Computing High Dimensional Integrals with Applications to Finance

High-dimensional integrals are usually solved with Monte Carlo algorithms although theory suggests that low discrepancy algorithms are sometimes superior. We report on numerical testing which compares low discrepancy and Monte Carlo algorithms on the evaluation of nancial derivatives. The testing is performed on a Collateralized Mortgage Obligation (CMO) which is formulated as the computation of ten integrals of dimension up to 360. We tested two low discrepancy algorithms (Sobol and Halton) and two randomized algorithms (classical Monte Carlo and Monte Carlo combined with antithetic variables). We conclude that for this CMO the Sobol algorithm is always superior to the other algorithms. We believe that it will be advantageous to use the Sobol algorithm for many other types of nancial derivatives. Our conclusion regarding the superiority of the Sobol algorithm also holds when a rather small number of sample points are used, an important case in practice. We built a software system which runs on a network of heterogeneous workstations under PVM 3.2 (Parallel Virtual Machine). Since workstations are ubiquitous, this is a cost-e ective way to do very large computations fast. The measured speedup is at least :9N for N workstations, N 25. The software can also be used to compute high dimensional integrals on a single workstation. 2