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Abstract: A Greek weight associated to a parameterized random variable $Z\lambda$ isa random variable $\pi$ such that$ abla {\lambda}E\phiZ\lambda=E\phiZ\lambda\pi$ for any function$\phi$. The importance of the set of Greek weights for the purpose of MonteCarlo simulations has been highlighted in the recent literature. Our mainconcern 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 thedensity of $Z\lambda$ is not explicitly known. To do this, we randomize theparameter $\lambda$ by introducing an a priori distribution, and we useclassical kernel estimation techniques in order to estimate the score function.By an integration by parts argument on the limit of this first kernelestimator, we define an alternative simpler kernel-based estimator which turnsout to be closely related to the partial gradient of the kernel-based estimatorof $\mathbb{E}\phiZ\lambda$. Similarly to the finite differencestechnique, and unlike the so-called Malliavin method, our estimators arebiased, but their implementation does not require any advanced mathematicalcalculation. We provide an asymptotic analysis of the mean squared error ofthese estimators, as well as their asymptotic distributions. For adiscontinuous payoff function, the kernel estimator outperforms the classicalfinite differences one in terms of the asymptotic rate of convergence. Thisresult is confirmed by our numerical experiments.

Author: Romuald Elie, Jean-David Fermanian, Nizar Touzi


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