Abstracts Mathematics

Regime-switching models with mean reversion and applications in option pricing

by Jie Yu

In option pricing the underlying stock price is traditionally assumed to follow a geometric Brownian motion. However it is observed that the stock prices often switch between geometric Brownian motion and mean reversion behaviors. The contributions in this dissertation include constructing a regime-switching model incorporating both geometric Brownian motion and mean reversion models in which the switching is determined by a finite state Markov chain. This model leads to an effective mathematical framework for studying the valuation of the corresponding financial derivatives. Then we use a PDE method as well as a viscosity solution method to solve the non-smooth boundary value problem and to characterize the pricing of European options. In the second part of this dissertation we obtain a closed-form pricing formula for European call options using a successive approximation approach. A stochastic approximation method is used to estimate parameters under this model. Numerical experiments are carried out to compare our results with that of Monte Carlo simulation. Our effort is also devoted to providing applications involving model calibration and prediction of stock market trends using option market data. Finally, the pricing of perpetual American put options under the mean reversion model is studied. We obtain a closed-form solution in this case.