Futures daily settlement and convexity

Definition A futures contract is a standardized, exchange-traded agreement on an underlying asset with expiry $T$, quoted through a futures price $F^{\text{fut}}_t$ that changes over time. Opening or closing a position at the current futures price costs nothing apart from collateral. Instead of one payment at $T$, each change in the futures price is paid in cash at the next settlement date, and the contract ends with a final settlement whose price, in the idealized contract of this lesson, equals the spot price $S_T$. A contract covers $m$ units of the underlying, its multiplier, so one long contract gains $m$ currency units for each unit rise in the futures price. A clearing house takes the other side of every trade and collects margin, and in this lesson default is ignored.

Definition Let the settlement dates be $0 = t_0 < t_1 < \dots < t_n = T$, often one per business day, and write $F^{\text{fut}}_k$ for the settlement price at $t_k$, with $F^{\text{fut}}_n = S_T$. Under daily settlement, one long contract with multiplier $m$ receives at each $t_k$ the amount

where a negative amount is a payment by the holder, and one short contract receives $-\Delta M_k$. After the payment the contract continues at the new price $F^{\text{fut}}_k$, so a position that has just been settled is worth zero. Without interest on the payments, their sum telescopes,

which is the payoff of a long forward on $m$ units with delivery price $F^{\text{fut}}_0$. The total is the same, but it is paid in pieces on different dates.

Definition Variation margin is the cash paid at each settlement, the amounts $\Delta M_k$ of the previous card. It is separate from initial margin, the collateral posted when a position is opened, which is not a payment for the underlying and is returned when the position is closed, with interest on it as the exchange’s rules provide. Suppose each variation margin payment is invested or financed until $T$ in the money-market account $B$, with $B_0 = 1$ and the same rate for borrowing and lending. The terminal wealth of one long contract is then

With zero interest it reduces to the telescoping sum $m(S_T - F^{\text{fut}}_0)$. With positive rates an early gain is worth more at $T$ than the same gain received late, and an early loss costs more than a late one. Two price paths with the same start and the same end can therefore leave different wealth. This is a difference in timing and financing, not an arbitrage.

Definition Consider settlement dates $t_0 < \dots < t_n = T$ and a money-market account $B$ with $B_0 = 1$, where the interest for each interval $(t_{k-1}, t_k]$ is known at $t_{k-1}$, so that $B_{t_k}$ is $\mathcal{F}_{t_{k-1}}$-measurable. Assume borrowing and lending at that rate, no default, no cost of posting margin, and an equivalent pricing measure $\mathbb{Q}$ for the numeraire $B$, so that a payment $X$ at $t_k$ has the value

at $t_{k-1}$. Assume also that $\mathbb{E}^{\mathbb{Q}}[\lvert S_T \rvert] < \infty$, that the settlement prices are integrable and that the final settlement price is $S_T$. If opening a futures position at any settlement date costs nothing, then

and in particular $F^{\text{fut}}_0 = \mathbb{E}^{\mathbb{Q}}[S_T]$. The reason is that a position opened at $t_{k-1}$ and closed at $t_k$ pays only $\Delta M_k$ at $t_k$ and costs nothing, so $\mathbb{E}^{\mathbb{Q}}[\Delta M_k / B_{t_k} \mid \mathcal{F}_{t_{k-1}}] = 0$. Since $B_{t_k}$ is known at $t_{k-1}$, it comes out of the conditional expectation, which leaves $\mathbb{E}^{\mathbb{Q}}[F^{\text{fut}}_k - F^{\text{fut}}_{k-1} \mid \mathcal{F}_{t_{k-1}}] = 0$. The futures price is therefore a $\mathbb{Q}$-martingale that ends at $S_T$. For continuous settlement with a short-rate process, the same formula is a standard result under compatible financing and the regularity that its martingale argument needs, and it is stated here without proof. The expectation is taken under the pricing measure, so it is not a forecast of $S_T$ under $\mathbb{P}$.

Definition In the model of the previous card, let $D_T = 1/B_T > 0$ be the random discount factor of the money-market account to $T$, and assume $\mathbb{E}^{\mathbb{Q}}[D_T] < \infty$, $\mathbb{E}^{\mathbb{Q}}[\lvert S_T \rvert] < \infty$ and $\mathbb{E}^{\mathbb{Q}}[D_T \lvert S_T \rvert] < \infty$. A forward with delivery price $K$ paid at $T$ is worth $\mathbb{E}^{\mathbb{Q}}[D_T(S_T - K)]$ at time $0$, and since $K$ is fixed this is zero exactly for

Writing both quotes over the positive denominator $\mathbb{E}^{\mathbb{Q}}[D_T]$ gives the forward-futures difference

If the zero-coupon bond maturing at $T$ is priced by $\mathbb{Q}$, then $\mathbb{E}^{\mathbb{Q}}[D_T] = P(0,T)$, and for a traded asset without income, priced by $\mathbb{Q}$, also $\mathbb{E}^{\mathbb{Q}}[D_T S_T] = S_0$, so the forward price is $S_0/P(0,T)$ as in the previous lesson. The difference is zero exactly when $D_T$ and $S_T$ are uncorrelated under $\mathbb{Q}$, in particular when interest rates are deterministic, since $D_T$ is then a constant. It is negative exactly when the covariance is negative, for example in a model where the states with high final prices are the states with high rates over $[0, T]$, which make $D_T$ small.

Definition A convexity adjustment is the amount that converts a futures-based quote into the forward quote for the same underlying payoff and dates. With the sign convention forward minus futures, for an asset-delivery contract in the model above it is the covariance ratio of the previous card. Its size and sign depend on the joint law of rates and the underlying under $\mathbb{Q}$, on the settlement schedule and on how margin is financed. The term is used mostly for interest rate futures. Their quoted price is an affine function of a reference rate, for example $100$ minus a rate in percent, and their daily margin follows a different timing from the single settlement of a forward rate agreement. The adjustment between such a futures rate and the corresponding forward rate has to be derived for the actual payoff in a specified rate model, and the covariance identity of this lesson is not a general formula for it.