Sum of All Black Scholes

This study considers pricing options under multivariate Black- Scholes models such as basket options, spread options and discretely monitored Asian options. We present an ecient multidimensional integration method against the common view that the exact pricing is
computationally prohibitive due to the curse of dimensionality. The key of our approach is to nd an optimal rotation of the factor matrix under which the rst dimension is approximately perpendicular to the exercise boundary. Then we integrate the option payo , analytically on the fi rst dimension (with a numerically solved exercise point) and with Gauss-Hermite quadratures on the remaining dimensions. The numerical examples show that the quadrature integration requires only a few points per dimension or can be even truncated for low-varying factors. In essence we express the price of an option on a weighted sum of asset prices as a weighted sum of a minimal but accurate set of Black-Schole-like prices, thereby generalizing the Black-Scholes formula to the multivariate cases.