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
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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.