  The importance of collateralization through the change of funding cost is now
well recognized among practitioners. 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. We have
shown that it allows us to decompose the price of generic contract into three
parts: market benchmark, bilateral credit value adjustment (CVA), and the
collateral cost adjustment (CCA) independent from the credit risk. We have
studied each term closely, and demonstrated the significant impact of
asymmetric collateralization through CCA using the numerical examples.
