Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models. Issue 4 (3rd April 2022) - Asmussen, Søren; Bladt, Mogens
Abstract : The Gram–Charlier expansion of a target probability density, f ( x ), is an L 2 -convergent series f ( x ) = ∑ 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 Lévy 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 Lévy processes modelling stochastic volatility.