XVA

Historically,[http://www.aims.ac.za/assets/files/Workshops/XVA.pdf XVAs: Funding, Credit, Debit & Capital in pricing]. Massimo Morini, Banca IMI{{citation |last1=Brigo |first1=Damiano |title=Nonlinear valuation and XVA under credit risk, collateral margins and Funding Costs |id=[Course notes: Doctoral course, Université catholique de Louvain, 19–20 Nov 2015] |date=November 2015 |location=Université catholique de Louvain |url=https://cdn.uclouvain.be/public/Exports%20reddot/csam/documents/Damiano_Brigo_Nonlinear_Valuation_and_XVA_under_Credit_Risk_Collateral_Margins_and_Funding_Costs.pdf}}Claudio Albanese, Simone Caenazzo and Stephane Crepey (2016). [https://arxiv.org/abs/1603.03012 Capital Valuation Adjustment and Funding Valuation Adjustment]. Risk Magazine, May 2016.{{Cite journal|first=Damiano |last=Brigo |date=November 5, 2011 |title=Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending|journal=Department of Mathematics, King's College, London |arxiv=1111.1331 |url= http://frank-oertel-math.de/CCR_CVA_Basel_III_Dialogue_Damiano_Brigo_Nov_2011.pdf }} (OTC) derivative pricing has relied on the Black–Scholes risk neutral pricing framework which assumes that funding is available at the risk free rate and that traders can perfectly replicate derivatives so as to fully hedge.See {{section link|Black–Scholes equation|Derivation of the Black–Scholes PDE}}; {{section link|Rational pricing|The replicating portfolio}}

This, in turn, assumes that derivatives can be traded without taking on credit risk. During the 2007–2008 financial crisis, many financial institutions failed, leaving their counterparts with claims on derivative contracts that were paid only in part. Therefore it became clear that counterparty credit risk must also be considered in derivatives valuation,{{Cite web |first1=Christian |last1=Kjølhede |first2=Anders |last2=Bech |url=http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf |title=Post-Crisis Pricing of Swaps using xVAs |publisher=Aarhus University |access-date=2017-02-02 |archive-date=2016-09-17 |archive-url=https://web.archive.org/web/20160917015231/http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf |url-status=dead }} and the risk neutral value is to be adjusted correspondingly.

When a derivative's exposure is collateralized, the "fair-value" is computed as before, but using the overnight index swap (OIS) curve for discounting. The OIS is chosen here as it reflects the rate for overnight secured lending between banks, and is thus considered a good indicator of the interbank credit markets.

When the exposure is not collateralized then a credit valuation adjustment, or CVA, is subtracted from this value[https://ssrn.com/abstract=2511585 Derivatives Pricing after the 2007–2008 Crisis: How the Crisis Changed the Pricing Approach], Didier Kouokap Youmbi, Bank of England – Prudential Regulation Authority (the logic: an institution insists on paying less for the option, knowing that the counterparty may default on its unrealized gain). This CVA is the discounted risk-neutral expectation value of the loss expected due to the counterparty not paying in accordance with the contractual terms, and is typically calculated under a simulation framework;{{Cite web |author=John Hull |author-link=John C. Hull (economist) |date=May 3, 2016 |url=https://fincad.com/blog/valuation-adjustments-1 |title=Valuation Adjustments 1 |website=fincad.com}}John C. Hull and Alan White (2014). [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2212953 Collateral and Credit Issues in Derivatives Pricing]. Rotman School of Management Working Paper No. 2212953

see {{slink|Credit valuation adjustment#Calculation}}.

When transactions are governed by a master agreement that includes netting-off of contract exposures, then the expected loss from a default depends on the net exposure of the whole portfolio of derivative trades outstanding under the agreement rather than being calculated on a transaction-by-transaction basis. The CVA (and xVA) applied to a new transaction should be the incremental effect of the new transaction on the portfolio CVA.

While the CVA reflects the market value of counterparty credit risk, additional Valuation Adjustments for debit, funding cost, regulatory capital and margin may similarly be added.[http://www.prmia.org/sites/default/files/PRMIA%20XVA%20Collateral%20Event.pdf XVA and Collateral: pricing and managing new liquidity risks]. Andrew Green[https://math.maths.univ-evry.fr/crepey/papers/opinions%20REVISED.pdf XVA: About CVA, DVA, FVA and Other Market Adjustments], Discussion paper: Louis Bachelier Finance and Sustainable Growth Labex. Stephane Crepey

As with CVA, these results are modeled via simulation as a function of the risk-neutral expectation of (a) the values of the underlying instrument and the relevant market values, and (b) the creditworthiness of the counterparty. This approach relies on an extension of the economic arguments underlying standard derivatives valuation.

