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As @Kermittfrog said in the comment, in Black formula for options on futures price you need to insert the futures price $F$: $$C = e^{-rT}[FN(d_1) - KN(d_2)]$$ where $r$ is the discounting rate. Here, $d_1$ depends only on $F$ (no rate involved). For Black-Scholes formula for options on spot price (assume asset pays no dividend to keep it clean), we have: $$...


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