Martingale representation for contingent claims with regime switching

Abstract

We derive a martingale representation for a contingent claim under a Markov-modulated version of the Black-Scholes economy. The mar- tingale representation for the price of the claim is established with respect to an equivalent martingale measure chosen by the Esscher transform. Under some dieren tiability conditions for the coecien ts of the price processes, we shall identify explicitly the integrands in the martingale representation us- ing stochastic o ws. We shall introduce a zero-coupon bond to minimize the residual risk due to incomplete hedging. Recently, regime-switching models have played an important role in dieren t branches of modern nancial economics. The origin of regime-switching models in econometrics goes back to the original work of Hamilton (1989) in which a discrete-time Markov-switching autoregressive time series models was proposed. Applications of regime-switching models penetrate dieren t areas in modern - nancial economics. Some works on these applications include Elliott and van der Hoek (1997) for asset allocation, Pliska (1997), and Elliott, Hunter and Jamieson (2001) for short rate models, Elliott and Hinz (2002) for portfolio analysis and chart analysis, Niak (1993), Guo (2001) and Bungton and Elliott (2002a,b) for option valuation and Elliott, Malcolm and Tsoi (2003) for volatility estimation. The representation of martingales as stochastic integrals and the identication of the integrand in the representation are important topics in stochastic calculus and its applications. In mathematical nance, the integrand in the martingale representation of the discounted price process of a contingent claim may be used to construct a hedging policy for the claim. Colwell, Elliott and Kopp (1991) use the martingale representation theorem and stochastic o ws to identify a hedging policy for a contingent claim in the context of a multi-dimensional diusion model. Colwell and Elliott (1993) use a martingale representation result to construct the local risk-minimizing strategy explicitly. Their result provides an alternative mo- tive for the concept of the minimal martingale measure. In this paper, we shall derive a martingale representation for a contingent claim written on a risky asset under a Markov-modulated version of the Black-Scholes economy. We assume that the market parameters, including the market interest 2000 Mathematics Subject Classic ation. Primary 91B70; Secondary 91B28.