# 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

• In the BS formulas, replace every occurrence of $S$ with $F e^{-rT}$ and you have the corresponding Black76 formulas. They are the result of a mechanical substitution of symbols, followed by simplification if appropriate (so it is not that "d1 does not use interest rate", rather "+r and -r cancel out in the expression for d1"). May 18, 2018 at 15:04

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.

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.

• Options on futures have no discounting. As long as interest rates are not zero the models cannot give the same price. The other five year old answer has mentioned that. I recommend to address blatand contradictions before posting. Aug 4, 2023 at 8:38
• Options on futures do have discounting [link] (en.wikipedia.org/wiki/Black_model#The_Black_formula ). I think you mean future-style options which don't have discounting. Primary application of black76 model is for pricing options on future contracts and that is what the above answer talks about. Aug 5, 2023 at 10:44
• Your answer is of low quality because it claims something that obviously contradicts another answer in this post and other posts. Possible resolution: different market conventions regarding settlement. See quote in this question. Looking forward hearing from you. Aug 6, 2023 at 5:12