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 nth-best-of-N contract. The result is highly accurate and almost instantaneous.