Before revision:  In this article, we have extended the previous studies of collateralized derivative pricing to more generic situation, that is asymmetric and imperfect collateralization as well as the associated CVA. We have presented approximate expressions for various cases using Gateaux derivative which allow straightforward numerical analysis.
After revision:  In this article, we have extended the previous studies of collateralized derivative pricing to more generic situation, that is asymmetric and imperfect collateralization with the associated counter party credit risk. By introducing the collateral coverage ratio, our framework can handle these issues in an unified manner. Although the resultant pricing formula becomes non-linear FBSDE and cannot be solve exactly, the fist order approximation is provided using Gateaux derivative which allow straightforward numerical analysis.