CVA and FVA to Derivatives Trades Collateralized by Cash
29 Pages Posted: 5 Feb 2013 Last revised: 28 Jul 2015
Date Written: July 28, 2015
In this article, we consider replication pricing of derivatives that are partially collateralized by cash. We let the issuer replicate the derivatives payout using shares and cash, and let the buyer replicate the the loss given the counterparty default using credit default swaps. The costs of funding for replication and collateral posting are accounted for in the pricing process. A partial differential equation (PDE) for the derivatives price is established, and its solution is provided in a Feynman-Kac formula, which decomposes the derivatives value into the risk-free value of the derivative plus credit valuation adjustment (CVA) and funding valuation adjustment (FVA). For most derivatives, we show that CVAs can be evaluated analytically or semi-analytically, while FVAs as well as the derivatives values can be solved recursively through numerical procedures due to their interdependence. In numerical demonstrations, continuous and discrete margin revisions are considered, respectively, for an equity call option and a vanilla interest-rate swap.
Keywords: CVA, FVA, default risk, funding risk, derivatives
JEL Classification: C51, C61
Suggested Citation: Suggested Citation