Derivative pricing under asymmetric and imperfect collateralization and CVA

The importance of collateralization through a change of funding cost is now well recognized among practitioners. In this article, the authors have extended previous studies of collateralized derivative pricing to more generic situations, i.e. asymmetric and imperfect collateralization with the associated counterparty credit risk. By introducing the collateral coverage ratio, their framework can handle these issues in a unified manner. Although the resultant pricing formula becomes a nonlinear forward--backward stochastic differential equation and cannot be solved exactly, its first-order approximation is provided using the Gateaux derivative. The authors have shown that it allows one to decompose the price of a generic contract into three parts: a market benchmark, a bilateral credit value adjustment (CVA), and a collateral cost adjustment (CCA) independent of the credit risk. Each term is studied closely, and the significant impact is demonstrated of asymmetric collateralization through CCA using numerical examples.