Credit valuation adjustment and wrong way risk

Definition Let $\tau$ be the default time of the counterparty, and consider one netting set, the trades whose obligations can be offset against each other at closeout under the assumed enforceable agreement. The closeout value $V^{\text{close}}_{\tau}$ is the signed amount that the closeout rule assigns to all trades of the netting set when the default ends the contract, seen from our side and positive when the counterparty owes us. This lesson uses the risk-free closeout convention, under which the closeout value is $V_{\tau}$, the default-free value at $\tau$ of the remaining cash flows, plus any contractual payments that fell due earlier and were not made. Other conventions, such as a replacement closeout that includes the adjustments a replacement counterparty would charge, give a different amount and a different loss. Let $C_{\tau}$ be the signed collateral balance that is usable at closeout, positive when we hold collateral and negative when we have posted it, after eligibility, haircuts and segregation are taken into account. The net claim after collateral and the residual exposure at default are

With several netting sets, each set has its own closeout value and collateral, and the residual exposures of the sets are added. With a margin period of risk, the collateral is the balance left by the last effective margin call before closeout and the closeout value is measured at the end of that period, so the value can move while the collateral cannot. Both are still indexed by $\tau$ for brevity.

Definition Assume that we cannot default, and let $R \in [0, 1]$ be the recovery fraction paid on an unsecured claim against the defaulted counterparty, treated as paid at $\tau$. Write $x^{-} = \max(-x, 0)$, so that $x = x^{+} - x^{-}$. At default the usable collateral is applied, a positive net claim $X_{\tau}^{+}$ becomes an unsecured claim that pays the fraction $R$, and a negative net claim is paid by us in full. What we end up with is therefore

where the loss at default is

The identity follows by substituting $X_{\tau}^{-} = X_{\tau}^{+} - X_{\tau}$ and $X_{\tau} = V^{\text{close}}_{\tau} - C_{\tau}$. The loss is positive for an adverse outcome and never negative, and the factor $1 - R$ is the loss given default. It vanishes when $R = 1$ or when $X_{\tau} \le 0$. Recovery here is a fraction of the net claim after collateral, and a recovery paid later than $\tau$ or measured on a different base is a different cash flow. A random recovery stays inside the expectation that defines CVA.

Definition Fix a netting set with final maturity $T$, a pricing measure $\mathbb{Q}$ with the money-market account $B$ as numeraire and $B_0 = 1$, and the closeout value, collateral and recovery of the previous cards. Assume that we cannot default, which is the unilateral setting. The unilateral credit valuation adjustment is the discounted expected loss at default,

With deterministic rates, $1/B_{\tau} = D(0,\tau)$, the deterministic discount factor evaluated at the random default time. The integrand is nonnegative, so CVA is nonnegative, and it is zero exactly when $\mathbf{1}_{\{\tau \le T\}} L_{\tau} = 0$ almost surely, for example when $R = 1$, when $\mathbb{Q}(\tau \le T) = 0$, or when the residual exposure is zero in the default states. The name comes from the following identity. Assume the risk-free closeout convention, a default time that is a stopping time of the information used for pricing, integrable cash flows, every contractual payment before $\tau$ made in full, and cash collateral remunerated at the short rate of the numeraire, so that collateral transfers before default have zero value. The defaultable netting set then pays the default-free cash flows up to $\tau \wedge T$ and, on $\{\tau \le T\}$, the settlement $V_{\tau} - L_{\tau}$ in place of the value $V_{\tau}$. By the tower rule the default-free value $V_0$ equals the value of the cash flows up to $\tau \wedge T$ plus $\mathbb{E}^{\mathbb{Q}}[\mathbf{1}_{\{\tau \le T\}} V_{\tau}/B_{\tau}]$, so the value including counterparty default is $V_0 - \operatorname{CVA}$. The default law and the recovery must be those of $\mathbb{Q}$, not a forecast of default frequency under $\mathbb{P}$.

When we can default too, the valuation is bilateral. It adds an own-default term on negative exposure, the debt valuation adjustment, and the closeout depends on which party defaults first, so the bilateral CVA counts only the states in which the counterparty defaults first. Adding a standalone unilateral CVA to a standalone own-default term can charge a second default in a state where the first default has already ended the contract. The first-to-default order has to enter the joint calculation.

