An Analytical Method for Multi-Asset Option Pricing Based on a Single-Factor Model

Abstract

Securities with payoffs determined by multiple correlated stochastic factors present some of the hardest valuation problems in derivatives. Except for rare cases in which a closed-form solution is available, the “curse of dimensionality” causes serious problems for Monte Carlo simulation and other numerical methods as the number of assets, N , grows. This article presents an approach that can simplify such problems enormously with relatively little lost in terms of accuracy for common cases. The idea is quite simple: Model each variable’s risk as being composed of exposure to a single common factor (e.g., the market portfolio) plus an independent idiosyncratic shock. With this assumption, the distribution of the payoff then becomes a realization of the single common factor, which can be easily simulated numerically, plus a draw from the composite density for the sum of the independent idiosyncratic shocks. The article illustrates the approach for pricing rainbow options and the n th-best-of- N contract. The result is highly accurate and almost instantaneous.