Monte Carlo Simulation of Stochastic Integrals when the Cost of Function Evaluation Is Dimension Dependent

Abstract

In mathematical finance, pricing a path-dependent financial derivative, such as a continuously monitored Asian option, requires the computation of \(\mathbb{E}[g(B(\cdot))]\), the expectation of a payoff functional, g, of a Brownian motion, B(t). The expectation problem is an infinite dimensional integration which has been studied in 1, 5, 7, 8, and 10. A straightforward way to approximate such an expectation is to take the average of the functional over n sample paths, B 1,…,B n . The Brownian paths may be simulated by the Karhunen-Loeve expansion truncated at d terms, \(\hat{B}_{d}\). The cost of functional evaluation for each sampled Brownian path is assumed to be \({\mathcal{O}}(d)\). The whole computational cost of an approximate expectation is then \({\mathcal{O}}(N)\), where N=nd. The (randomized) worst-case error is investigated as a function of both n and d for payoff functionals that arise from Hilbert spaces defined in terms of a kernel and coordinate weights. The optimal relationship between n and d given fixed N is studied and the corresponding worst-case error as a function of N is derived.