Seminario The Carr Copula

Abstract

The Carr copula is recovered by the dependence structure of the running maximum and the running drawdown of a standard Brownian motion with no drift. The copula is recovered from the reflection principle, it is Archimedean and at most bivariate, i.e. 2-monotone. The generator is the Williamson transform of the Maxwell-Boltzmann distribution. The copula has no parameters and the Kendall tau is equal to 1-4 /π, approximately-0.27324. In a generalization, the Azéma-Yor martingale and C-convolution allow to specify the dependence structure of  the running maximum and drawdown of any process. An invariance principle allows to extend the same copula and C-convolution analysis to the geometric process for the analysis of speculative price dynamics and option pricing applications based on the Azéma-Yor principle. An application to stock index data gives mixed results: once the innovation process is extracted from a CEV model with constant geometric drift, we find that dependence is quite consistent with the Carr copula for the S&P 500 index, while it is not for the Nasdaq indexes, for which the running maximum and drawdown are close to independent.