Variance Reduction Techniques

Abstract

The evaluation of the expectation of a given function of a solution of an SDE with jumps provides via the Feynman-Kac formula, see Sect. 2.7, the solution of a partial integro differential equation. In many applications it is of major interest to obtain numerically these expectations, in particular in multi-dimensional settings. Monte Carlo simulation appears to be a method that may be able to provide answers to this question under rather general circumstances. However, raw Monte Carlo estimates of the expectation of a payoff structure, for instance for derivative security prices, can be very expensive in terms of computer resource usage. In this chapter we investigate the problem of constructing variance reduced estimators for the expectation of functionals of solutions of SDEs that can speed up the simulation enormously. We follow again closely Heath (1995). As we will see, variance reduction is more of an art and can be applied in many ways. This chapter shall enable the reader to design her or his own variance reduction method for a given problem at hand.