Paris-Princeton Lectures on Mathematical Finance 2003

The goal of this chapter 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.

In this chapter, which is a substantial extension of an earlier essay [3], we give an overview of some recent work on the geometric properties of the evolution of the forward rate curve in an arbitrage free bond market. The main problems to be discussed are as follows. We consider interest rate models of Heath-Jarrow-Morton type, where the forward rates are driven by a multidimensional Wiener process, and where he volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Within this framework we give necessary and sufficient conditions for consistency, as well as for the existence of a finite dimensional realization, in terms of the forward rate volatilities. We also study stochastic volatility HJM models, and we provide a systematic method for the construction of concrete realizations.

We survey recent developments in finance that analyze how heterogeneous beliefs among investors generate speculation and trading. We describe the joint effects of heterogeneous beliefs and short-sales constraints on asset prices, using both static and dynamic models, discuss the no-trade theorem in the rational expectations framework, and present investor overconfidence as a potential source of heterogeneous beliefs. We review recent results of Scheinkman and Xiong (2003) modeling the resale option that is embedded in share prices in the presence of short-sale constraints and heterogeneous beliefs, highlighting the implied correlation between stock prices and trading volume. Finally, we discuss the survival of investors with incorrect beliefs.