Unbiased and efficient Greeks of financial options

Abstract

The price of a derivative security equals the discounted expected payoff of the security under a suitable measure, and Greeks are price sensitivities with respect to parameters of interest. When closed-form formulas do not exist, Monte Carlo simulation has proved very useful for computing the prices and Greeks of derivative securities. Although finite difference with resimulation is the standard method for estimating Greeks, it is in general biased and suffers from erratic behavior when the payoff function is discontinuous. Direct methods, such as the pathwise method and the likelihood ratio method, are proposed to differentiate the price formulas directly and hence produce unbiased Greeks (Broadie and Glasserman, Manag. Sci. 42:269–285, 1996). The pathwise method differentiates the payoff function, whereas the likelihood ratio method differentiates the densities. When both methods apply, the pathwise method generally enjoys lower variances, but it requires the payoff function to be Lipschitz-continuous. Similarly to the pathwise method, our method differentiates the payoff function but lifts the Lipschitz-continuity requirements on the payoff function. We build a new but simple mathematical formulation so that formulas of Greeks for a broad class of derivative securities can be derived systematically. We then present an importance sampling method to estimate the Greeks. These formulas are the first in the literature. Numerical experiments show that our method gives unbiased Greeks for several popular multi-asset options (also called rainbow options) and a path-dependent option.