Collateral discounting and multiple curves

Definition Work in one currency with a derivative whose value to its holder is $V_t = V(t, X_t)$, where $X$ is a Markov state with generator $\mathcal{L}$ under the pricing measure $\mathbb{Q}$, and $\mathcal{L}$ already contains the financing of the hedge instruments, for example a stock financed by repo drifts at its repo rate less its dividend yield. Assume continuous, symmetric variation margin in cash, with collateral balance $C_t$ that the holder has received when $C_t > 0$ and posted when $C_t < 0$, interest on it at the collateral rate $c_t$, paid by the party that holds the collateral, unsecured borrowing and lending at one rate $f_t$, reuse of received collateral, and no thresholds, minimum transfer amounts, margin gaps or defaults. In this simplified replication ledger the part $V - C$ of the value that collateral does not finance is funded at $f$, and the collateral itself costs $c$, so

which is the same as

With full collateral, $C = V$, the two terms with $f$ cancel and

Under the regularity and growth conditions of the Feynman-Kac theorem, the solution with terminal value $V_T = H$ is

so a fully collateralized payoff is discounted at its collateral rate. Write $B^c_t = \exp(\int_0^t c_s\, ds)$ for the collateral account. Today’s collateral discount factor $D^c(0,T)$ is the value of a fully collateralized payment of $1$ at $T$,

which equals $\exp(-\int_0^T c(s)\, ds)$ when the collateral rate is deterministic. The cancellation removes $f$ from this equation only. It does not change the drifts in $\mathcal{L}$, which still need their own financing assumptions, and it fails as soon as one of the assumptions above fails.

Definition Let an accrual period contain the business days $j = 1, \dots, n$, let $r_j$ be the overnight rate published for day $j$ as an annual simple rate, and let $d_j \ge 1$ be the number of calendar days that $r_j$ applies to, more than one over weekends and holidays. Under the actual/360 day count the accrual factor and the compounded rate are

A loan or swap leg with notional $N$ then pays $N(G - 1) = NRD/360$ for the period. The rate is backward-looking. It is known only after the last fixing of the period, while a term rate is fixed at the start. Lookbacks, observation shifts, lockouts, payment lags and the day-count basis, such as $360$ or $365$, come from the contract. Expanding the product shows that $G - 1$ is the sum $\sum_j r_j d_j/360$ plus products of two or more of the terms $r_j d_j/360$. So if every $r_j \ge 0$, the compounded rate is at least the day-weighted arithmetic rate $\sum_j r_j d_j / D$, with equality exactly when at most one $r_j$ is positive. With fixings of mixed sign the inequality can fail. An example of an overnight benchmark is SOFR, a secured rate based on overnight repurchase agreements collateralized by Treasury securities. The benchmark is the daily rate, and the compounding of its observations is defined by the contract.

Definition An overnight indexed swap, or OIS, on the schedule $0 = T_0 < T_1 < \dots < T_n$ with accrual fractions $\delta_k$ exchanges a fixed rate $K$ for the compounded overnight rate. At each $T_k$ the fixed payer pays $\delta_k K$ and receives $\delta_k R_k$ per unit of notional, where $1 + \delta_k R_k$ is the factor $G$ of the period $[T_{k-1}, T_k]$. OIS discounting is collateral discounting for trades whose cash collateral earns that same overnight rate, with the discount curve built from OIS quotes. Assume the hypotheses of collateral discounting with this collateral rate, and in addition that the collateral account grows over each period by exactly the factor that the floating leg compounds, $B^c_{T_k}/B^c_{T_{k-1}} = 1 + \delta_k R_k$, with no payment lag. Since $B^c_0 = 1$,

so the floating leg telescopes to $1 - D^c(0,T_n)$, and the par OIS rate is

Solving these equations one maturity at a time,

bootstraps the collateral curve from quoted OIS rates, as in Bootstrapping discount curves. The phrase OIS discounting names this matched setup and nothing more. A trade with other collateral terms, such as collateral in another currency, securities as collateral, a spread on the collateral rate or a threshold, needs its own discount curve or further valuation terms.

Definition Fix a floating index $x$ whose fixing $L^x_k$ for the period $[T_{k-1}, T_k]$ is a simple rate, paid as $\delta_k L^x_k$ at $T_k$ under the collateral terms of the discount curve $D^c$, with $\mathbb{E}^{\mathbb{Q}}[\lvert L^x_k \rvert / B^c_{T_k}] < \infty$. The projected forward $F^x_k$ is the number that makes the payment’s value equal to its accrual times its discount factor times the forward,

Equivalently $F^x_k = \mathbb{E}^{\mathbb{Q}^{c,T_k}}[L^x_k]$, the expectation under the collateralized $T_k$-forward measure, whose density with respect to $\mathbb{Q}$ is $(1/B^c_{T_k})/D^c(0,T_k)$. The projection curve of $x$ is the collection of these forwards, often stored as pseudo discount factors with $P^x(0,T_0) = 1$ and

These pseudo discount factors are not prices of traded bonds, and absence of arbitrage does not require them to be. Projection curves are usually bootstrapped from swaps on the index after the discount curve is known. Under the OIS hypotheses the compounded overnight index has $F^{\mathrm{ON}}_k = (D^c(0,T_{k-1})/D^c(0,T_k) - 1)/\delta_k$, so its projection curve is the discount curve itself, while a term index generally has its own.

Definition Consider a vanilla swap with notional $N$ on the schedule above, fixed rate $K$ and floating index $x$, with both legs paying at the period ends under the collateral terms of $D^c$ and with the projected forwards $F^x_k$ taken as given. Then

and the par rate, at which the two legs have equal value, is

A payer swap with fixed rate $K$ is worth $N A^c (K_{\text{par}} - K)$ and a receiver swap the negative of that. The floating leg reduces to $N(1 - D^c(0,T_n))$ when every $F^x_k$ equals the forward read from $D^c$ itself, as for the overnight index under the OIS hypotheses, and not in general. This is cash-flow accounting with a declared projection convention. A random basis, and the convexity adjustments needed when a payment date differs from the end of its accrual period, are further model ingredients, and giving two curves two names does not supply them.

Definition The tenor basis of index $x$ over index $y$ for the period $[T_{k-1}, T_k]$ is the difference of their projected forwards,

in rate units and usually quoted in basis points, with $1$ bp $= 0.0001$. Most often $y$ is the compounded overnight index, whose forwards come from $D^c$ under the OIS hypotheses, or a term index of another tenor in the same currency. A basis swap exchanges the two floating legs, one of them with a spread $s$ added. When both legs share one schedule and the discount curve $D^c$, the par spread on the $y$ leg is

a weighted average of the period bases with the positive weights $\delta_k D^c(0,T_k)/A^c$, which add up to one. Legs with different payment frequencies need their own schedules in each sum. In this lesson the basis is an input read from quotes. Its economic sources and its random dynamics lie outside the model.