Efficient Quasi-Monte Carlo Methods for Multiple Integrals in Option Pricing

In the present paper we consider European style options with an exponential payoff function. The problem is transformed to evaluation of multidimensional integrals of the exponential function over the unit cube where the values of the parameters involved in the formula depend the values of the European options. We compare the performance of quasi-Monte Carlo methods based on lattice rules for multiple integrals up to 30 dimensions. The performance of a lattice rule depends on the choice of the generator vectors. When the integrand is sufficiently regular the lattice rules outperform not only the standard Monte Carlo methods, but also other types of methods using low discrepancy sequences. We consider "rank 1" rules whose lattices have a a single generator vector. The advantages and disadvantages of the different quasi-Monte Carlo methods for multidimensional integrals related to evaluation of European options are studied in the paper.