A Note on Monte Carlo Greeks using the Characteristic Function

Abstract

We consider the derivation of generic Monte Carlo estimators for Greeks for (path-dependent) options with discontinuous payoffs in the case where only the characteristic function is known. In Kienitz (2008) we have shown how to derive such Greeks for a wide range of models under the assumption that the transition probability is known in closed form. Unfortunately, this is not always the case. For example when considering exponential Levy models with stochastic volatility such as the Variance Gamma model with a Gamma Ornstein-Uhlenbeck or CIR stochastic clock. The characteristic function in this case the density is only given through its characteristic function. We give an algorithm to compute the probability density from the characteristic function and show that computing the transition density in this way gives the same results as in Kienitz (2008) but works for very general models. In this paper we focus on the Variance Gamma model and the same model with a Gamma Ornstein-Uhlenbeck stochastic clock. Since the methods are very general we can cope with other complex models like the Normal Inverse Gaussian model, considering other types of stochastic clocks or other classes of models where the characteristic function is known.

Keywords: Monte Carlo, Greeks, Levy, Characterisitc Function, Fourier Transform, Proxy