Exposure netting and collateral dynamics

Definition Fix one counterparty and a closeout valuation convention. For trade $i$ with that counterparty, let $V_i(t)$ be its closeout value at time $t$ seen from our side, in currency units. It is the value at $t$ of the trade’s remaining cash flows, together with any contractual amounts that fell due and were not paid, positive when the trade is an asset to us and negative when it is a liability. For a group $S$ of trades, the signed portfolio value is

and the same trades have the signed value $-V_S(t)$ from the counterparty’s side. At a future date $V_S(t)$ is a random variable whose law depends on the measure in use, a pricing measure $\mathbb{Q}$ for valuation adjustments or the physical measure $\mathbb{P}$ for risk limits. Which closeout value applies, for example a default-free value or a replacement value that includes the charges of a new counterparty, is a contractual convention that has to be stated.

Definition Let $V_t$ be the signed value of trades that are settled as a single claim at closeout. Without collateral, the exposure at $t$ is

the amount we stand to lose if the counterparty defaults at $t$ and pays nothing on the claim, before any recovery. A negative value gives zero exposure, because we would then owe the closeout amount to the counterparty and have no credit loss on these trades. With a usable collateral balance $C_t$, described in a later card, the residual exposure is $(V_t - C_t)^{+}$. At a future date the exposure is random, and its mean under a stated measure, $\mathbb{E}[(V_t - C_t)^{+}]$, is the expected exposure at $t$. The positive part is convex, so Jensen’s inequality gives

often with strict inequality. The positive part has to be taken in each scenario, for example on each path of a Monte Carlo simulation, before averaging.

Definition A netting set is a group of trades with one counterparty whose obligations are assumed to be offset into a single net claim at closeout under an enforceable netting agreement. A trade covered by no such agreement forms a netting set of its own. With disjoint netting sets $S_1, \dots, S_K$ and no collateral, the exposure is

so values are added inside a set before the positive part is taken, and positive parts are added across sets. Whether an agreement is enforceable in a given jurisdiction is a legal question. The model takes the grouping as an input and draws no legal conclusion from it. A collateral balance belongs to the agreement that governs it and reduces exposure only in the netting set that agreement covers, so one balance cannot be counted in full against two separate sets.

Definition For any real numbers $V_1, \dots, V_m$,

Each value satisfies $V_i \le V_i^{+}$, so $\sum_i V_i \le \sum_i V_i^{+}$, and the right side is nonnegative. The positive part of the sum is either the sum itself or $0$, and both are at most $\sum_i V_i^{+}$, which proves the inequality. Equality holds exactly when the values do not include both a strictly positive and a strictly negative entry. If they include one of each and the sum is nonnegative, the negative entry makes the sum strictly smaller than the sum of positive parts, and if the sum is negative, its positive part is $0$ while the sum of positive parts is positive. The inequality holds scenario by scenario, so it also holds for expected exposures at every date. Applied to subtotals it shows that merging two netting sets never raises exposure, $(A + B)^{+} \le A^{+} + B^{+}$, and applied to the values $-V_i$ it shows that netting also lowers the counterparty’s exposure to us. It compares groupings of the same trades at the same values.

Definition The signed collateral balance $C_t$ is the value in currency units, after haircuts, of the collateral that is usable at closeout, positive when we hold collateral from the counterparty and negative when we have posted collateral to it. Eligibility, haircuts, segregation and the right to use collateral decide how much counts, so they belong in the definition of $C_t$. If $C_t$ can be applied immediately at closeout, the residual exposure is

Held collateral lowers the claim. Posted collateral that is not segregated adds to it, since a defaulting counterparty may not return it. Variation margin follows the current mark. Initial margin is meant to cover adverse moves over a future closeout period and is often segregated, so it may not be available for ordinary use. This lesson uses a one-sided variation margin rule with threshold $H \ge 0$ and minimum transfer amount $M > 0$, under which only the counterparty posts. At a margin date with balance $C_{t^-}$ before the call, the target is

If $\lvert C^{*}_t - C_{t^-} \rvert \ge M$, a transfer brings the balance to the target, and otherwise the balance stays at $C_{t^-}$. The steps come in this order, first the target, then the comparison, then the transfer. A two-sided agreement adds the mirror rule for collateral we post, with target $(V_t - H)^{+} - (-V_t - H')^{+}$, where $H'$ is the threshold that applies to us. Because small transfers are skipped, $C_t$ depends on the path of past marks, and the residual exposure depends on the joint path of $V$ and $C$. If $C_t > V_t$, the exposure is $0$, but the excess $C_t - V_t$ is owed back to the counterparty, an obligation that still matters for our own credit risk and funding.

Definition A margin call protects against the mark at the time of the call and does not freeze the future value. The margin period of risk $\delta > 0$ is the modeled time from the last margin call that is effectively met to the closeout after a default, during which no further collateral arrives. Under the convention that the balance stays at $C_t$ throughout, the gap exposure is

where $V_{t+\delta}$ is the closeout value at the end of the period and includes, consistently with the contract, the coupons, interest and other cash flows that fall due or go unpaid during it. Even with $C_t = V_t$, a rise of the value during the period creates exposure. Suppose that, given the information $\mathcal{F}_t$, the change $V_{t+\delta} - V_t$ is normal with mean $0$ and standard deviation $s = \sigma_V \sqrt{\delta}$, where $\sigma_V$ is a value volatility in currency units per square root of a year and $\delta$ is in years. Then

so the expected gap exposure with $C_t = V_t$ grows like the square root of the period, not linearly. The normal law is only a local diffusion approximation. Jumps, market moves linked to the default and large cash flows inside the period change the result. Initial margin that is usable at closeout enters the usable balance and lowers the gap exposure, the risk it is meant to cover.