These XVA include the following;{{Cite web|url=https://www.numerix.com/xvas-defined-profitability-puzzle|title=XVAs Defined: The Profitability Puzzle |website=www.numerix.com|date=5 July 2018 }}

and will require{{Cite web |title=Xva PDF {{!}} PDF {{!}} Hedge (Finance) {{!}} Arbitrage |url=https://www.scribd.com/document/405136644/XVA-pdf |access-date=2023-09-25 |website=Scribd |language=en}} careful and correct aggregation to avoid double counting:

• DVA, Debit Valuation Adjustment: analogous to CVA, the adjustment (increment) to a derivative price due to the institution's own default risk. DVA is basically CVA from the counterparty’s perspective. If one party incurs a CVA loss, the other party records a corresponding DVA gain.{{Cite web |title=CVA, DVA And Hedging Earnings Volatility {{!}} Quantifi |url=https://www.quantifisolutions.com/cva-dva-and-hedging-earnings-volatility-1/ |access-date=2023-08-24 |language=en-US|website=www.quantifisolutions.com}} (Bilateral Valuation Adjustment, BVA = DVA-CVA.Christopher Foot. [https://users.physics.ox.ac.uk/~Foot/Phynance/CDS2012.pdf Credit Default Swaps])
• FVA, Funding Valuation Adjustment, due to the funding implications of a trade that is not under Credit Support Annex (CSA), or is under a partial CSA; essentially the funding cost or benefit due to the difference (variation margin) between the funding rate of the bank's treasury and the collateral rate paid by a clearing house.{{Cite web |date=2014-03-20 |title=Funding Valuation Adjustment (FVA), Part 1: A Primer {{!}} Quantifi |url=https://www.quantifisolutions.com/funding-valuation-adjustment-fva-part-1-a-primer/ |access-date= |language=en-US}}
• MVA, Margin Valuation Adjustment, refers to the funding costs of the initial margin specific to centrally cleared transactions. It may be calculated according to the global rules for non-centrally cleared derivatives rules.{{citation |author1=((Basel Committee on Banking Supervision)) |author2=((Board of the International Organization of Securities Commissions)) |title=Margin requirements for non-centrally cleared derivatives |date=March 2015|url=https://www.bis.org/bcbs/publ/d317.htm |publisher=Bank for International Settlements (BIS) |location=Basel |isbn=978-92-9197-063-6}}
• KVA, the Valuation Adjustment for regulatory capital that must be held by the Institution against the exposure throughout the life of the contract (lately applying SA-CCR).

Other adjustments are also sometimes made including TVA, for tax, and RVA, for replacement of the derivative on downgrade. FVA may be decomposed into FCA for receivables and FBA for payables – where FCA is due to self-funded borrowing spread over Libor, and FBA due to self funded lending. Relatedly, LVA represents the specific liquidity adjustment, while CollVA is the value of the optionality embedded in a CSA to post collateral in different currencies. CRA, the collateral rate adjustment, reflects the present value of the expected excess of net interest paid on cash collateral over the net interest that would be paid if the interest rate equaled the risk-free rate.

{{See also|Credit valuation adjustment#Function of the CVA desk|Financial risk management#Banking}}

Per the IFRS 13 accounting standard, fair value is defined as "the price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date."{{cite web|title=IFRS 13.9 and IFRS 13 Defined Terms |publisher=ifrs.org|url=http://www.ifrs.org/issued-standards/list-of-standards/ifrs-13-fair-value-measurement/}}

Accounting rules thus mandateErnst & Young (2014). [https://assets.ey.com/content/dam/ey-sites/ey-com/en_gl/topics/ifrs/ey-applying-fv-april-2014.pdf?download Credit valuation adjustments for derivative contracts]

the inclusion of CVA, and DVA, in mark-to-market accounting.

