Short Course: Dynamic Copula - Part I: Basics

Umberto Cherubini and Sabrina Mulinacci (University of Bologna, Italy)

Abstract: Copula functions represent the most general tool to describe the dependence structure among random variables. Almost all applications of copula functions to Finance, both in asset pricing and risk management, have been devoted to modeling in a static setting the dependence across markets, Financial products and risk factors. Only recently copula functions have been used to model dynamic dependence, that is the dependence across time in a stochastic process. Having in mind financial applications, relevant restrictions on the classes of stochastic processes to be considered must be imposed: the most typical ones are the Markov property and the martingale requirement.

This short course is based on Cherubini et al. (2012). After introducing the basic notion and the main properties of copula functions, we will analyze the seminal result due to Darsow et al. (1992): it allows to express the Markov property in terms of a specific relation among the copula functions representing the dependence structure among the random variables of the stochastic process at different times. Therefore, one can implement a technique to build stochastic processes modeling the increments and the dependence structure between levels and increments in order to disentangle processes with independent and dependent increments. This technique turns out to be well suited to provide a discrete time representation of the dynamics of innovations to financial prices under the restrictions imposed by the efficient market hypothesis and the martingale condition.

References:

Cherubini, U., Gobbi, F., Mulinacci, S. and S. Romagnoli (2012). Dynamic Copula
Methods in Finance. Wiley

Darsow, Z.F., Nguyen, B. and E.T. Olsen (1992). Copulas and Markov Processes,
Illinois Journal of Mathematics 36, 600-642.