Fast Pricing of Basket Default Swaps

Abstract

A basket default swap is a derivative security tied to an underlying basket of corporate bonds or other assets subject to credit risk. The value of the contract depends on the joint distribution of the default times of the underlying assets. Valuing a basket default swap often entails Monte Carlo simulation of these default times. For baskets of high-quality credits and for swaps that require multiple defaults to trigger payment, pricing the swap is a rare-event simulation problem. The JoshiKainth algorithm is an innovative importance-sampling technique for this problem that forces a predetermined number of defaults to occur on each path. This paper analyzes, extends, and improves the Joshi-Kainth algorithm. We show that, in its original form, the algorithm can actually increase variance; we present an alternative that is guaranteed to reduce variance, even when defaults are not rare. Along the way, we provide a rigorous underpinning in a setting sufficiently general to include both the original method and the version proposed here.

Sensitivity estimates for portfolio credit derivatives using Monte Carlo

“…Joshi and Kainth [14] and Chen and Glasserman [5] developed importance sampling techniques for basket default swaps that force at least n defaults to occur on every path. They accomplish this by sequentially increasing the default probabilities for the names in the basket.…”

“…Chen and Glasserman [5] call their version the Conditional Probability (CP) method. The JK method uses somewhat arbitrary default probabilities for importance sampling; in the CP method, the default probability for each name is set equal to its conditional probability of default, given that at least n names default.…”

“…The JK method uses somewhat arbitrary default probabilities for importance sampling; in the CP method, the default probability for each name is set equal to its conditional probability of default, given that at least n names default. Chen and Glasserman [5] show that these importance sampling probabilities are, in a sense, optimal and guarantee variance reduction.…”

“…As in Chen and Glasserman [5], or in Joshi and Kainth [14] with signs reversed, we write the discounted payoff of the swap as the difference between the discounted payoffs of the payments made between the parties, called the protection leg and the value leg: (τ 1 , . .…”

Monte Carlo Computation in Finance

“…This makes it worthwhile to pursue variance reduction: see [46,Ch. 9] for general techniques and [8,25,48,49,51] for techniques specific to credit risk.…”