Generalizing the Black–Scholes formula to multivariate contingent claims

René Carmona, Valdo Durrleman

Abstract

ABSTRACT

This paper provides approximate formulas that generalize the Black–Scholes formula in all dimensions. Pricing and hedging of multivariate contingent claims are achieved by computing lower and upper bounds. These bounds are given in closed form in the same spirit as the classical one-dimensional Black–Scholes formula. Lower bounds perform remarkably well. Similar to the one-dimensional case, Greeks are also available in closed form. We discuss an extension to basket options with barrier.