We develop a consistent, arbitrage-free framework for valuing derivative
trades with collateral, counterparty credit risk, and funding costs. Credit, debit, liquidity,
and funding valuation adjustments (CVA, DVA, LVA, and FVA) are simply
introduced as modifications to the payout cash-flows of 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 – all aspects which are often neglected. The
generalized valuation equation takes the form of a forward-backward SDE or semilinear
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 implement
numerically the case of an equity option and show how its valuation changes
when including the above effects.
In the paper 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 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 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. The value of a trade to a counterparty
will not be just the opposite of the value seen by the bank. 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.