8.8 Black-76: options on futures, Nifty and MCX

Motivation

Nifty options are hedged with Nifty futures, and MCX options are written on futures. Black (1976) prices both with a formula that takes the futures price as the underlying and has no dividend or carry input.

The Black-76 formula#

Setup

Let $F_t$ be the price at $t$ of a futures contract expiring at $T_F$ . Entering a futures position costs nothing and gains or losses are paid daily (M07 · 7.2). Under deterministic interest rates futures and forward prices coincide (M07 · 7.2), and under $\mathbb Q$ the futures price is a martingale: a position that costs nothing and is marked to market must have zero expected gain under the measure that prices everything at $r$ . If $F_t=S_te^{(r-q)(T_F-t)}$ and $S$ follows (8.20), Itô's lemma gives

dF_t=\sigma F_t\,d\widetilde W_t ,

the same $\sigma$ as the spot and zero drift. A European option expiring at $T\le T_F$ with payoff $(F_T-K)^+$ is worth $e^{-r\tau}\mathbb E^{\mathbb Q}[(F_T-K)^+]$ .

Proposition 8.8.1Black-76

\begin{gathered} c=e^{-r\tau}\big[FN(d_1)-KN(d_2)\big],\qquad p=e^{-r\tau}\big[KN(-d_2)-FN(-d_1)\big],\\ d_{1,2}=\frac{\ln(F/K)\pm\frac12\sigma^2\tau}{\sigma\sqrt\tau}. \end{gathered} \tag{8.25}

Proof

$F$ is a GBM with drift zero under $\mathbb Q$ , which is a stock with $q=r$ (its "dividend yield" exactly offsets the risk-free drift). Substitute $S\to F$ and $q\to r$ in (8.22)–(8.24): $Fe^{-r\tau}N(d_1)-Ke^{-r\tau}N(d_2)$ , with $d_1=[\ln(F/K)+\frac12\sigma^2\tau]/(\sigma\sqrt\tau)$ .∎

Pricing Nifty options off the future#

When the option and the future expire together (Nifty monthly options and futures, both on 29 Dec 2026 for the December series), $F_T=S_T$ and (8.25) with $F=Se^{(r-q)\tau}$ equals (8.23)–(8.24) exactly. The test test_black76_equals_bsm_on_forward checks this to $10^{-12}$ . Parity for (8.25) is (8.5).

Worked example 8.8Nifty December options off the future

On 5 Nov 2026: $F=24{,}719.10$ , $\tau=54/365$ , $r=5.5\%$ . The implied carry is $\ln(F/S)/\tau=\ln(24{,}719.10/24{,}536.60)/0.147945=5.01\%$ ; with $r=5.5\%$ that corresponds to $q=0.49\%$ for this period. You do not need to believe that number. Black-76 uses $F$ directly. Take $\sigma=12.5\%$ as an assumed volatility (close to the 12.4% implied by that day's settlement prices, 8.10) and $K=24{,}700$ :

\begin{aligned} d_1&=\frac{\ln(24{,}719.10/24{,}700)+\frac12(0.125)^2(0.147945)}{0.125\sqrt{0.147945}}\\ &=\frac{0.000773+0.001156}{0.048079}=0.04012,\qquad d_2=-0.00796 . \end{aligned}

$N(d_1)=0.51600$ , $N(d_2)=0.49682$ , and $c=0.991896\,[24{,}719.10\times0.51600-24{,}700\times0.49682]=479.60$ points; the put is $460.66$ . One lot of the call costs $479.60\times65=\text{₹}31{,}174$ . The settlement prices were 476.90 and 454.95, so 12.5% is slightly above the market's implied volatility for that strike.

Options on commodity futures#

MCX options on futures MCX lists options on commodity futures (gold, silver, crude oil, natural gas, copper, zinc and others; the product list is on mcxindia.com, which blocked automated access, so check it there). Under SEBI's design circular SEBI/HO/CDMRD/DMP/CIR/P/2017/55 of 13 Jun 2017, the options are European, an option's lot equals the underlying future's lot, options expire before the future's tender period, and an ITM option at expiry devolves into a futures position at the strike rather than paying cash. M10 · 10.15.4 gives the MCX execution details and costs. Devolvement does not change the valuation: receiving a long future at strike $K$ when the futures price is $F_T$ is worth $F_T-K$ immediately through the next mark-to-market. So Black-76 with $\tau$ equal to the time to option expiry applies.

Worked example 8.9MCX gold

MCX Gold futures (lot 1 kg, quoted in ₹ per 10 g, so ₹100 per ₹1 move) traded at ₹1,38,839 for December on 2 Nov 2026 (news reports of that day; secondary, since the primary record, the MCX bhavcopy, could not be fetched automatically). Price a call and put struck at ₹1,40,000 with 20 days to option expiry, $r=5.5\%$ , and an assumed volatility of 25%. Then $d_1=-0.1130$ , and (8.25) gives a call of ₹2,698.71 and a put of ₹3,856.22 per 10 g, or ₹2,69,871 and ₹3,85,622 per lot. The futures delta of the call, $\partial c/\partial F=e^{-r\tau}N(d_1)=0.454$ , means a writer of one call hedges by buying 0.454 of a gold futures lot.

Practitioner notes

For Nifty, Black-76 on the same-expiry future is standard. For weekly options there is no same-expiry future; desks build a synthetic forward from put–call parity at strikes near the money (the $F^{\text{impl}}$ of 8.2) and feed that into (8.25).
The futures delta $e^{-r\tau}N(d_1)$ differs from the spot delta $e^{-q\tau}N(d_1)$ by the factor $e^{-(r-q)\tau}$ . When you hedge with futures, use the futures delta (for the example, 0.5118 against a spot delta of 0.5156).