New methodologies for valuing derivatives

Abstract

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 financial 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 financial 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 have built a software system called FINDER for computing high dimensional integrals. FINDER runs on a heterogeneous network of workstations under PVM 3.2 (Parallel Virtual Machine). Since workstations are ubiquitous, this is a cost-effective way to do large computations fast. Of course, FINDER can also be used to compute high dimensional integrals on a single workstation. A routine for generating Sobol points may be found, for example, in Numerical Recipes in C, Second Edition, by Press et al. However, we incorporated major improvements in FINDER and we stress that the results reported in this paper were obtained using FINDER. One of the improvements was developing the table of primitive polynomials and initial direction numbers for dimensions up to 360.