Using CDS Implied Default Risk in Counterparty Valuation Adjustments

# CVA - Where does the default probability (PD) come from?


# CVA - Where does the default probability (PD) come from?












Some authors use CDS from the market to derive the implied default probability (from a risk-neutral point of view).

I wonder: how exactly does a CDS reflect counterparty risk?

Let me put an example: let's say I would like to price a derivative instrument (rate) that I am negotiating with counterparty $A$. I need to include the CVA (counterparty credit risk). Should I look for the CDS of $A$ in the market? I thought CDS reflect the debt issuer default risk, but maybe it also reflects counterparty credit risk.

Can someone clarify this please?

## Answer by Daneel Olivaw (score 5)

https://quant.stackexchange.com/a/42479

"Debt issuer default risk" and "counterparty risk" are very similar. From Risk magazine:

> Counterparty Risk The risk that a counterparty to a transaction or contract will default (fail to perform) on its obligation under the contract. Counterparty risk is not limited to credit risk (the risk that the counterparty cannot fulfill its contractual obligations for payment) but may also result from other problems associated with a counterparty unwilling to honor the contract.

In practice, it is implicitly assumed the material part of CVA can be captured by CDS prices. Quantifying other sources of counterparty risk would be particularly challenging due to the absence of legal precedents (bear in mind that the CDS market is only 25 years old).

Now suppose you have entered into some interest rate derivative contract with your counterparty $A$. The contract has maturity $T$ and its value at $t<T$ is given by $V(t)$. Your exposure $E(t)$ at time $t$ is equal to the positive value of the contract: $$E(t)=\max(0,V(t))$$ CVA is the price of the risk of default of your counterparty thus you need to look at the future expected exposure under the risk-neutral measure. Enter the risk-neutral discounted expected exposure $EE^Q(t,u)$ for a time $t<u<T$: $$EE^Q(t,u) = E^Q_t[D(t,u)E(u)]$$ where $Q$ is the risk-neutral measure and $D(t,u)$ the discount factor from $u$ to $t$. This corresponds to the risk-neutral value at time $t$ of the future value to you of the derivative contract, namely what you stand to lose if your counterparty defaults (or is "unwilling to honor the contract"). The default can occur at any time $u$ between $t$ and $T$, hence CVA for counterparty $A$ at time $t$ is equal to (assuming a recovery rate $\text{Rec}$): $$\text{CVA}_A(t)=(1-\text{Rec})\int_t^TEE^Q(t,u)\color{blue}{\text{d}P_t^Q(u)}$$ where $\text{d}P_t^Q(u)$ is the risk-neutral probability of default of $A$ on the infinitesimal time interval $[u,u+\text{d}u]$, conditional on the current ($t$) information.

To obtain estimates of default probabilities, you can extract information from the CDS market for counterparty $A$. To understand this, consider the following toy example: assume the risk-free rate is constant equal to $r$ and there exists a contract with a $1$-year maturity which pays $1\times(1-\text{Rec})$ if counterparty $A$ defaults within the year in exchange for a CDS premium $s_A$. By risk-neutral theory: $$\begin{align} s_A &= E^Q[e^{-r}(1-\text{Rec})1_{\{A \text{ defaults within 1 year\}}}] \\[3pt] & = e^{-r}(1-\text{Rec})\color{blue}{P^Q(A \text{ defaults within 1 year})} \end{align}$$ Thus: $$\color{blue}{P^Q(A \text{ defaults within 1 year})}=e^r\frac{s_A}{1-\text{Rec}}$$ You observe that CDS premiums encapsulate expected default information. You could also consider that distributional information $P^Q(\cdot)$ contained by CDS premiums is not limited to default but includes any failure to pay: this would probably depend on the design of the CDS contract and its payment triggers.

Anyway, in practice to price CVA we only look at the probability information contained by CDSs and assume it captures any material event that might result in no payment.

[Edit] In practice you also need to consider which CDS you choose: different CDSs might have different contractual designs and definitions of "default", thus some CDSs might not be relevant to quantify the probability of default/no payment/etc. for a particular derivative contract $V(t)$. See comments from @Mehness.

References

Pykhtin, M. and Zhu, S. (2007). "A Guide to Modelling Counterparty Credit Risk", GARP Risk Review.

## Answer by achirikhin (score 1)

https://quant.stackexchange.com/a/79522

That's an excellent question, and it illustrates the common confusion of most authors regarding the true nature of CVA.

CVA is cost of counterparty credit risk in a marginal liability, a marginal stochastic notional liability in case of a derivative. Marginal is the key word.

If liability was not marginal for the counterparty, e.g. two counterparties agreed that one of them is paid $\max(0, V(t))$ for some $V(t)$, assuming it is tradeable in broad sense, contingent of a default of some reference entity, then this would be a "contingent CDS", a generalization of a regular CDS, where $V(t)=1$. That is a true derivative. This is what is actually derived above.

CVA is different in that reference entity is one of the counterparties and entering into a derivative with the value process $V(t)$. Such derivative alters credit worthiness of at least one of the counterparties involved.

Understanding this is key to answering your question. Pre-trade CDSs of the counterparties, even if they exist, contain no information on the post-trade credit-worthiness of the counterparties, hence, strictly speaking, they cannot be used to determine default probabilities to be used in CVA valuation. It is most obvious in the case when both counterparties doing the derivative trade were not leveraged, hence there would be no CDS spreads to use. This is the problem many prime brokers face when dealing with hedge funds in estimating the residual CVA.

If, however, CDSs are traded and the marginal credit exposure created by the new trade is "small" in some sense, then one can argue that cost of marginal debt is same as average cost of debt, which is exactly what CDSs trade. Such assumption is usually true for large professional counterparties hence this is the trick used by CVA desks on daily basis. Then you can indeed reduce valuation of CVA to that of a CCDS. This is what is happening in practice most of the time.

p.s. From this point of view, market tradeable CDSs are supposed to price in the expected loss due to both non-stochastic and stochastic debt (derivatives).