Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea

Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E[g(B(·))], the expectation of a payoff functional, g, that depends on a Brownian motion, (B(t))t=0. The expectation corresponds to an infinite dimensional integral, which is approximated by the sample average of a d-dimensional approximation to the integrand. In this article, a multilevel algorithm with low discrepancy designs is used to improve the convergence rate of the worst case error with respect to a single level algorithm. The worst case error is derived as a function of each level l’s sample size, nl, and truncated dimension, dl, for payoff functionals that arise from certain Hilbert spaces with moderate smoothness. If the error in approximating an infinite dimensional expectation by a d-dimensional integral is O(d), and the error for approximating a d-dimensional integral by an n-term sample average is O(n), independent of d, then it is shown that the error in computing the infinite dimensional expectation may be as small as N for a well-chosen multilevel algorithm, where N , the cost of the algorithm is defined as N = n1d s 1 + · · · + nLd s L for some s ≥ 0. This optimal convergence rate is achieved for either small or large q for rank-1 lattice rule designs, or alternatively for Niederretier net designs for large q.