Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea

Abstract. 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))Tt=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 discrep-ancy 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 di-mension, dl, for payoff functionals that arise from certain Hilbert spaces with moderate smoothness. If the error in approximating an infinite di-mensional expectation by a d-dimensional integral is O(d−q), and the error for approximating a d-dimensional integral by an n-term sample average is O(n−p), independent of d, then it is shown that the error in computing the infinite dimensional expectation may be as small as N−min(p,q/s) 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