Nonlinear Valuation Under Collateralization, Credit Risk, and Funding Costs - Brigo,D; Liu,Q; Pallavicini,A; Sloth,D (Springer, 2016)
We develop a consistent, arbitrage-free framework for valuing derivativetrades with collateral, counterparty credit risk, and funding costs. Credit, debit, liquidity,and funding valuation adjustments (CVA, DVA, LVA, and FVA) are simplyintroduced as modifications to the payout cash-flows of the trade position.The framework is flexible enough to accommodate actual trading complexitiessuch as asymmetric collateral and funding rates, replacement close-out, and rehypothecationof posted collateral – all aspects which are often neglected. Thegeneralized valuation equation takes the form of a forward-backward SDE or semilinearPDE. Nevertheless, it may be recast as a set of iterative equations which can beefficiently solved by our proposed least-squares Monte Carlo algorithm. We implementnumerically the case of an equity option and show how its valuation changeswhen including the above effects.In the paper we also discuss the financial impact of the proposed valuation frameworkand of nonlinearity more generally. This is fourfold: Firstly, the valuationequation is only based on observable market rates, leaving the value of a derivativestransaction invariant to any theoretical risk-free rate. Secondly, the presenceof funding costs makes the valuation problem a highly recursive and nonlinear one.Thus, credit and funding risks are non-separable in general, and despite common practice in banks, CVA, DVA, and FVA cannot be treated as purely additive adjustmentswithout running the risk of double counting. To quantify the valuation errorthat can be attributed to double counting, we introduce a ’nonlinearity valuation adjustment’(NVA) and show that its magnitude can be significant under asymmetricfunding rates and replacement close-out at default. Thirdly, as trading parties cannotobserve each others’ liquidity policies nor their respective funding costs, the bilateralnature of a derivative price breaks down. The value of a trade to a counterpartywill not be j