First-passage Time Of Brownian Motion With Regime Switching And Application In Finance

The first-passage time is the moment at which a random variable firstly crosses a given threshold number, has an important role in the stochastic theory. On the study of financial problems, especially of pricing, hedging and risk management of those pass-dependent derivative, a lot of these problems can be based on the research of first-passage time. Thus, first-passage time is vital in the finance field. Otherwise, classical Black-Scholes model has some defects, such as it can not explain the "volatility smile" phenomenon. To overcome this disadvantage, a series of expanded models are proposed, for example, Regime Switching model has been extensively used in the finance field. Therefore, the study on first-passage time of Brownian motion with Regime Switching is meaningful in both theory and applications.There are 3 main approaches to study the first-passage time-martingale method, renewal-type equations and Wiener-Hopf factorization. In this paper, Wiener-Hopf fac-torization is used in discussing the first-passage time of Brownian motion with 2-state Regime Switching. By means of the Wiener-Hopf factors, this paper lastly obtains the explicit expression of Laplace transform of the first-passage time.Lookback option is a kind of exotic options, whose payoffs depend on all underly-ing asset price attained during the option’s life. As the feature, it’s more complex to pricing lookbak options. This paper applies the conclusion of the first-passage time to price lookback call option with floating strike price under 2-state Regime Switching. The Laplace transform of the option price can be easily obtained with the Laplace transform of the responding first-passage time. Then, with the numerical inversion of Laplace trans-forms, the numerical solution can be obtained. Because option price is always on real line, Gaver-Stehfest algorithm is more suitable. Thus, this paper chooses Gaver-Stehfest algorithm to do numerical simulation. From the numerical simulation, this paper gives out the numerical solution, the relationship between option price and option’s life, and the relationship between option price and volatility of underlying asset price.