Double Kernel Estimation of Sensitivities

Abstract

In this paper we address the general issue of estimating the sensitivity of the expectation of a random variable with respect to a parameter characterizing its evolution. In finance, for example, the sensitivities of the price of a contingent claim are called the Greeks. A new way of estimating the Greeks has recently been introduced in Elie, Fermanian and Touzi (2007) through a randomization of the parameter of interest combined with nonparametric estimation techniques. In this paper we study another type of estimator that turns out to be closely related to the score function, which is well known to be the optimal Greek weight. This estimator relies on the use of two distinct kernel functions and the main interest of this paper is to provide its asymptotic properties. Under a slightly more stringent condition, its rate of convergence is the same as the one of the estimator introduced in Elie, Fermanian and Touzi (2007) and outperforms the finite differences estimator. In addition to the technical interest of the proofs, this result is very encouraging in the dynamic of creating new types of estimator for the sensitivities.