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Analysis Of Nonlinear Valuation Equations Under Credit And Funding Effects

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Title: Analysis Of Nonlinear Valuation Equations Under Credit And Funding Effects
Authors: Brigo, D
Francischello, M
Pallavicini, A
Item Type: Conference Paper
Abstract: We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011) [11]. These equations take the form of semi-linear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk free rate. Indeed, our final semi-linear PDE or FBSDEs and their classical solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk free rate with a real market rate, since it acts as an instrumental variable. The equations derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including [11], [12], [10], [6] and [4].
Issue Date: 31-Dec-2016
Date of Acceptance: 4-Oct-2016
URI: http://hdl.handle.net/10044/1/41720
DOI: https://dx.doi.org/10.1007/978-3-319-33446-2_2
ISBN: 9783319334455
ISSN: 2194-1009
Publisher: Springer
Start Page: 37
End Page: 52
Journal / Book Title: Innovations in Derivatives Markets. Fixed Income Modeling, Valuation Adjustments, Risk Management, and Regulation
Volume: 165
Copyright Statement: This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated. The images or other third party material in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material.
Conference Name: Challenges in Derivatives Markets
Publication Status: Published
Start Date: 2015-04-30
Conference Place: Munich
Appears in Collections:Financial Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences