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

The Gram-Charlier expansion of a target probability density, f (x), is an L-2-convergent series f (x) = Sigma(infinity)(0) c(n)P(n)(x)f * (x) in terms of a reference density f* (x) and its orthonormal polynomials p(n) (x). We implement this for the density of a regime-switching Levy process at a given time horizon T. The main step is the evaluation of moments of all orders of f (x) 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 f * (x) as normal with the same mean and variance as f (x) only works for the regime-switching Black-Scholes model. Outside the scope of Black-Scholes, f* (x) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Levy processes modelling stochastic volatility.