  A Greek weight associated to a parameterized random variable $Z(\lambda)$ is
a random variable $\pi$ such that
$\nabla_{\lambda}E[\phi(Z(\lambda))]=E[\phi(Z(\lambda))\pi]$ for any function
$\phi$. 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 $Z(\lambda)$ is not explicitly known. To do this, we randomize the
parameter $\lambda$ 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 $\mathbb{E}[\phi(Z(\lambda))]$. 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.
