Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks

Abstract

The introduction of Central Clearing Counterparties (CCPs) in most derivative transactions will dramatically change the landscape of derivatives pricing, hedging and risk management, and, according to the TABB Group, will lead to an overall liquidity impact of about USD 2 trillions. In this paper, we develop for the first time a comprehensive approach for pricing under CCP clearing, including variation and initial margins, gap credit risk and collateralization, showing concrete examples for interest rate swaps. This framework stems from our 2011 framework on credit, collateral and funding costs in Pallavicini et al. (Pallavicini, A., D. Perini and D. Brigo, 2011, Funding Valuation Adjustment: FVA consistent with CVA, DVA, WWR, Collateral, Netting and Re-hypothecation, arxiv.org, ssrn.com). Mathematically, the inclusion of asymmetric borrowing and lending rates in the hedge of a claim, and a replacement closeout at default, lead to nonlinearities showing up in claim dependent pricing measures, aggregation dependent prices, nonlinear Partial Differential Equations (PDEs) and Backward Stochastic Differential Equations (BSDEs). This still holds in presence of CCPs and CSA. We introduce a modeling approach that allows us to enforce rigorous separation of the interconnected nonlinear risks into different valuation adjustments where the key pricing nonlinearities are confined to a funding costs component that is analyzed through numerical schemes for BSDEs. We present a numerical case study for Interest Rate Swaps that highlights the relative size of the different valuation adjustments and the quantitative role of initial and variation margins, of liquidity bases, of credit risk, of the margin period of risk and of wrong-way risk correlations.