Pricing American Option Using A Modified Fractional Black–Scholes Model Under Multi-State Regime Switching

The American option pricing problem is examined in this work using a regime switching finite moment log-stable model. The option prices under this model are governed by a coupled system of fractional partial differential equations. Combination of the coupled system and the spatial-fractional derivative makes it extremely difficult to find an analytic solution. We have constructed a numerical algorithm to numerically solve such problems. The developed predictor-corrector type method is highly efficient and reliable in solving coupled system in each regime having different volatility and interest rates. Two-sided Riesz space fractional diffusion term is approximated using fractional finite difference scheme whereas the classical space derivative term is approximated using central difference formula. Splitting technique is utilized to construct a highly efficient scheme which can also be implemented on parallel processors. Stability and error analysis of the scheme is proved analytically and demonstrated through numerical experiments. Effect of the order of the fractional derivative (also called tail index) on the option prices is shown through graphs by performing numerical experiments for different values of the tail index.