CVA, FVA (and DVA?) with stochastic spreads. A feasible replication approach under realistic assumptions.

In this paper we explore the different components that should be incorporated in the price of uncollateralized derivatives. We do so by putting special focus on the hedge-ability of every term. In order to reflect the most realistic situation, we assume stochastic credit spreads for both counterparties. In such a framework, the counterparty acting as the hedger will be concerned about market risk (movements in the price of the underlying asset), both sources of the credit risk of the investor (spread changes and default event) and also his own credit risk. Regarding his own credit risk, we assume that the derivatives hedger has no incentive to hedge the change in value of the derivative upon his own default, since the hedger will not be exposed to this change in value. Nevertheless, we assume that the hedger has a strong incentive to hedge the changes in the derivative’s price due to changes in his credit spread curve, which is a source of risk that the derivatives hedger will be exposed to during the replication process. We also suggest a hedging strategy for this risk factor (spread changes of the investor) as we do for the other (market risk, spread changes and default event of the issuer). We conclude that under these assumptions CVA (a unilateral version of it that does not depend on the hedger’s funding curve) and FVA (a funding adjustment that does only depend on the investor’s default indicator and not on the hedger’s) are the only components to be incorporated in the price of financial derivatives.