One notable impact of this standard, is that bank earnings are subject to XVA volatility, (largely) a function of changing counterparty credit risk. A major task of the XVA-desk, therefore,James Lee (2010). [https://web.archive.org/web/20120417014447/https://www.boj.or.jp/announcements/release_2010/data/fsc1006a5.pdf Counterparty credit risk pricing, assessment, and dynamic hedging], Citigroup Global Markets is to hedge this exposure; see {{slink|Financial risk management#Banking}}. This is achieved by buying, for example, credit default swaps: this "CDS protection" applies in that its value is driven, also, by the counterparty's credit worthiness.[https://www.oreilly.com/library/view/counterparty-credit-risk/9781118316665/c16anchor-2.html "Components of CVA Hedging"]; §16.2 in John Gregory (2014). "Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge for Global Financial Markets", 2nd Edition. Wiley. {{ISBN|9781118316672}}

Hedges can also counter the variability of the exposure component of CVA risk, offsetting PFE at a given quantile.

Under Basel III banks are required to hold specific regulatory capital on the net CVA-risk.Bank for International Settlements (2020). [https://www.bis.org/basel_framework/chapter/MAR/50.htm MAR 50 - Credit valuation adjustment framework]

(To distinguish: this charge for CVA addresses the potential mark-to-market loss, while the SA-CCR framework addresses counterparty risk itself.Bank for International Settlements (2018). [https://www.bis.org/fsi/fsisummaries/ccr_in_b3.htm Counterparty credit risk in Basel III - Executive Summary])

Two approaches are available for calculating the CVA required-capital: the standardised approach (SA-CVA) and the basic approach (BA-CVA). Banks must use BA-CVA unless they receive approval from their relevant supervisory authority to use SA-CVA.

The XVA-desk is then responsible for managing counterparty risk as well as (minimizing) the capital requirements under Basel.Kenneth Kapner and Charles Gates (2016). [https://www.gfmi.com/articles/long-short-overview-xva/ "The Long and Short of It: An Overview of XVA"]. GFMI

The requirements of the XVA-desk differ from those of the Risk Control group and it is not uncommon to see institutions use different systems for risk exposure management on one hand, and XVA pricing and hedging on the other, with the desk employing its own quants.

{{Reflist}}

• {{cite book | author= Andrew Green | title=XVA: Credit, Funding and Capital Valuation Adjustments | publisher=Wiley | year=2015| isbn=978-1-118-55678-8}}
• {{cite book | author= Jon Gregory | title=The xVA Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital| publisher=Wiley | year=2015| isbn= 978-1-119-10941-9| edition=3rd}}
• {{cite book | author= Chris Kenyon and Andrew Green (Eds) | title=Landmarks in XVA: From Counterparty Risk to Funding Costs and Capital| publisher=Risk Books| year=2016| isbn= 978-1782722557}}
• {{cite book | author= Roland Lichters, Roland Stamm and Donal Gallagher | title=Modern Derivatives Pricing and Credit Exposure Analysis: Theory and Practice of CSA and XVA Pricing, Exposure Simulation and Backtesting | publisher= Palgrave Macmillan| year=2015| isbn=978-1137494832}}
• {{cite book | author= Dongsheng Lu | title=The XVA of Financial Derivatives: CVA, DVA and FVA Explained | publisher=Palgrave Macmillan | year=2015| isbn=978-1137435835}}
• {{cite book | author= Ignacio Ruiz | title=XVA Desks – A New Era for Risk Management | publisher=Palgrave Macmillan UK| year=2015| isbn=978-1-137-44819-4}}
• {{cite book | author= Antoine Savine and Jesper Andreasen | title=Modern Computational Finance: Scripting for Derivatives and XVA | publisher=Wiley| year=2021| isbn=978-1119540786}}
• {{cite book | author= Donald J. Smith | title=Valuation in a World of CVA, DVA, and FVA: A Tutorial on Debt Securities and Interest Rate Derivatives | publisher=World Scientific| year=2017| isbn=978-9813222748}}
• {{cite book | author= Alexander Sokol | title=Long-Term Portfolio Simulation: For XVA, Limits, Liquidity and Regulatory Capital| publisher=Risk Books| year=2014| isbn=978-1782720959}}
• {{cite book | author= Osamu Tsuchiya | title=A Practical Approach to XVA| publisher=World Scientific | year=2019| isbn=978-9813272750}}

Category:Mathematical finance

Category:Credit risk

Category:Derivatives (finance)

Category:Financial risk modeling

Category:Monte Carlo methods in finance