Pricing and Hedging in a GBM market with Markov switching: A survey

Abstract

This current thesis aims to survey recent development on certain problems in Mathematical Finance. The geometric Brownian motion model for stock price was rst proposed by the renowned economist Samuelson in 1965. Later in 1973 Black, Scholes and Merton used that model to nd a formula for price of European options. This work commences the application of Stochastic calculus in the research eld of quantitative nance. But this model assumes that the basic market parameters, namely, growth rate, volatility and bank interest rate remain constant during the entire period of the option. Numerical data from actual market does not support these assumptions. To overcome these drawbacks, several alternative models are still being proposed in the literature and thereby new mathematical challenges are arising. In recent years a large amount of research is being carried out by considering the market parameters as Markov chains which evolve according to a prescribed transition rate. Markov modulated GBM model is one of that kind. This model can be regarded as straight forward generalization of B-S-M (Black, Scholes and Merton) model. Although such market is proved to have no arbitrage, but the cost paid for this generalization includes features like incompleteness of market, lack of analytic solution, non-uniqueness of option pricing etc. Nevertheless, consideration of the above model opens up a wide range of research topics. The existing literature, those assume above model and related to locally risk minimizing pricing, optimal hedging, portfolio optimization with risk sensitive cost, stability of numerical solutions of associated PDEs, computation of complexity of numerical schemes etc. are thoroughly being surveyed in this current project. Besides, a number of numerical experiments are carried out based on the theoretical results. During thorough study of Springer lecture note on Introduction to stochastic Calculus for Finance" by Dieter Sondermann, as part of prerequisite, a list of errata along with few corrections/suggestion is prepared and enclosed to this thesis.