Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models

Abstract

The Gram–Charlier expansion of a target probability density, , is an -convergent series in terms of a reference density and its orthonormal polynomials . We implement this for the density of a regime-switching Lévy process at a given time horizon T. The main step is the evaluation of moments of all orders of in terms of model primitives, for which we give a matrix-exponential representation. A number of numerical examples, in part involving pricing of European options, are presented. The traditional choice of as normal with the same mean and variance as only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.