Hedging of Defaultable Claims

The goal of these lectures is to present a survey of recent developments in the practically important and challenging area of hedging credit risk. In a companion work, Bielecki et al. (2004a), we presented techniques and results related to the valuation of defaultable claims. It should be emphasized that in most existing papers on credit risk, the risk-neutral valuation of defaultable claims is not supported by any other argument than the desire to produce an arbitrage-free model of default-free and defaultable assets. Here, we focus on the possibility of a perfect replication of defaultable claims and, if the latter is not feasible, on various approaches to hedging in an incomplete setting. These lecture notes are organized as follows. Chapter 1 is devoted to methods and results related to the replication of defaultable claims within the reduced-form approach (also known as the intensity-based approach). Let us mention that the replication of defaultable claims in the so-called structural approach, which was initiated by Merton (1973) and Black and Cox (1976), is entirely different (and rather standard), since the value of the firm is usually postulated to be a tradeable underlying asset. Since we work within the reduced-form framework, we focus on the possibility of an exact replication of a given de-faultable claim through a trading strategy based on default-free and defaultable securities. First, we shall analyze (following, in particular, Vaillant (2001)) various classes of self-financing trading strategies based on default-free and defaultable primary assets. Subsequently, we present various applications of general results to financial models with default-free and defaultable primary assets are given. We develop a systematic approach to replication of a generic defaultable claim, and we provide closed-form expressions for prices and replicating strategies for several typical defaultable claims. Finally, we present a few examples of repli-cating strategies for particular credit derivatives. In the last section, we present, by means of an example, the PDE approach to the valuation and hedging of defaultable claims within the framework of a complete model. In Chapter 2, we formulate a new paradigm for pricing and hedging financial risks in incomplete markets, rooted in the classical Markowitz mean-variance portfolio selection principle and first examined within the context of credit risk by Bielecki and Jeanblanc (2003). We consider an investor who is interested in dynamic selection of her portfolio, so that the expected value of her wealth at the end of the pre-selected planning horizon is …