Analytic Approximations for Multi-Asset Option Pricing

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option's price as a sum of prices of various compound exchange options, each with different pairs of sub-ordinate multi- or single-asset options. For some multi-asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single-asset option under a modified measure, and in such cases an almost exact analytic price exists for the multi-asset option. The underlying asset prices are assumed to follow lognormal processes, although the strong condition can be extended to certain other price processes for the underlying. More generally, approximate analytic prices for multi-asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the sub-ordinate basket options. The analytic formulae for multi-asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to sub-ordinate multi-asset options, and the deltas of these sub-ordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations, and we demonstrate how to calibrate these parameters to market data so that multi-asset option prices are consistent with the implied volatility and correlation skews of the assets.