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Modelling the Bid and Ask Prices of Illiquid CDSs

by Michael Walker of the University of Toronto

May 28, 2010

Abstract: CDS (credit default swap) contracts that were initiated some time ago frequently have spreads and/or maturities that are not available on the current market of CDSs, and are thus illiquid. This article introduces an incomplete-market approach to valuing illiquid CDSs that, in contrast to the commonly used complete-market risk-neutral approach to valuation, takes into account the risky nature of the illiquid CDS, and allows a dealer who buys an illiquid CDS from an investor to determine ask and bid prices (which differ) in such a way as to guarantee a minimum positive expected return on the deal. An alternative procedure, which replaces the expected return by an analogue of the Sharpe ratio, is also discussed. The approach to pricing just described belongs to the good-deal category of approaches, since the dealer decides what it would take to make an appropriate expected return, and sets the bid and ask prices accordingly. A number of different hedges are discussed and compared within the general framework developed in the article. These include the hedge that enforces the no-arbitrage bounds, a vanilla hedge making use of a single CDS from the market having the same maturity and notional as the illiqid CDS (but a different spread), and an optimal hedge that minimizes the capital at risk for the dealer (who is identified above), conditional on the dealer achieving a desired minimum expected return and on the bid or ask price for the transaction having a certain definite value. The approach is implemented numerically, and example plots of important quantities are given.

JEL Classification: G12, D52.

AMS Classification: 91G40.

Keywords: credit defaults swaps, CDSs, hedging, valuation, incomplete markets.

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