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

The Gram–Charlier expansion of a target probability density, $ f(x) $ f(x), is an $ L_2 $ L2-convergent series $ f(x)=\sum _0^\infty c_np_n(x)f^*(x) $ f(x)=∑0∞cnpn(x)f∗(x) in terms of a reference density $ f^*(x) $ f∗(x) and its orthonormal polynomials $ p_n(x) $ pn(x). 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 $ f(x) $ 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) $ f∗(x) as normal with the same mean and variance as $ f(x) $ f(x) only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, $ f^*(x) $ f∗(x) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.