Valuation of a Basket Loan Credit Default Swap

This paper provides a methodology for valuing a basket Loan CDS (LCDS) by considering both default and prepayment risks. Under “top down” and intensity framework, using a single-factor model, correlated default and prepayment risks are considered, where the stochastic interest rate is used to be their common factor. All stochastic processes in the model are assumed to follow CIR processes. Through Feynman-Kac formula, we obtain a PDE problem and its closed-form solution. Numerical examples are provided.


Introduction
Loan-only Credit Default Swaps, called LCDS in simple, are financial instruments that provide the buyer an insurance against the default of the underlying syndicated secured loan. Its markets were launched in 2006 both in US and Europe. It is developed on the basis of standard Credit Default Swaps (CDS). A standard Credit Default Swap 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 the credit event occurs, the seller pays compensation to the buyer. If there is no credit event occurs during the term of swap, the buyer continues to pay the premium until the CDS maturity. Hull-White (Hull and White, 2000) first considered the valuation of a standard credit default swap when there is no counterparty default risk. And they extended their study to the situation where there is possibility of counterparty default risk and obtaining a pricing formula with Monte Carlo simulation (Hull and White, 2001). Furthermore, they develop two fast procedures for valuing tranches of collateralized debt obligations and th n to default CDS without Monte Carlo simulation. (Hull and White, 2004) Comparing to a standard CDS, a LCDS contract is almost the same, except that 1. Its reference obligation is limited on loans; 2. It can be cancelled. So that, the pricing of LCDS must take into account not only default probabilities with recovery rate, but also the prepayment probabilities. These two probabilities are negative correlated. The stronger the relationship between default and prepayment is, the higher the LCDS spread will be.
In the literature, different models for LCDS have been developed. These models can be classified into two main categories known as structural models and reduced form models. In these models, default and prepayment are modeled as a function of a set of state variables. The intensity based model is reduced form one. For the pricing LCDS, Zhen Wei (Zhen Wei, 2007)  problem. Based on Zhen Wei's work, Peter Dobranszky (Dobranszky, 2008) times the prepayment intensity p  with a coefficient variation to describe the relationship between LCDS and CDS.
For pricing a basket LCDS, the reduced form model is used more frequently due to the scale and complexity of the pool. The reduced form model can be classified to two categories: "bottom up" and "top down". In a bottom up model, the portfolio intensity is an aggregate of the constituent intensities. In a top down model, the collateral portfolio is modeled as a whole, instead of drilling down to individual constituents; the portfolio intensity is specified without reference to the constituents. The constituent intensities are recovered by random thinning. The benefit of a top-down approach is its simplicity as a result of not having to model the individual constituents of the underlying portfolio. Kay Giesecke (Giesecke, 2008) contrasts these two modeling approaches. It emphasized the role of the information filtration as a modeling tool. S.Wu, LS.Jiang and J.Liang (Jiang and Liang, 2008) use top-down model to pricing of MBS with repayment risk. Using bottom up framework, Shek, H., S. Uematsu and Zhen Wei (Uematsu and Wei, 2007) studied pricing a CDS referenced a pool loan, described the default and prepayment by single-factor Gaussian Copula model. They obtained the spread of the LCDX through Monte Carlo simulation. Dobranszky and Schoutens (Dobranszky and Schoutens, 2007) used single-name Lévy copula to describe the relationship between default and prepayment.
In this study, under "top down" framework, we consider the valuation of a LCDS, where the references pool is considered as an entity. Thus, in this model, the default and the prepayment here become "loss" of the pool by different way, while the inverse prepayment behaviors as "gain". In the following section, we introduce two processes of default and prepayment, and construct the model of LCDS in the continuous time situation. In the third section, we use single-factor model to describe the relationship between default and prepayment through interest rate and other CIR processes. Moreover, we develop a two dimensional partial differential equation. Under the assumption that the PDE has radiation structure solution, the equation can be separated into one ODE problems and one Riccati equation, which can obtain a closed-form of the spread. In the fourth section, numerical examples are shown.

Modeling LCDS
As mentioned before, the main point of LCDS is a probability that the loan prepays earlier and hence the instrument is cancellable. During the life of an LCDS contract, two kinds of events may be triggered, either the underlying loan is prepaid or the loan-taker goes to default. If a prepayment event were triggered first, the LCDS would have been cancelled. If during the life of the LCDS contract a default event were triggered first, the LCDS would have defaulted, when the LCDS issuer would have to pay the recovery adjusted notional amount to the LCDS buyer.
We consider a basket of loans, from the idea of "top-down". We denote by t A the outstanding principal balance at time t. Without loss of generality, we assume that the outstanding balance in pool at time Any default event or prepayment event will affect, in general reduce, the total principal in the pool. We denote by t D as the accumulative amount of default and t P as the accumulative amount of prepayment which result in the decrease of total principal at time t . Then for any 0 In our model, a continuous time model is considered. We model default and prepayment by introducing stochastic processes   0 Plus the equation (1) and (2) Solving the equation (3) B t  denotes the risk-free discount factor from time  to time t .
The protection leg, the expected present value of the losses in case of default equals where R is the recovery rate, which assumed to be a positive constant smaller than 1.
Taking equation (4) and equation (6) into equation (5), we get the spread is This is the pricing formula. Now the problem turns to find those expectations.

The correlation between default and prepayment
The default and prepayment rates are negative correlated. The more relevant is, the higher spread rate will be. Now we use single-factor model to describe the relationship between default rate and prepayment rate. That is 24 three independent processes.
The motivation and factors that affect the prepayment are complex. Researchers have shown that the prepayment of loans is affected by interest rate, macroeconomic factors, and seasonal factors and so on. Among them, the interest rate is the most important one. When interest rate decreases, borrowers refinance their loans, which led to higher prepayment rate. This means that when the current mortgage interest rate below the contract rate to a certain extent, the borrowers can take loans from other banks to repay the existing loans. As long as the spread reached a certain level, and covered the transaction costs sufficiently, the borrower will have a strong incentive to refinance.
But there is an opposite situation for default events. The cost of borrowing increases with the interest rate increasing, which increases the likelihood of default. Also default has close relation with the borrower's financial condition, macroeconomic and so on.
Measuring these factors, it is suitable to use the interest rate as the common factor of default rate and prepayment rate. So we take If t r and d t X are positive processes, the d t  will be a nonnegative process, but p t  may be negative some time as we mentioned before.

The solution under CIR process
Now, we give a model to describe the behavior of t r , p t X follows the stochastic process of CIR, i.e. ( ) , Taking equation (8) and equation (9) into equation (7) and considering the independence among t r , (1 ) The expectation in equation (10) can be expressed as these two expectations, for example, If we can value these two expectations, the pricing equation (10) (2 ) ( 2 ) (2  ) , <Figure 1 about here> However, Figure 2 shows that the shape of LCDS term structures when we fix the common factor t r ,