Funding and Collateral in Counterparty Valuation and Derivatives Pricing
Summary
The paper develops a risk-neutral framework for pricing collateralized counterparty trades while accounting jointly for credit and debit valuation adjustments, margining, and funding costs. Its central pricing equation is recursive, so funding effects generally cannot be represented as a simple additive adjustment. The authors recast the problem as iterative relationships that can be solved with standard least-squares Monte Carlo methods.
The framework permits asymmetric collateral and funding rates, external liquidity and hedging policies, re-hypothecation risk, and close-out amount considerations, without imposing narrow assumptions on margining or netting. It argues that funding costs and debit valuation adjustments cannot generally be identified separately, and that cases where funding costs vanish rely on special assumptions. Examples connect the general setup to established discounting results. The document describes a modeling framework rather than empirical trading performance; conclusions depend on the chosen market conventions, policies, and risk-neutral assumptions.
Key ideas
- The pricing equation incorporates counterparty credit, debit, collateral, margining, and funding effects together.
- Recursive funding dependence makes a purely additive funding valuation adjustment difficult in general.
- Iterative pricing relationships can be computed using least-squares Monte Carlo techniques.
- Funding costs and debit valuation adjustments are not generally separable.
- Symmetric rates and restrictive margining assumptions are not required by the broader framework.
Tags
Full text
# Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments # Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments 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 risk-neutral 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.
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