1 , , , , .
Set = 0 t , we get the probability of  which was defined in Section 2 Thus, the CDS spread (2) can be rewritten as Remark 1 It is a special case of our model that the CDS with the counterparty default when the correlation of the counterparty and reference entity are independent.
Remark 2 The same method can be used to the pricing the CDS for a basket reference entities.In this case, the PDE model is simpler as the boundaries condition are all equal to 0. So that, it has no problem caused by the singularity near (0,0).However, if the basket has a big number of reference entities, the closed form solution of the PDE is difficult to be obtained.

Main Result
Combine the previous two subsection, we obtain the all probabilities required in Formula (36).Therefore we ob-tain the main theorem of this paper presented as follows: Theorem 2 (main theorem) Under the Basic Assumption (1)-( 4), the credit default swap spread with counterparty default risk is given by (36), where, in the formula, the probability     are given by (30) solving the problem (4);     are solved as     in the same way;  is given by (35).

Numerical Analysis
So far, we have derived the three probabilities in Section 2. With these, we can calculate the CDS spread by (29).
Even though we have a so called closed or semi-closed form solution, but the calculation of the form is still not trivial.The expression of the form includes integration and infinite serial, as well as a special function.The direct calculation is not easy to undertaken and the result is usually not satisfying.This because that the value of the integrand concentrates in a very small area and this area is moving as the change of the time t.So that, the difference approximation, in general, will make the result value very small.
Here we introduce an algorithm of Monte Carlo method to evaluate the form (29).It sounds that there is no difference from the one to calculate CDS spread by direct Monte Carlo method, however, it is really different with and without the closed form solution.We will see it in the later.
Using Monte Carlo to calculate the closed form, in fact, we only need to know how to calculate the first integration of the Formula (26).The steps to do it are as follows: 1) Representing the integration with the exception of the integrand.
 as a density function, where where   where G is defined in (25).
In our case, the three- the following way: a) First, the marginal density function respect to 0 r is and the marginal distribution function is The marginal density function respect to 0  is and its distribution function is Then generate uniform random number 2i U and set  In Figure 1, method 1 means the CDS spread is obtained by simulating directly on the original Formula (2), method 2 means the CDS spread is calculated by our closed form solution with the integral evaluated by Monte Carlo method.The method 1 is repeated 10000 times using computer time 157.6858second, while the method 2 is repeated 1000 times using computer time 178.7021 second.We can see that the calculation by our solution converges much faster than directly simulate the original formula.As less as 1/10 times, the result of the method 2 is much better than the method 1. Figure 3 confirms that CDS spread increases with expired time T and decreases with recover rate R, when the correlation is fixed.
Figure 4 show what kind of the rules for the volatilities of the two companies.The behaviors of them affect to the CDS spread in different way.Suppose that the other parameters are fixed.If the volatility of the Company B is larger, which means the probability of the default goes larger as well, it results that the CDS spread is more expansive.On the other hand, if the volatility of the Company C is larger, which means the probability of the failure of the CDS payoff is larger, it results CDS spread is cheaper.

Conclusions
In this paper, we have introduced a PDE methodology for modeling default correlations.We assume that the value of companies follow correlated geometric brownian motions.When the asset value of a company reaches a predefined barrier, a credit event called default occurs.The essential part is to derive the joint default probability as the solution to a partial differential equation.This solution is more computationally efficient than traditional simulation for original formula or lattice techniques to the equation.We applied the default probabilities solved from the PDE to the valuation of credit default swaps with counterparty default risk.The model can be extended to the valuation of any credit derivative when the payoff is based on defaults by two companies.
The shortage of the model is limited by the dimension, it is difficult to extend the method to a basket CDS with a large portfolio.
25) can be simplified into (


is defined in (28).By simple calculation, we obtain Now rewrite the integration, E is measured respect to  f : , the joint marginal distribution function with respect to 0 r and 0  is


n times (e.g.= 1000,10000 n  as required), then find the mean value, to find approximated the expectation.It may argue that if use Monte Carlo method, why just simulate directly on the original Formula (2)?The Fig- ure 1 can answer this question.Consider practical examples.Assuming there are two companies B and C with initial values of and the default barriers are $40 million and $60 million respectively.
Now use the closed form solution, by Monte Carlo simulate 1000 times to calculate the integral, we can analysis the parameters of R,  ,  and T respectively.The other parameters are chosen as above.The left figure and the right one show the impact of correlation coefficient  , maturity time T and recovery rate R on CDS spread.Two figures in Figure 2 show their relationship.In the upper figure of Figure 2, CDS spreads are greater for swaps with longer maturities.The lower one illustrates the extent to which CDS spreads depend on the recovery rate.When the recovery rate becomes larger, the payoff will get smaller.Hence the CDS spread is getting smaller when recovery rate getting larger.Both of them show that the spread goes down as the correlation goes up.

Figure 1 .
Figure 1.CDS spread with counterparty risk by two methods.

Figure 3 .
Figure 3. CDS spread with counterparty risk vs. time T, varying R.

Figure 5
Figure 5 is a three-dimensional surface of the value for the probability of      0 , 0 ,0 = 5 B C T u V V respect to   0 B V and

Figure 5 .
Figure 5. is a three-dimensional surface of the value for the probability of       0 , 0 ,0 = 5 B C T u V V respect to Case 3. neither the reference entity nor the counterparty defaults prior to maturity time T.This time the present value of the payments is t  and   t  , the total expected present value of the premium leg is