Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Chair in Mathematical Finance







805Weeks BuildingSouth Kensington Campus






BibTex format

author = {Brigo, D and Liu, Q and Pallavicini, A and Sloth, D},
title = {Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes},
url = {},
year = {2014}

RIS format (EndNote, RefMan)

AB - We develop an arbitrage-free framework for consistent valuation of derivativetrades 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 pricingequation where Credit, Debit, Liquidity and Funding Valuation Adjustments (CVA,DVA, LVA and FVA) are introduced by simply modifying the payout cash-flows ofthe deal. Funding costs and specific close-out procedures at default break thebilateral nature of the deal price and render the valuation problem anon-linear and recursive one. CVA and FVA are in general not really additiveadjustments, and the risk for double counting is concrete. We introduce a newadjustment, called a Non-linearity Valuation Adjustment (NVA), to addressdouble-counting. The theoretical risk free rate disappears from our finalequations. The framework can be tailored also to CCP trading under initial andvariation margins, as explained in detail in Brigo and Pallavicini (2014). Inparticular, we allow for asymmetric collateral and funding rates, replacementclose-out and re-hypothecation. The valuation equation takes the form of abackward stochastic differential equation or semi-linear partial differentialequation, and can be cast as a set of iterative equations that can be solved byleast-squares Monte Carlo. We propose such a simulation algorithm in a casestudy involving a generalization of the benchmark model of Black and Scholesfor option pricing. Our numerical results confirm that funding risk has anon-trivial impact on the deal price, and that double counting matters too. Weconclude the article with an analysis of large scale implications ofnon-linearity of the pricing equations.
AU - Brigo,D
AU - Liu,Q
AU - Pallavicini,A
AU - Sloth,D
PY - 2014///
TI - Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes
UR -
ER -