Evaluating Stochastic Correlation Processes and Copulas, Comparing Dependence Structures: Mean-Reverting vs. Constant Correlation Models

Abstract

The linear correlation coefficient is a widely used dependence measure in the financial world. Another measure that is gaining popularity is the lower tail dependence coefficient, which indicates the likelihood that, given one return is far in the left tail of its distribution, another return will also be far in the left tail of its distribution. A high coefficient can be interpreted as a high-risk situation, which requires more attentiveness. However, a challenge is that tail dependence is difficult to compute numerically, making it beneficial to derive the distribution of this measure analytically.

Alfonsi and Ahdida introduced a mean-reverting correlation (MRC) process, which shares similar dynamics with the one-dimensional Jacobi process. This process will be used to introduce tail dependence. We will cover an in-depth explanation of the MRC process, detailing its properties and its connection to the Jacobi process. We will also discuss the estimation method used for the MRC process and address its simulation limitations, particularly those arising from Euler discretization with projection.

To address these limitations, we explore alternative methods for generating stochastic correlation dynamics, with a focus on copulas. We assess the dependence metrics of the MRC process compared to constant correlation models, which are represented by a Gaussian copula. Our analysis aims to determine whether the copula of the MRC process deviates from the Gaussian copula and to investigate the implications of parameter variations on this comparison.