Pricing and Hedging of Derivatives in Markov-Modulated Markets Through Benchmark Approach

Abstract

The aim of the thesis is to study the pricing and hedging problems for contingent claims for various Markov-modulated models through the benchmark approach. This approach is based on a speciDc benchmark portfolio known as the growth optimal portfolio (GOP). GOP has been obtained for diDerent market models using the stochastic control method. When used as a numeraire, GOP ensures that all the benchmarked price processes are supermartingales. Using this supermartingale nature of benchmarked price, a fair price has been deDned. Since the GOP is closely related to a martingale density or state-price density, it can be used as a tool to price derivatives in complete or incomplete markets under the real world probability measure itself. The major advantage of this method is that it can be used in situations where the conventional methods do not work. We consider in this thesis Markov-modulated jump-diDusion models, where the stock price processes are modulated by irreducible continuous-time Markov processes. A Markov- modulated market model incorporates the stochastic volatility in a simple and empirically tractable way. Fair prices for the contingent claims are derived using the benchmark approach. For the case of complete markets, the hedging strategies are determined using the martingale representation theorem. We also derive the FDollmer-Schweizer decomposition for incomplete market models to get the risk-minimizing hedging strategy. For each model, it has been shown that the pricing under the minimal-martingale measure, if it exists, is equivalent to the benchmarked fair price. We analyze the models under both the scenarios, one where all the information are available to the investor and the one where the investor can only observe the stock price processes (not the underlying uncertainties). In a market with incomplete (or partial or imperfect) information, the investor has to Dlter the unobservable processes from the available information. The markets with incomplete information is tackled in two ways. In one, the FDollmer-Schweizer decomposition assuming complete information is derived first.and its projection to the smaller Dltration (with incomplete information) is constructed to derive the hedging strategy. In another, we convert the market with incomplete information to a complete information case by using the innovation process method in Dltering theory and derive the hedging strategy using the FDollmer-Schweizer decomposition. The derivative securities (or contingent claims) that we consider are of two types: default-free and defaultable, mostly European-style derivatives. A Markov-modulated de- faultable Brownian market is considered and the defaultable contingent claims are priced with the intensity-based methodology. The recovery processes are assumed to have ran- dom payments at the default time as well as at the maturity of the claims. We consider two classes of models, one where the parameters are modulated by an observable Dnite state Markov process and another where the parameters are modulated by an unobserv- able Ornstein-Uhlenbeck process. The representation theorems for defaultable claims are established and, using this representation, the locally-risk-minimizing hedging strategies for the defaultable contingent claims are derived under the benchmark approach...