7.2 Futures: daily settlement and the futures–forward relation

7.2.1Exchange mechanics#

• Standardisation. The exchange fixes the underlying, the lot size, the expiry dates and the settlement method. Prices are the only thing traders negotiate.
• Central counterparty. After a trade is matched, the clearing corporation (NSE Clearing Ltd for NSE, MCX Clearing Corporation for MCX) becomes the buyer to every seller and the seller to every buyer. This substitution is called novation. Counterparty risk becomes risk against the clearing corporation, which protects itself with margins and a default fund.
• Daily settlement price. Each evening the exchange publishes a settlement price $F_t$. On NSE the daily settlement price of an index future is the volume-weighted average price (VWAP) of the futures in the last half hour of trading (as described in brokers' documentation; verify with the exchange). The final settlement price on expiry day is the closing value of the underlying index.
• Mark-to-market (MTM). Every open position is revalued at $F_t$ and the gain or loss is paid in cash.

7.2.2Margin accounts and the cash flows of a futures position#

A long position of $Q$ units opened at price $F_{\text{entry}}$ on day 1 and held to day $n$ generates the variation margin cash flows

with the opposite sign for a short. They telescope: $\sum_t m_t = Q(F_n - F_{\text{entry}})$, the same total as a forward. Only the timing differs.

To ensure losses can be paid, the exchange requires an initial margin deposit. In the textbook (CME-style) model there is also a maintenance margin below the initial margin: if the account balance falls below it, the trader receives a margin call and must top the account back up to the initial level. In India the mechanics are stricter. MTM losses are collected in cash from the member every day (settled the next working day), and the full initial margin requirement must be met on the current position at all times, including at random intraday "peak margin" snapshots. The Indian regime is the special case "maintenance = initial": every loss is called. Failure to pay leads the broker to close the position.

7.2.3Futures and forward prices under deterministic interest rates#

Write $G_0$ for the forward price and $F_0$ for the futures price, both for delivery at $T$. Let the futures be settled at times $0 = t_0 < t_1 < \dots < t_n = T$ (trading days). Interest rates may vary over time but are known in advance: $A(t,T) = \exp\left(\int_t^T r_s\,ds\right)$ is deterministic.

Proof

Consider the strategy: over the interval $(t_i, t_{i+1}]$ hold

$$\delta_i = \frac{1}{A(t_{i+1},T)}$$

long futures contracts. The variation margin $\delta_i(F_{t_{i+1}} - F_{t_i})$ is received (or paid) at $t_{i+1}$ and invested (or financed) at the risk-free rate until $T$, where it is worth

$$\delta_i (F_{t_{i+1}} - F_{t_i})\,A(t_{i+1}, T) = F_{t_{i+1}} - F_{t_i}.$$

Summing over $i$, the futures strategy is worth $F_T - F_0 = S_T - F_0$ at $T$, using convergence $F_T = S_T$ (7.3). The holdings $\delta_i$ are known at $t_i$ because rates are deterministic, so the strategy can actually be executed. It costs nothing to enter. Add an investment of $F_0/A(0,T)$ in the risk-free asset at time 0, which grows to $F_0$. The combined portfolio costs $F_0/A(0,T)$ and pays $S_T$ at $T$.

Now consider a long forward at $G_0$ plus $G_0/A(0,T)$ invested at the risk-free rate. It costs $G_0/A(0,T)$ and pays $S_T - G_0 + G_0 = S_T$ at $T$.

Two portfolios with the same terminal payoff and no intermediate net cash flows must cost the same, so $F_0/A(0,T) = G_0/A(0,T)$ and $F_0 = G_0$.∎

The holdings $\delta_i = e^{-r(T - t_{i+1})}$ (for constant $r$) are the interest tail of a futures hedge (7.6.4). The test test_tailed_futures_replicate_forward_on_any_path checks the identity $\sum_i \delta_i(F_{i+1}-F_i)e^{r(T-t_{i+1})} = F_T - F_0$ on simulated Nifty paths.

