Simulation and Calibration of Options Prices under a Levy-Type Stochastic Dynamic and Semi Markov Market Switching Regimes Processes

Abstract

This work mainly highlights the benefits of derivative pricing in a semi Markov switching market. We explore the main differences between Markov and Semi Markov regime switching models. The three main problems we deal with are, (1) historical parameter calibration through the recently developed LLGMM method, (2) effects of semi Markov parameters on option prices and (3) comparison of Heston model, semi Markov regime model and Markov regime model calibration performances over both sequential option price calibration and the overall implied volatility surface of the market. Employing the LLGMM method, parameters of the spot price process described by a linear Levy-type stochastic differential equation under semi-Markov structural perturbations are calibrated to observed prices. From Fourier space time stepping and Carr and Madan methods, risk neutral parameters of the spot price are calibrated and interpreted. American and European style vanilla and exotic option prices are simulated. The presented results are shown in the context of a piecewise constant semi-Markov intensity matrix approximating a Weibull intensity matrix. We obtain that under the conditional minimum entropy martingale measure, option prices predictably increase as the regime risk increases through the intensity matrix of the semi Markov process. Calibration and simulation results demonstrate noticeable effects of semi-Markov parameters on option prices and a demonstrably better calibration fit of the Black Scholes model over the entire volatility surface, in a market with semi Markov regimes.