The Valuation of Default-Triggered Credit Derivatives

Abstract

Credit derivatives are among the fastest growing contracts in the derivatives market. We present a simple, easily implementable model to study the pricing and hedging of two widely traded default-triggered claims: default swaps and default baskets. In particular, we demonstrate how default correlation (the correlation between two default processes) impacts the prices of these claims. When we extend our model to continuous time, we find that, once default correlation has been taken into consideration, the spread dynamics have very little explanatory power. THE VALUATION OF DEFAULT-TRIGGERED CREDIT DERIVATIVES Credit derivatives are among the fastest growing contracts in the derivatives market. They are exercised either upon the occurrence of a default event or when spreads exceed certain pre-defined boundaries. Among default-triggered credit derivatives, default swaps (the most popular) and default baskets are the most widely traded. Spurred by the recent Asian financial crises, the trade volume in default swaps has increased dramatically. According to Reyfman and Toft (2001), "Globally, more than $1 billion in default swap notional exposures to more than several dozen names are traded daily." Over the years, many default swap contracts have been standardized in the over-thecounter market and there have emerged specialized broker-dealers (e.g., Tullet and GFI Inc.) who provide two-way quotes. Default-triggered credit derivatives, such as default swaps and default baskets, are regarded as "cash product" which can be priced off the default probability curve. Like the risk-free forward rate curve, the default probability curve describes default probabilities for various future points in time. While deriving the single default probability is simple, the derivation of the joint default probability curve is very complex. Correlations among various default processes play a dominant role in deriving the joint default probability curve. Unfortunately default swaps and default baskets both involve more than one default process. Default swaps involve two default processes: counterparty default and issuer default. Default baskets involve multiple default processes: counterparty default and defaults of multiple issuers. Default can be modeled in a variety of ways: one popular way is through the use of a Poisson distribution. However, correlation is not clearly defined between two Poisson processes. In this paper, we present a simple model for the default correlation. The proposed correlation model not only preserves the desired features for defaulttriggered contracts but also is very easy to implement. With our model, we are able to demonstrate, in a single-period setting, how the default correlation plays a dominant role in the pricing of default products. We then extend the model to multiple periods and incorporate random spreads and random interest rates. In this setting, we demonstrate that correlations among spreads have literally no impact on prices, instead we demonstrate that default correlation is the relevant (domina...

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