Numerical Valuation of High Dimensional Multivariate American 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

The Stochastic Grid Bundling Method: Efficient pricing of Bermudan options and their Greeks

“…This method is motivated by the stratified state aggregation method by Barraquand and Martineau (1995) [3], where rather than partitioning the actual state space in which the value function resides, a reduced state space obtained by some mapping function is partitioned. We, like the authors of [3], use the payoff as the mapping function and bundle the grid points based on proximity of the reduced state space h(S tm−1 ).…”

“…We, like the authors of [3], use the payoff as the mapping function and bundle the grid points based on proximity of the reduced state space h(S tm−1 ). The bundling scheme is then similar to the recursive bifurcation, except that now the effective dimension d is equal to 1.…”

“…Another approach is based on approximating the transition probabilities using either bundling, as in Tilley (1990) [21], partitioning, as in Barraquand and Martineau (1997) [3] and Jin et al (2007) [14], or quantization, as in Bally and Pages (2004) [2], of the state space; or computing weights to approximate these conditional probabilities, as in the stochastic mesh method Broadie and Glasserman (2004) [6].…”

Pricing American Options: A Duality Approach

“…Tilley (1993) was the first to demonstrate that American options could be priced using simulation techniques. Other important work in this literature includes Barraquand and Martineau (1995), Carriere (1996), Raymar and Zwecher (1997), Ibanez and Zapatero (1999) and Garcia (1999). Longstaff and Schwartz (2001) and Van Roy (1999, 2000) have proposed an approximate dynamic programming approach that can compute good price estimates and is very fast in practice.…”

On the use and improvement of Hull and White's control variate technique

“…For example, Barraquand and Martineau (1995) and Longstaff and Schwartz (2001) have proposed a method for pricing multidimensional American options by Monte Carlo simulation. Given the flexibility for pricing exotic options under complex asset price processes, combining their methods with the control variate technique may provide a powerful tool for both academics and practitioners.…”