Pricing Index Options in a Multivariate Black & Scholes Model

The price of an index option depends on the distributions of the marginals and the dependence structure. In this paper we assume that the stocks composing the index can be described by a multivariate Black & Scholes model, which is the most straightforward extension of the one-dimensional Black & Scholes model. In this multivariate stock price model, vanilla option prices can be determined in closed form, but the price of an index option as well as the cdf of the index cannot be determined analytical. In this paper, we use the theory of comonotonicity to derive approximate index option prices which can be determined using the Black & Scholes formula. Furthermore, we show that the approximate index option curve can be interpreted as an index option curve of a modified, synthetic, index and the cdf of this synthetic index is given in closed form