Calibration of jump-diffusion option-pricing models: a robust non-parametric approach.∗

We present a non-parametric method for calibrating jump-diffusion models to a finite set of observed option prices. We show that the usual formulations of the inverse problem via nonlinear least squares are illposed and propose a regularization method based on relative entropy: we reformulate our calibration problem into a problem of finding a risk neutral jump-diffusion model that reproduces the observed option prices and has the smallest possible relative entropy with respect to a chosen prior model. Our approach allows to conciliate the idea of calibration by relative entropy minimization with the notion of risk neutral valuation in a continuous time model. We discuss the numerical implementation of our method using a gradient based optimization algorithm and show via simulation tests on various examples that the entropy penalty resolves the numerical instability of the calibration problem. Finally, we apply our method to data sets of index options and discuss the empirical results obtained. ∗Preliminary versions of this work were presented at the Bernoulli Society International Statistical Symposium (Taipei 2002), Maphysto (Aarhus & Copenhagen), University of Freiburg, University of Warwick and INRIA. We thank Marco Avellaneda, Frédéric Bonnans, Stephane Crépey, Dilip Madan and Monique Pontier for helpful remarks.