Black-76 model (options on futures)
The Black-76 model is the variant of Black-Scholes used to price European options on futures and forwards: the futures price replaces the spot price, no dividend or carry term is needed, and the risk-free rate only discounts the payoff. It is the standard quoting model for options on index futures such as the E-mini S&P 500.
Senzoukria · Glossary · Updated September 2026
At a glance
- Call price
- C = e^(−rτ)·[F·N(d1) − K·N(d2)]
- Put price
- P = e^(−rτ)·[K·N(−d2) − F·N(−d1)]
- d1
- [ln(F/K) + σ²τ/2] / (σ√τ), d2 = d1 − σ√τ
From Black-Scholes to Black-76
A futures price already embeds the cost of carry of the underlying: financing minus dividends until the future's expiry. Fischer Black's 1976 model takes the futures price F as the input and treats it as having zero drift under the pricing measure, so the rate and dividend terms disappear from d1. The payoff is discounted at the risk-free rate, which is why e^(−rτ) multiplies the whole bracket. Black-Scholes with dividends and Black-76 give the same price when F equals S·e^((r−q)τ) and the option expires with the future.
Worked example
An at-the-money one-week option on a future trading at 5,000, strike 5,000, implied volatility 18%, rate 4%: d1 = 0.0125, d2 = −0.0125, and the call is worth 49.7 index points. For an E-mini S&P 500 option, whose underlying is one futures contract at 50 dollars per index point, that is about 2,484 dollars. The 5,050 call of the same expiry is worth 28.9 points.
What changes for traders
- Deltas are expressed in futures contracts, not in shares or index units, so hedging is direct: a delta of 0.5 on one option is half a future.
- Rho is only a discounting effect, −τ times the option value, for calls and puts alike.
- Put-call parity becomes C − P = e^(−rτ)·(F − K), which lets the futures price be implied from the options.
- American-style futures options need a lattice or approximation method; Black-76 prices the European case.
In Senzoukria
Senzoukria's GEX and Option Flow modules read listed US option chains through the configured provider, such as SPY or QQQ, not CME options on futures, so Black-76 is not the model behind their greeks. Levels computed on an ETF chain can be drawn on an ES or NQ chart, where the app converts them with a ratio between the futures chart's last close and the ETF spot, and labels them with the source underlying so a converted level is never presented as a native futures-option level.
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Frequently asked questions
- Why is there no dividend term in Black-76?
- Because the futures price already reflects expected dividends and financing up to the future's expiry. The model only needs the distribution of the futures price, which it assumes lognormal with zero drift under the pricing measure.
- Should I use Black-Scholes or Black-76 for ES options?
- Black-76, with the price of the underlying futures contract as input. Using the cash index with Black-Scholes requires the right rate and dividend assumptions to reproduce the futures price; using the futures price directly avoids those estimates.