Markov switching and jump diffusion models with applications in mathematical finance.

摘要

In this thesis, we study some jump diffusion models with Markov switching and transition densities for Markov switching diffusion processes with and without an absorbing barrier. We work out some analytical results, which have useful applications in mathematical finance and other related fields. The first-passage time problem for a Markov switching model is also studied and European type options and lookback options are computed in closed-form as examples to show that these models can be applied in practice. We apply optimization methods and kernel smoothing techniques to produce some important numerical results that show that jump diffusion with Markov switching models successfully capture the empirical feature of the market implied volatility of stock prices. We also use a path integral approach for a two-state Markov switching diffusion model, and it turns out that the transition probability density is a weighted average of gaussian densities for this model. As we will see in this thesis, the models can be extended to the multi-state case, but two-state models have particular applicability in the sense of economic cycles - expansion and contraction. As an interesting application, the two-state Markov switching jump diffusion model can be used for modelling insurance surplus with pricing cycles. In this case, the ruin probability is easily obtained.;Keywords: Option Pricing, Path Integral Approach, Markov Regime Switching, Jump Diffusion, Implied Volatility, First-Passage Time, Risk Process, Ruin Probability