Abstracts Mathematics

Asset pricing and portfolio optimization under regime switching models

by Yang Shen

Recently, there has been a considerable interest in applications of regime-switching models in various aspects of finance and insurance. One of the main features of these models is that some model parameters are modulated by a finite-state Markov chain. This makes regime-switching models very useful to describe structural changes in macro-economic conditions, periodical fluctuations in business cycles and sudden transitions in market modes. In this thesis, a continuous-time, finite-state, observable Markov chain is adopted to model the regime switches. Our regime-switching models are a set of diffusion models, jump-diffusion models or Levy models coupled by the underlying Markov chain. Under this modeling set up, the financial market is incomplete. So asset pricing and portfolio optimization problems are more involved. Roughly speaking, this thesis can be divided into two parts. The first part is devoted to asset pricing problems under regime-switching models. Due to the market incompleteness, the equivalent martingale measure is not unique. Therefore, we either choose a particular equivalent martingale measure using the Esscher transform or start directly from a risk-neutral measure. We present analytical pricing formulae for European options and variance swaps in Chapters 2 and 3, respectively. Numerical and empirical implementations of these formulae show that the regime-switching effect is material for asset pricing problems. In the second part of this thesis, we apply the stochastic optimal control theory to discuss portfolio optimization problems under regime-switching models. In Chapter 4, we use the dynamic programming principle approach to solve a mean-variance portfolio selection problem with uncertain investment horizon. Explicit expressions of the efficient portfolio and the efficient frontier are obtained. In Chapter 5, the stochastic optimal control theory for portfolio optimization problems is borrowed to investigate an fundamental issue in asset pricing problems, i.e. the selection of equivalent martingale measures. We derive and compare equivalent martingale measures selected by three different approaches, that is, the stochastic differential game approach, the Esscher transformation approach and the general equilibrium approach.