Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models
研究了Gram-Charlier展开在机制转换Lévy过程密度估计中的应用，通过矩阵指数表示计算各阶矩，并给出欧式期权定价的数值例子，对随机波动率的时间变换Lévy过程也做了类似分析。
The Gram–Charlier expansion of a target probability density, f(x), is an L2-convergent series f(x)=∑0∞cnpn(x)f∗(x) in terms of a reference density f∗(x) and its orthonormal polynomials 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) 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.