Closed-Form Expression of Geometric Brownian Motion with Regime-Switching and Its Applications to European Option Pricing

Abstract

Mathematical difficulty remains in many classical financial problems, especially for a closed-form expression of asset value. The European option evaluation problem based on a regime-switching has been formally modeled since early 2000, for which a recursive algorithm was developed to solve it. The key mathematical difficulty of this problem relies on the expectation IE[h(YT)], where h is a payoff function and {Yt}t∈[0,T] denotes a geometric Brownian motion with Markovian regime-switching. It is long since attempted to conclude this problem with closed form formulas. Towards the same target, this paper applies some novel techniques to draw explicit formulas for cases with more states for regime-switching (whereas the former deals the cases with two states) for any integrable function h (whereas the former only applies to the payoff of a European option). This paper combines the technique of occupation time of Markov chains and inverse Laplace transform to achieve the density function of geometric Brownian motion with Markovian regime-switching. Extension along this technique creates potential for probabilistic computations in addition to European option pricing. The reflection from the inverse Laplace transform to the expression of a moment-generating function is the core technique developed by this paper, and it is presented in symmetric forms.