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Nonlinear valuation under credit, funding, and margins: existence, uniqueness, invariance, and disentanglement

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Title: Nonlinear valuation under credit, funding, and margins: existence, uniqueness, invariance, and disentanglement
Authors: Brigo, D
Francischello, M
Pallavicini, A
Item 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.
Issue Date: 16-Apr-2019
Date of Acceptance: 28-Oct-2018
URI: http://hdl.handle.net/10044/1/64598
DOI: 10.1016/j.ejor.2018.10.046
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/Funder: Engineering and Physical Sciences Research Council
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EPSRC Mathematics Platform Grant EP/I019111/1
EP/I019111/1
Keywords: 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
Online Publication Date: 2018-10-31
Appears in Collections:Financial Mathematics
Mathematics
Faculty of Natural Sciences