MULTILEVEL MONTE CARLO METHODS IN FINANCE.

Abstract

This project brings the frontier of the Multilevel literature to the context of option pricing. A key element is the computation of complex high-dimensional integrals when evaluating the expected payoff. Such integrals often do not have closed form solutions, hence requiring numerical evaluation. We adopt novel Monte Carlo methods to estimate these integrals. We examine the reduction in variance by comparing the ordinary Monte Carlo (MC) Method against Multilevel Monte Carlo (MLMC). MLMC is implemented by sampling coupled discretised di usion processes. The variance of the MLMC estimate is given by O( T ) where T is the time to maturity and is some constant greater than 1. To reduce the variance to O(T), we employ Particle Filters and the Multilevel Particle Filter (MLPF). We then explore using Transport Maps to approximate our target distribution, which we postulate has a variance of O(T). Finally, we evaluate the methods on the S&P500 index.