7.2.4Stochastic interest rates: the futures–forward gap#

When rates are random, the gains on a futures position are reinvested at unknown rates, and the proof above fails because $\delta_i$ would have to depend on future rates. To get the price we need the pricing principle that M08 develops in full: in a market without arbitrage there exists a risk-neutral measure $\mathbb{Q}$, equivalent to the real-world measure, under which the price at $t$ of any cash flow $X$ paid at $T$ is $B_t\,\mathbb{E}^{\mathbb{Q}}_t[X/B_T]$ (Harrison and Pliska 1981; M08 · 8.3 derives it in the binomial model and 8.7 in continuous time). We take it as given here.

Work in discrete time. Day $i$'s interest rate $r_i$ is known at the start of the day, so $B_{t_{i+1}} = B_{t_i}e^{r_i\Delta t}$ is known at $t_i$.

Proof

Futures. Entering a futures position at $t_i$ costs nothing, and holding it over one day produces the single cash flow $F_{t_{i+1}} - F_{t_i}$ at $t_{i+1}$ (after which it can be closed at no cost). Its value at $t_i$ must therefore be zero:

$$0 = B_{t_i}\,\mathbb{E}^{\mathbb{Q}}_{t_i}\!\left[\frac{F_{t_{i+1}} - F_{t_i}}{B_{t_{i+1}}}\right] = \frac{B_{t_i}}{B_{t_{i+1}}}\,\mathbb{E}^{\mathbb{Q}}_{t_i}[F_{t_{i+1}} - F_{t_i}],$$

where we pulled $B_{t_{i+1}}$ out because it is known at $t_i$. Hence $\mathbb{E}^{\mathbb{Q}}_{t_i}[F_{t_{i+1}}] = F_{t_i}$: the futures price is a $\mathbb{Q}$-martingale. With $F_T = S_T$, iterating gives $F_t = \mathbb{E}^{\mathbb{Q}}_t[S_T]$.

Forward. A forward struck at $G_0$ costs nothing and pays $S_T - G_0$ at $T$, so $0 = \mathbb{E}^{\mathbb{Q}}[D_T(S_T - G_0)]$, giving $G_0 = \mathbb{E}^{\mathbb{Q}}[D_TS_T]/\mathbb{E}^{\mathbb{Q}}[D_T]$.

Gap. $\mathbb{E}[D_TS_T] = \mathbb{E}[D_T]\mathbb{E}[S_T] + \operatorname{Cov}(D_T, S_T)$. Divide by $P(0,T) = \mathbb{E}[D_T]$ and subtract from $F_0 = \mathbb{E}[S_T]$.∎

Intuition. A long futures position receives gains in cash during the contract. If those gains arrive when rates are high, they are reinvested at high rates; if losses arrive when rates are low, they are financed cheaply. That is worth something to the long, so the long must pay a higher futures price to compensate. The reverse correlation makes the futures cheaper than the forward.

How big is it? Take a Gaussian (Vasicek) short rate $dr = \kappa(\theta - r)dt + \eta\,dW^r$ and $dS/S = r\,dt + \sigma\,dW^S$ under $\mathbb{Q}$ with $d\langle W^S, W^r\rangle = \rho\,dt$. Then $\ln S_T$ and $\int_0^T r_s\,ds$ are jointly Gaussian, and Problem 7.7 shows

The first term is positive even when $\rho = 0$, because under $\mathbb{Q}$ the asset drifts at the short rate, so high-rate paths are also high-$S_T$ paths. For an index future with three months to run ($\sigma = 15\%$, $\eta = 1\%$ a year, $\kappa = 0.3$, $\rho = 0.3$), the ratio is $1 + 1.4\times10^{-5}$: about 0.35 Nifty points, far below the day-to-day noise in the end-of-day basis. For a two-year contract with $\sigma = 25\%$, $\eta = 2\%$, $\kappa = 0.3$, it is about $+0.82\%$ for $\rho = 0.9$ and $-0.67\%$ for $\rho = -0.9$ (asymmetric because the $\eta^2$ term is positive for either sign). The code m07_mtm.futures_forward_mc reproduces both signs by Monte Carlo and matches the closed form. The gap matters for long-dated interest-rate futures, where it is called the convexity adjustment (M11), and is negligible for Indian equity index futures, which never run beyond three months.