Numerical Valuation of High Dimensional Multivariate European Securities

We consider the problem of pricing a contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the discounted expected value of future cash flows under the modified risk-neutral information process. Although analytical solutions have been developed in the literature for a few particular option pricing problems, computing the arbitrage prices of securities under several sources of uncertainty is still an open problem in many instances. In this paper, we present efficient numerical techniques based upon Monte Carlo simulation for pricing European contingent claims depending on an arbitrary number of risk sources. We introduce in particular the method of quadratic resampling (QR), a new powerful error reduction technique for Monte Carlo simulation. Quadratic resampling can be efficiently combined with classical variance reduction methods such as importance sampling to further improve the accuracy of the estimate. Our numerical experiments show that the method is practical for pricing claims depending on up to one hundred underlying assets. We also describe an implementation of the method on a massively parallel supercomputer, yielding two orders of magnitude of performance improvement over the same implementation on a desktop workstation.