Numerical Valuation of High Dimensional Multivariate European Securities

Abstract

an acknowledgement of the authors and individual contributors to the work; and all applicable portions of the copyright notice. Copying, reproducing, or republishing for any other purpose shall require a license with payment of fee to the Paris Research Laboratory. All rights reserved. ii AbstractWe consider the problem of pricing an American 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 maximum over all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process.Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the number of risk sources.In this paper, we present an efficient numerical technique that combines Monte Carlo simulation with a particular partitioning method of the underlying assets space, which we call Stratified State Aggregation (SSA). Using this technique we can compute accurate approximations of prices of American securities with an arbitrary number of underlying assets. Our numerical experiments show that the method is practical for pricing American claims depending on up to 400 risk sources. On all problems for which we could compare the method with known optimal solutions, the price computed through stratified state aggregation was indistinguishable from the optimal theoretical price. Several numerical examples are presented and discussed.iii Acknowledgements This research benefited from discussions with Bruno Langlois. We wish to thank Thierry Pudet for reviewing an earlier draft.iv

Numerical Valuation of High Dimensional Multivariate American Securities

“…In particular, the method has been used for pricing mortgage-backed securities (see Schwartz and Torous (1989), Hutchinson and Zenios (1991)). Barraquand (1993) presents the method of Quadratic Resampling for Monte Carlo valuation of European securities with many underlying assets. The Quadratic Resampling method presented in this paper is an extension of this earlier work to the American pricing problem.…”

“…The computation time is proportional to Mn 2 d+k 2 d, the first term corresponding to the drawing of the M Monte Carlo sample paths, and the second to the backwards integration. Hence the memory and time complexities of the SSAP method are polynomial in n. This is to be contrasted with classical PDE methods which are exponential in n. Barraquand (1993) for reducing the variance of multivariate Monte Carlo integration.…”

Multi-Level Monte Carlo Simulations with Importance Sampling

“…Barraquand (1995) introduces quadratic resampling and combines it with the importance sampling. Avramidis (2002) proposes an algorithm that selects the importance sampling density as a mixture of multivariate Normal densities.…”

Basket Options on Heterogeneous Underlying Assets

“…In the numerical techniques category, Rubinstein (1994) has developed a multivariate binomial tree, but this is not a very efficient way to price basket options, as the number of nodes grows exponentially with the number of assets in the basket. Barraquand (1995) and Pellizzari (1998) have used Monte Carlo simulations with different variance-reduction techniques. The former used quadratic resampling and the latter developed two control variates based, respectively, on unconditional and conditional expectations of assets.…”