This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X t for small t.
We present a non-parametric method for calibrating jump-diffusion and, more generally, exponential Lévy models to a finite set of observed option prices. We show that the usual formulations of the inverse problem via non-linear least squares are ill-posed and propose a regularization method based on relative entropy: we reformulate our calibration problem into a problem of finding a risk-neutral exponential Lévy model that reproduces the observed option prices and has the smallest possible relative entropy with respect to a chosen prior model. Our approach allows us to reconcile the idea of calibration by relative entropy minimization with the notion of risk-neutral valuation in a continuoustime model. We discuss the numerical implementation of our method using a gradient-based optimization algorithm and show by 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.
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