Sensitivity estimates for compound sums

Abstract

We derive unbiased derivative estimators for expectations of functions of random sums, where differentiation is taken with respect to a parameter of the number of terms in the sum. As the number of terms is integer valued, its derivative is zero wherever it exists. Nevertheless, we present two constructions that make the sum continuous even when the number of terms is not. We present a locally continuous construction that preserves continuity across a single change in the number of terms and a globally continuous construction specific to the compound Poisson case. This problem is motivated by two applications in finance: approximating Lévydriven models of asset prices and exact sampling of a stochastic volatility process. © Springer-Verlag Berlin Heidelberg 2009.