Kernel estimation of Greek weights by parameter randomization

Abstract

A Greek weight associated to a parameterized random variable $Z(λ)$ is a random variable $π$ such that $\nabla_λE[ϕ(Z(λ))]=E[ϕ(Z(λ))π]$ for any function $ϕ$. The importance of the set of Greek weights for the purpose of Monte Carlo simulations has been highlighted in the recent literature. Our main concern in this paper is to devise methods which produce the optimal weight, which is well known to be given by the score, in a general context where the density of $Z(λ)$ is not explicitly known. To do this, we randomize the parameter $λ$ by introducing an a priori distribution, and we use classical kernel estimation techniques in order to estimate the score function. By an integration by parts argument on the limit of this first kernel estimator, we define an alternative simpler kernel-based estimator which turns out to be closely related to the partial gradient of the kernel-based estimator of $\mathbb{E}[ϕ(Z(λ))]$. Similarly to the finite differences technique, and unlike the so-called Malliavin method, our estimators are biased, but their implementation does not require any advanced mathematical calculation. We provide an asymptotic analysis of the mean squared error of these estimators, as well as their asymptotic distributions. For a discontinuous payoff function, the kernel estimator outperforms the classical finite differences one in terms of the asymptotic rate of convergence. This result is confirmed by our numerical experiments.