Nonlinear valuation under credit, funding, and margins: existence, uniqueness, invariance, and disentanglement
File(s)
Accepted version
Author(s)
Brigo, Damiano
Francischello, Marco
Pallavicini, Andrea
Type
Journal Article
Abstract
Since the 2008 global financial crisis, the banking industry has been using valuation adjustments to account for default risk and funding costs. These adjustments are computed separately and added together by practitioners as if the valuation equations were linear. This assumption is too strong and does not allow to model market features such as different borrowing and lending rates and replacement default closeout. Hence we argue that the full valuation equations are nonlinear, and this paper is devoted to studying the nonlinear valuation equations introduced in Pallavicini et al (2011).
We illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then we show how to obtain semi-linear PDEs or Forward Backward Stochastic Differential Equations (FBSDEs) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these PDEs and FBSDEs in a viscosity and classical sense.
Moreover, from a financial perspective, we discuss cases where classical valuation adjustments (XVA) can be disentangled. We show how funding costs are offset by treasury valuation adjustments when one takes a whole-bank perspective in the valuation, while the same costs are not offset by such adjustments when taking a shareholder perspective. We show that although we use a risk-neutral valuation framework based on a locally risk-free bank account, our final valuation equations do not depend on the risk-free rate. Finally, we show how to consistently derive a netting set valuation from a portfolio level one.
We illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then we show how to obtain semi-linear PDEs or Forward Backward Stochastic Differential Equations (FBSDEs) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these PDEs and FBSDEs in a viscosity and classical sense.
Moreover, from a financial perspective, we discuss cases where classical valuation adjustments (XVA) can be disentangled. We show how funding costs are offset by treasury valuation adjustments when one takes a whole-bank perspective in the valuation, while the same costs are not offset by such adjustments when taking a shareholder perspective. We show that although we use a risk-neutral valuation framework based on a locally risk-free bank account, our final valuation equations do not depend on the risk-free rate. Finally, we show how to consistently derive a netting set valuation from a portfolio level one.
Date Issued
2019-04-16
Date Acceptance
2018-10-28
Citation
European Journal of Operational Research, 2019, 274 (2), pp.788-805
ISSN
0377-2217
Publisher
Elsevier
Start Page
788
End Page
805
Journal / Book Title
European Journal of Operational Research
Volume
274
Issue
2
Copyright Statement
© 2018 Elsevier Ltd. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence http://creativecommons.org/licenses/by-nc-nd/4.0/
Sponsor
Engineering and Physical Sciences Research Council
Engineering & Physical Science Research Council (EPSRC)
Identifier
https://www.sciencedirect.com/science/article/pii/S0377221718309147?via%3Dihub
Grant Number
EPSRC Mathematics Platform Grant EP/I019111/1
EP/I019111/1
Subjects
Social Sciences
Science & Technology
Technology
Management
Operations Research & Management Science
Business & Economics
Pricing
Valuation adjustments
Backward stochastic differential Equations
Funding costs
Nonlinear valuation
STOCHASTIC DIFFERENTIAL-EQUATIONS
BILATERAL COUNTERPARTY RISK
DERIVATIVES
COLLATERALIZATION
SWAPS
Operations Research
Publication Status
Published
Date Publish Online
2018-10-31