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Under GBM $$ \frac {dS_t}{S_t} = \mu dt + \sigma dW_t $$ we get $$ S_T = S_0 e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma W_T} $$ suggesting that $$ S_T \sim \text{ln}\mathcal {N} ( \tilde {\mu}, \tilde {\sigma}) $$ where \begin{align} \tilde {\mu} &= \ln S_0 + (\mu - \frac{1}{2}\sigma^2)T \\ \tilde {\sigma} &= \sigma \sqrt {T} \end{align} Now if $X \...


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If you have a formula for options on stock, you can turn it into a formula for options on futures by using the relation $S=F e^{-(r-d)T}$. In other words you observe the price of the future F, you turn it into S by this relation and then you pass this pseudo stock price to the options on stock function or program that you have. When you do this to the Black-...


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Finally I have found the answer on my own. The problem was related to the trasformation of the dataset. The original code used: ${{y}_{t}}=400*(\log ({{P}_{t}})-\log ({{P}_{t-1}}))$ as dataset. Initially I did not care about multipling by 400 because I thought it was usless. Instead it makes a big difference. Now the two series are completely overlapped, I ...



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