A consistent framework for valuation under collateralization, credit risk and funding costs

Abstract

We develop a consistent, arbitrage-free framework for valuing derivative trades with collateral, counterparty credit risk, and funding costs. This is achieved by modifying the payout cash-flows for the trade position. The framework is flexible enough to accommodate actual trading complexities such as asymmetric collateral and funding rates, replacement close-out, and rehypothecation of posted collateral. We show also how the traditional self-financing condition is adjusted to reflect the new market realities. The generalized valuation equation takes the form of a forward-backward SDE or semi-linear PDE. Nevertheless, it may be recast as a set of iterative equations which can be efficiently solved by our proposed least-squares Monte Carlo algorithm. We numerically implement the case of an equity option and show how its valuation changes when including the above effects. We also discuss the financial impact of the proposed valuation framework and of nonlinearity more generally. This is fourfold: Firstly, the valuation equation is only based on observable market rates, leaving the value of a derivatives transaction invariant to any theoretical risk-free rate. Secondly, the presence of funding costs and default close-out makes the valuation problem a recursive and nonlinear one. Thus, credit and funding risks are non-separable in general, and despite common practice in banks, the related CVA, DVA, and FVA cannot be treated as purely additive adjustments without running the risk of double counting. To quantify the valuation error that can be attributed to double counting, we introduce a nonlinearity valuation adjustment (NVA) and show that its magnitude can be significant under asymmetric funding rates and replacement close-out at default. Thirdly, as trading parties cannot observe each others liquidity policies nor their respective funding costs, the bilateral nature of a derivative price breaks down. Finally, valuation becomes aggregation-dependent and portfolio values cannot simply be added up. This has operational consequences for banks, calling for a holistic, consistent approach across trading desks and asset classes.