Funding Valuation Adjustment: A Consistent Framework Including CVA, DVA, Collateral, Netting Rules and Re-Hypothecation

Abstract

In this paper we describe how to include funding and margining costs into a risk-neutral pricing framework for counterparty credit risk. We consider realistic settings and we include in our models the common market practices suggested by the ISDA documentation without assuming restrictive constraints on margining procedures and close-out netting rules. In particular, we allow for asymmetric collateral and funding rates, and exogenous liquidity policies and hedging strategies. Re-hypothecation liquidity risk and close-out amount evaluation issues are also covered.We define a comprehensive pricing framework which allows us to derive earlier results on funding or counterparty risk. Some relevant examples illustrate the non trivial settings needed to derive known facts about discounting curves by starting from a general framework and without resorting to ad hoc hypotheses.Our main result is a bilateral collateralized counterparty valuation adjusted pricing equation, which allows to price a deal while taking into account credit and debt valuation adjustments along with margining and funding costs in a coherent way. We find that the equation has a recursive form, making the introduction of an additive funding valuation adjustment difficult. Yet, we can cast the pricing equation into a set of iterative relationships which can be solved by means of standard least-square Monte Carlo techniques.

Analysis of Nonlinear Valuation Equations Under Credit and Funding Effects

“…(Funding valuation adjustment: a consistent framework including CVA, DVA, collateral, netting rules and re-hypothecation, 2011, [11]). These equations take the form of semi-linear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs).…”

“…The equations' derivations, their numerical solutions, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including [3,5,[10][11][12], and the monograph [6], which further summarizes earlier credit and debit valuation adjustment (CVA and DVA) results. We refer to such works and references therein for a general introduction to comprehensive nonlinear valuation and to the related issues with valuation adjustments related to credit (CVA), collateral (LVA), and funding costs (FVA).…”

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AbstractWe study conditions for existence, uniqueness, and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al(Funding valuation adjustment: a consistent framework including CVA, DVA, collateral, netting rules and re-hypothecation, 2011, [11]). These equations take the form of semi-linear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs).

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Negative Basis Measurement: Finding the Holy Scale

“…Generally speaking, this stream of literature reconsiders the pricing of derivatives under the new post-crisis fundamentals regarding funding, liquidity, and credit risk issues. Substantial contributions have been made, among others, by [5,7,12,13,23,27,29]. Loosely speaking, most references agree upon the fact that, at least under certain simplifying assumptions (full, bilateral, and continuous collateralization), derivative contracts can be evaluated in the traditional way, only the involved discount factors have to be adjusted by means of a spread accounting for funding and liquidity charges.…”

Efficient Option Pricing Under Levy Processes, with CVA and FVA

“…Debates for funding valuation adjustment continue in Burgard and Kjaer [2] Hull and White [14], Castagna [15]. Further generalizations appear in Burgard and Kjaer [16,17], Lu and Juan [18], Wu [19] which resort to a PDE approach, while Brigo et al [20,21], Pallavicini [22], Brigo and Pallavicini [23], Pallavicini et al [24], Brigo and Pallavicini [25], Brigo [26] use the no-arbitrage approach. The models in these papers are of the reduced type.…”