| Математика
Key words: CVA, credit valuation adjustment, credit value adjustment, DVA, debt valuation adjustment,
debt value adjustment, option, hedging, greeks, gamma, cross-gamma, hedge, credit risk, SPV01,
DV01, sensitivities, exposure, EAD, billion trader, ENE, EPE, FVA, Basel III

Note: you can download this article in PDF here.
The previous article on credit valuation adjustment and counterparty credit risk was concluded with the following questions: should the CVA always be treated as a charge or can it be assessed as a gain?
How are the main sensitivities that the CVA desks monitor are being hedged (exposure DV01, the credit SPV01 and gamma)?
What role does the collateral play in CVA?
This article continues the overview of the most arguable topic in financial risk management and quantitative analysis.

Charge or gain? The increasing size of exposure in the derivative position and deterioration in the counterparty’s credit quality will mean a substantial adjustment in the market position value to properly reflect the credit risk.
An easy way to understand it is to treat CVA as a cost, that you are willing to pay provided the counterparty defaults.
But what if our creditworthiness is worse than that of our counterparties?
Recall that DVA reflects the adjustment that is to be made during position valuation based on the entity’s own credit quality.
This means that CVA management involves asset and liability fair value adjustments.
The EPE and ENE both form EAD profiles for the entity and its counterparty.
We can view the prototype of the bilateral exposure in a swap trade at a hypothetical time (t) on the chart below.

The sum of CVA and DVA provide us with the net adjustment that should be made to the mark-to-market position.
If the EPE profile increases its present value on the MTM axis, the entity should treated as a charge (figure A), whereas a skew towards the ENE (larger DVA) will result in a bilateral CVA gain (figure B).

Statistics says that a lot of banks still do not fully include DVA into their adjustments because it is considered to be more subjective.
For example, collateralized trades with counterparties of equal credit quality may be given a DVA relief, whereas in case of an inferior credit quality or an uncollateralized position the DVA could be ignored.

Hedging and sensitivity analysis.
Is there a standard list of greeks that CVA trading desks need to monitor throughout their trades? And which ones play the most important role in hedging?
The typical set of derivatives used in sensitivity analysis of CVA trades are exposure DV01, credit SPV01, delta, gamma, cross-gamma, correlation, vega and jump to default (risk of a credit default happening before the market reflects it in the credit spreads).
That is quite a list to monitor, isn’t it? Rebalancing the position may become problematic if you use all these to hedge.
Hence banks would normally monitor all these sensitivities on a daily basis and in some cases more seldom.

To make it simple enough, let’s derive the payoff of the CVA option.

On the far left we can notice a payoff diagram that resembles a call option.
This is virtually true.
Just like any other option the CVA is subject to time decay (i.e. there are less possible outcomes and the expected loss will decline over time).
But unlike a vanilla instrument this option is more complex as it represents a credit claim on default, and hence has specific greeks and sensitivities.

The most popular indicators are the exposure DV01 (dollar value of 1 basis point) and the credit SPV01 (spread present value of an 01).
The exposure DV01 is the sensitivity of the position value to a 1 basis point change in the underlying interest rate of the swap.
This sensitivity tends to decrease as time passes. It is possible to hedge against the DV01 risk and underlying volatilities using options.
And the same applies to the default probability using CDS, which calls for SPV01 management.
In this case the CVA trading desk needs to rebalance their position in the CDS and options as a response to movements in credit spreads, interest rates and volatility.
At this stage another important derivative indicator comes into play – the cross-gamma of the CVA option (measures the change in credit delta in response to a change in the underlying market rates).
Managing cross-gamma risk is not an easy subject that is typical for most structured products which have more than one underlying asset.
As the number of assets in the underlying basket increases, the cross-gamma may exceed the gamma of the position, and become relatively stochastic with when the volatility is low.
This is why hedging volatility risk and exploiting vega as a sensitivity of the CVA position is also important.
There are numerous constraints and problems linked to calculating and managing the greeks CVA.
For example dimensionality: numerous credit and risk-free yield curves (like the OIS) will need to be employed for projection and discounting of the positive and negative exposures; and the problem of negative gamma is also present whenever we deal with wrong-way risk.
There can also be a liquidity issue with single-name CDS contracts.
Some counterparties may have no traded CDS issued on their credit.
Therefore CDS indices (like the iTraxx or CDX), contingent CDS or default swap proxies like basket default swaps, are more common for default probability hedging.

And finally, a few words on FVA. What is “risk-free”? How do you define “risk-free”?
FVA is a correction that is to be made to the risk-free value of the derivative.
There has been a lot of debate on whether this adjustment needs to be made to reflect CCR.
It would be correct to take FVA into account when pricing uncollateralized OTC derivative positions, since risks of liquidity for instance should also be considered and reflected in the risk-free rates, which are likely used for funding.