Definition Assume deterministic discount factors $D(0,t)$, a deterministic default intensity $\lambda(t) \ge 0$ under $\mathbb{Q}$, as in Default intensity survival and recovery, with survival probability

so that $\tau$ has the density $\lambda(t) S(t)$ on $[0, T]$. Assume also a constant recovery fraction $R$ and a residual exposure process $E_t = (V^{\text{close}}_t - C_t)^{+}$, measurable in time and state, that is independent of $\tau$. Write $\operatorname{EE}(t) = \mathbb{E}^{\mathbb{Q}}[E_t]$ for the expected exposure. Then

Independence gives $\mathbb{E}^{\mathbb{Q}}[E_{\tau} \mid \tau = t] = \operatorname{EE}(t)$, and integrating the discounted loss against the density of $\tau$ gives the formula, with no further integrability condition because the integrand is nonnegative. An interval $[t, t + dt]$ contributes its default probability $\lambda(t) S(t)\,dt$ times the discounted expected loss there. The intensity is a rate per year, not a probability. The independence assumption is exactly the step that replaces the exposure expected given default at $t$ by its unconditional mean, and it does not hold for a contract whose value to us rises as the counterparty’s credit worsens. With constant $\operatorname{EE}$, $\lambda$ and a continuously compounded rate $r$, the integral is elementary,

when $r + \lambda > 0$, with the fraction replaced by $T$ when $r + \lambda = 0$.

Definition Consider one horizon at which a default during the period is settled, ignoring the timing of default within the period, with a deterministic discount factor $D$ to the settlement date and a constant recovery fraction $R$. Let $I$ be the indicator of counterparty default by the horizon and $E \ge 0$ the residual exposure at settlement, integrable under $\mathbb{Q}$. Then $\operatorname{CVA} = (1 - R)\,D\,\mathbb{E}^{\mathbb{Q}}[E I]$. The exposure-default covariance $\operatorname{Cov}^{\mathbb{Q}}(E, I) = \mathbb{E}^{\mathbb{Q}}[E I] - \mathbb{E}^{\mathbb{Q}}[E]\,\mathbb{E}^{\mathbb{Q}}[I]$ is finite because $I$ is bounded, and rearranging its definition gives

With $\operatorname{EE} = \mathbb{E}^{\mathbb{Q}}[E]$ and $\operatorname{PD} = \mathbb{Q}(I = 1)$, the adjustment splits into an independent part and a dependence part,

When $\operatorname{PD} > 0$, the identity $\mathbb{E}^{\mathbb{Q}}[E I] = \operatorname{PD} \cdot \mathbb{E}^{\mathbb{Q}}[E \mid I = 1]$ shows that the covariance equals $\operatorname{PD}$ times the excess of the exposure expected given default over the overall expected exposure. Independence makes the covariance zero, and in this one-date model zero covariance is all that the factorization needs. When $\operatorname{PD}$ is $0$ or $1$ the indicator is almost surely constant and the covariance is zero.

Definition Wrong-way risk is positive dependence between the exposure to a counterparty and its default, so that the exposure expected in the states where default happens exceeds the exposure expected overall. In the one-date model it means $\operatorname{Cov}^{\mathbb{Q}}(E, I) > 0$, which raises CVA above the factorized value $(1 - R)\,D\,\operatorname{EE} \cdot \operatorname{PD}$, and negative covariance, right-way risk, lowers it. Specific wrong-way risk comes from the structure of the trade, for example a put option on the counterparty’s own shares bought from that counterparty. General wrong-way risk comes from common factors, such as a recession that raises both the exposure and the default probability. Over a time horizon a Cox model keeps the dependence. Let $\lambda_t \ge 0$ be an intensity process adapted to the market information with $\Lambda_t = \int_0^t \lambda_s\,ds$, and let $\tau$ be the first time $\Lambda_t$ reaches $\xi$, a unit exponential random variable independent of the market information. If the residual exposure $E_t$ is adapted to the market information, discounting is deterministic and $R$ is constant, then

because, given the whole market path, $\tau$ has the density $\lambda_t e^{-\Lambda_t}$. Wrong-way risk enters through the joint law of $E_t$ and $\lambda_t$ under $\mathbb{Q}$. When the two processes are independent, the formula reduces to the independence formula with the default density $\mathbb{E}^{\mathbb{Q}}[\lambda_t e^{-\Lambda_t}]$. In this model the default itself does not move market prices, so an exposure that jumps at the default event, as a currency can when its sovereign issuer defaults, needs a model in which default feeds back into prices.