XVA: back to CVA?

Fundamental questions on CVA remain unanswered, writes mathematical finance head

Revisit mathematics
Look back: some basic CVA issues still need to be addressed

While many professionals are chasing the latest derivatives valuation adjustments (XVAs) and trying to introduce new ones, we should make sure we deal with the fundamental ones first.

Many basic issues with credit valuation adjustment (CVA) – the first of the family – still remain unaddressed. We should be cautious in considering it dead due to the effects of margining for non-cleared swaps, especially given the blunt methodology adopted for the industry’s standard initial margin model (Simm), and we should also pay attention to potential double counting in XVAs due to non-linearities.

CVA answers the following question: “How much discount do I get on the price of this deal due to the fact that you, my counterparty, can default?” It has traditionally been calculated with a risk-neutral valuation approach (Brigo and Masetti, 2006), with all the pros and cons discussed in my previous column (Risk Magazine, February 2018).

But despite having been around for years, a question remains: can CVA really be hedged? Challenges range from finding good liquid instruments for default probabilities and recoveries, to modelling the dynamics and option prices of complex netting sets with option maturities given by the counterparty random default time. Wrong-way risk is also quite model-dependent and hard to assess (Brigo and Pallavicini, 2008).

Then there is debit valuation adjustment (DVA), which is the CVA seen from the other counterparty’s perspective (Brigo and Capponi, 2008). DVA answers the question: “How much markup do I need to pay over the price of this deal to my counterparty due to the fact I can default?”

The mark-to-market value of DVA goes up when a company’s credit quality goes down. Companies could therefore profit from their debt deterioration – we have seen banks report $2.5 billion DVA gains in a quarter in the past.

Given that a party cannot sell protection on itself, DVA is notoriously difficult to hedge. It is typically hedged by proxy, which is not ideal when jump-to-default risk is included in the picture. Furthermore, bilateral CVA and DVA introduce a first-to-default time that embeds a default correlation that is difficult to hedge. The Basel Committee on Banking Supervision hasn’t recognised DVA, whereas the accounting standards do.

The blunt methodology adopted by the Simm implies it does not always address the real gap risk, which will be always quite model-dependent

Some ask will margin and clearing kill CVA and DVA? In a 2014 paper on bilateral counterparty risk valuation (Brigo, Capponi and Pallavicini, 2014), it is shown that even under continued collateralisation, contagion and gap risk at default may lead to a residual CVA that in some cases is as large as CVA as for the uncollateralised trade. In a 2014 paper on cleared and bilateral swap pricing (Brigo and Pallavicini, 2014), we also study liquidation delays coming from possible disputes.

The additional initial margin might limit the problem, but the blunt methodology adopted by the Simm implies it does not always address the real gap risk, which will be always quite model-dependent (Brigo and Pallavicini, 2014). 

FVA is the next in the XVA family. This accounts for all the borrowing costs and lending benefits one faces in servicing the trade accounts. It can be sizeable – JP Morgan, for example, declared an FVA of $1.5 billion in a single quarter in 2013.

FVA is linked to the CVA and DVA that come from the external borrowing and lending operation of the bank, so in a way it is also driven by credit risk. These three XVAs may introduce non-linear features in valuation when borrowing and lending rates are not equal, or when replacement closeout at default is used in case of early default.

The mathematical tools needed in this case range from semi-linear partial differential equations to backward stochastic differential equations. These tools are quite advanced and rarely used in the industry. They can also be used to assess the cost of trading via a clearing house or a standardised credit support annex with variation and initial margin (Brigo and Pallavicini, 2014). Approximating non-linearities by linearising can lead to double counting, and the issue needs to be investigated further.

The final valuation adjustment is capital valuation adjustment (KVA). Initially, a classic replication approach was proposed in 2014 (Green, Kenyon and Dennis, 2014). However, as I mentioned in the February column, there is a different approach: in a 2017 paper (Brigo, Francischello, Pallavicini, 2017), we propose an indifference approach for KVA, finally moving beyond the continued and unrealistic stretch of risk-neutral valuation and replication.

However, while further research on KVA and XVA is needed, we should not forget the unsolved, fundamental challenges around CVA.

Damiano Brigo is head of the mathematical finance research group at Imperial College, London, and part of the stochastic analysis research group.


Brigo D and Masetti M, 2006

Risk Neutral Pricing of Counterparty Risk

Counterparty Credit Risk Modelling: Risk Management, Pricing and Regulation (edited by Michael Pykhtin)

Risk Books

Brigo D and Pallavicini A, 2008

Counterparty Risk and Contingent CDS under correlation

Risk February, 2008

Brigo D and Capponi A, 2008

Bilateral counterparty risk with stochastic dynamical models


Brigo D, Capponi A and Pallavicini A, 2014

Arbitrage-free bilateral counterparty risk valuation under collateralization and application to Credit Default Swaps

Mathematical Finance, Vol 24, No 1, pages 125–146

Brigo D and Pallavicini A, 2014

Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks

Journal of Financial Engineering, 1 (1):1–60

Green A, Kenyon C and Dennis CR, 2014

KVA: Capital Valuation Adjustment by Replication

Risk 27(12)

Brigo D, Francischello M and Pallavicini A, 2017

An indifference approach to the cost of capital constraints: KVA and beyond


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