Replication & XVA: The Unique Counting Approach

Abstract

In the large context of valuation adjustments, collectively referred to as “XVA”, we treat replication strategies as a unique way to calculate fair price equations. We consider different replication strategies (Piterbarg, Burgard-Kjaer and our contribution) and identify the valuation adjustments for the different pricing equations. We reduce a (large) list of pricing PDE’s to two main cases and calculate the XVAs for each case. We also consider the funding cost counting issue raised by Hull and White and treat it mathematically. This issue relates to the fact that any funding cost adjustment associated with unsecured funding should, in theory, be offset by a corresponding debit valuation adjustment. The framework is generalized to allow for stochastic hazard rates, as has been treated by others, and we also consider the case where collateralized vanilla derivatives are used as hedging instruments, demonstrating that there is no impact upon pricing outcomes. The special case of a “back-to-back” hedge with another institution is examined, and it is seen that the funding valuation adjustment is driven by differences between the CSA with the trade counterparty and the CSA with the hedge counterparty. We also provide numerical results for an interest rate swap and a related Bermudan swaption.