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 E[g(B(·))], 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, B1, . . .,Bn. The Brownian paths may be simulated by the Karhunen-Lóeve expansion truncated at d terms, B̂d . The cost of functional evaluation for each sampled Brownian path is assumed to be O(d). The whole computational cost of an approximate expectation is then 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. © Springer-Verlag Berlin Heidelberg 2009.