Funding, Collateral and Hedging: Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustments

Abstract

The main result of this paper is a collateralized counterparty valuation adjusted pricing equation, which allows to price a deal while taking into account credit and debit valuation adjustments (CVA, DVA) along with margining and funding costs, all in a consistent way. Funding risk breaks the bilateral nature of the valuation formula. We find that the equation has a recursive form, making the introduction of a purely additive funding valuation adjustment (FVA) 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. As a consequence, we find that identifying funding costs and debit valuation adjustments is not tenable in general, contrary to what has been suggested in the literature in simple cases. The assumptions under which funding costs vanish are a very special case of the more general theory.We define a comprehensive framework that allows us to derive earlier results on funding or counterparty risk as a special case, although our framework is more than the sum of such special cases. We derive the general pricing equation by resorting to a riskneutral approach where the new types of risks are included by modifying the payout cash flows. We consider realistic settings and include in our models the common market practices suggested by 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. Finally, relevant examples of non-trivial settings illustrate how to derive known facts about discounting curves from a robust general framework and without resorting to ad hoc hypotheses.

Nonlinearity Valuation Adjustment

“…Full mathematical subtleties are left for other papers and may motivate slightly different versions of the cash flows, see for example Brigo et al [15]. More details on the origins of the cash flows used here are in Pallavicini et al [33,34].…”

“…Morini and Prampolini [31], Fries [25] and Castagna [19] consider basic implications of funding in presence of default risk. However, the most comprehensive attempts to develop a consistent valuation framework are those of Burgard and Kjaer [16,17], Crépey [21][22][23], Crépey et al [24], Pallavicini et al [33,34], and Brigo et al [13,14].…”

“…We follow the works of Pallavicini et al [34], Brigo et al [13,14], and Sloth [37] and consider a general valuation framework that fully and consistently accounts for collateralization, counterparty credit risk, and funding risk when pricing a derivatives trade. We find that the precise patterns of funding-adjusted values depend on a number of factors, including the asymmetry between borrowing and lending rates.…”

Analysis of Nonlinear Valuation Equations Under Credit and Funding Effects

“…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).…”

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.…”