The study of dynamics for credit default risk by backward stochastic differential equation method

Abstract

In this paper, we discuss the dynamics of credit default risk for bilateral collateralized credit valuation adjusted (BCCVA) for counterparty credit risk with two positive collateral accounts by assuming default times for both investor and counterparty satisfying the density hypothesis, and the contagion risk between them being reflected by the density process. We first split the price process into three key parts, and then describe the dynamics of each part by using backward stochastic differential equations (BSDEs). As applications, we introduce the “double” Cox model, in which the BSDEs have their specific forms and thus results in this paper generalize and improve corresponding theoretic results in the existing literature. We also would like to point out that, in this paper, we do not pay attention to explore how the introduction of default contagion impacts with true market data for the BCCVA predicted by for a model without contagion versus a model with contagion, but will plan to conduct the study on the impact for BCCVA with or without contagion in a separate research project.