Nonlinear Valuation Under Collateralization, Credit Risk, and Funding Costs

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Title: Nonlinear Valuation Under Collateralization, Credit Risk, and Funding Costs
Authors: Brigo, D
Liu, Q
Pallavicini, A
Sloth, D
Item Type: Conference Paper
Abstract: We develop a consistent, arbitrage-free framework for valuing derivative trades with collateral, counterparty credit risk, and funding costs. Credit, debit, liquidity, and funding valuation adjustments (CVA, DVA, LVA, and FVA) are simply introduced as modifications to the payout cash-flows of the trade position. The framework is flexible enough to accommodate actual trading complexities such as asymmetric collateral and funding rates, replacement close-out, and rehypothecation of posted collateral – all aspects which are often neglected. The generalized valuation equation takes the form of a forward-backward SDE or semilinear PDE. Nevertheless, it may be recast as a set of iterative equations which can be efficiently solved by our proposed least-squares Monte Carlo algorithm. We implement numerically the case of an equity option and show how its valuation changes when including the above effects. In the paper we also discuss the financial impact of the proposed valuation framework and of nonlinearity more generally. This is fourfold: Firstly, the valuation equation is only based on observable market rates, leaving the value of a derivatives transaction invariant to any theoretical risk-free rate. Secondly, the presence of funding costs makes the valuation problem a highly recursive and nonlinear one. Thus, credit and funding risks are non-separable in general, and despite common practice in banks, CVA, DVA, and FVA cannot be treated as purely additive adjustments without running the risk of double counting. To quantify the valuation error that can be attributed to double counting, we introduce a ’nonlinearity valuation adjustment’ (NVA) and show that its magnitude can be significant under asymmetric funding rates and replacement close-out at default. Thirdly, as trading parties cannot observe each others’ liquidity policies nor their respective funding costs, the bilateral nature of a derivative price breaks down. The value of a trade to a counterparty will not be just the opposite of the value seen by the bank. Finally, valuation becomes aggregation-dependent and portfolio values cannot simply be added up. This has operational consequences for banks, calling for a holistic, consistent approach across trading desks and asset classes.
Issue Date: 31-Dec-2016
Date of Acceptance: 4-Oct-2016
URI: http://hdl.handle.net/10044/1/41718
DOI: https://dx.doi.org/10.1007/978-3-319-33446-2_1
ISBN: 9783319334455
ISSN: 2194-1009
Publisher: Springer
Start Page: 3
End Page: 35
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
Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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