Black-Scholes and Black-76: Pricing Inputs and Greek Interpretation
Summary
The document explains how Black-Scholes and Black-76 represent the underlying and carry when valuing European options. Black-Scholes is framed around spot, with rates and carry components represented in the model, while Black-76 uses the forward price, which already reflects financing and carry. Accordingly, the forward-based formula omits the separate interest-rate term in its d1 expression.
For options with matching expiry on spot and futures, the replies reason that their terminal payoffs coincide when futures converge to spot, so the models can produce the same value when inputs are consistent. Their Greeks still have different input interpretations: Black-Scholes delta measures sensitivity to spot, while Black-76 delta measures sensitivity to the futures price. The document is a conceptual comparison rather than a full formula reference, and the equivalence depends on matching contracts, expiry, and carry assumptions.
Key ideas
- Black-Scholes expresses the underlying as spot and models rates or carry separately, while Black-76 starts from the forward price.
- The interest-rate effect is reflected in the forward input in Black-76 rather than appearing separately in d1.
- With aligned expiry and consistent inputs, spot and futures options can share the same terminal payoff and value.
- Delta has different meanings across the models because it is measured against spot in Black-Scholes and the forward in Black-76.
- The stated price equivalence relies on matching contract and carry assumptions.
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Full text
# Option Greeks' Formulas for Black & Scholes vs Black 76 # Option Greeks' Formulas for Black & Scholes vs Black 76 I know Black76 uses forward prices instead of spot and that D1 calculation doesn't use the interest rate. Are there any other differences between the two? I'm calculating: theoretical value, delta, lambda, vega, theta, rho, gamma ## Answer by Hui (score 5) https://quant.stackexchange.com/a/39918 There is no fundamental/assumptional difference between these two models. The only difference is Black 76 reflects interest rate, cost of carries, dividend etc. on the forward price, while Black Scholes treats them as separate components of the model. In the formulas of calculating D1, the only difference in addition to the change of S - >F is that Black76 doesn't have "r" component in the nominator because r has already been priced in F. ## Answer by user2329744 (score 4) https://quant.stackexchange.com/a/76313 Black-Scholes model is used to price options on spot while Black76 is used for pricing options on futures contracts. For European options both models will give exactly same price given options expiry is same as futures expiry. To see this, notice that value of futures contract at expiry is same as the underlying spot value. Therefore, at expiry both options on spot and options on futures will have the exact same payoff. As options price at any point is discounted expected value of payoff at expiry, both kind of options should be valued the same. Greeks computation will be different for both models. Using BSM one would get dividend/convenience yield term in greeks formulas while those are accounted for in futures price in case of Black. Also, greeks will be interpreted differently. Delta for instance in BSM is change in option price with respect to change in spot price while for Black it will be change in option price with respect to change in futures price.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.