Nonlinear Derivative Valuation with Credit, Collateral and Funding Costs
Summary
This document presents a risk-neutral framework for pricing derivatives when collateral terms, counterparty credit exposure and funding costs affect cash flows. It incorporates credit, debit, liquidity and funding adjustments, and explains that asymmetric funding terms and default close-out rules can make the price recursive and nonlinear. Because common adjustments such as CVA and FVA may overlap, simply adding them can double count risk; the authors introduce a nonlinearity adjustment to address this issue.
The valuation is expressed as a backward stochastic differential equation or a semilinear partial differential equation, then reformulated as iterative equations suitable for least-squares Monte Carlo. A generalized Black–Scholes option case study finds that funding risk can materially affect prices and that double counting matters. The article also discusses central clearing, margin arrangements and broader effects of nonlinear pricing. Its numerical evidence comes from a case study, so the document does not establish universal price impacts for all products or market conventions.
Key ideas
- Collateral, counterparty credit risk and funding costs can change derivative cash flows and valuation.
- Asymmetric funding and close-out terms can make bilateral pricing nonlinear and recursive.
- CVA and FVA are not generally independent additive adjustments, creating a risk of double counting.
- The pricing equations can be solved iteratively with least-squares Monte Carlo.
- A generalized Black–Scholes case study finds nontrivial funding effects and highlights double counting.
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Full text
# Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes # Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes We develop an arbitrage-free framework for consistent valuation of derivative trades with collateralization, counterparty credit gap risk, and funding costs, following the approach first proposed by Pallavicini and co-authors in 2011. Based on the risk-neutral pricing principle, we derive a general pricing equation where Credit, Debit, Liquidity and Funding Valuation Adjustments (CVA, DVA, LVA and FVA) are introduced by simply modifying the payout cash-flows of the deal. Funding costs and specific close-out procedures at default break the bilateral nature of the deal price and render the valuation problem a non-linear and recursive one. CVA and FVA are in general not really additive adjustments, and the risk for double counting is concrete. We introduce a new adjustment, called a Non-linearity Valuation Adjustment (NVA), to address double-counting. The theoretical risk free rate disappears from our final equations. The framework can be tailored also to CCP trading under initial and variation margins, as explained in detail in Brigo and Pallavicini (2014). In particular, we allow for asymmetric collateral and funding rates, replacement close-out and re-hypothecation. The valuation equation takes the form of a backward stochastic differential equation or semi-linear partial differential equation, and can be cast as a set of iterative equations that can be solved by least-squares Monte Carlo. We propose such a simulation algorithm in a case study involving a generalization of the benchmark model of Black and Scholes for option pricing. Our numerical results confirm that funding risk has a non-trivial impact on the deal price, and that double counting matters too. We conclude the article with an analysis of large scale implications of non-linearity of the pricing equations.
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