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In my case (and I work mostly with natural gas) what I do in the calibration is to use the real value of $\Delta t$ from the historical data, and measure the time in days. In this way, $\Delta t=1$ in most cases, and $\Delta t=1$ in the weekends, so that you take into account the invisible changes in the markets during the weekends. It also help after the ...

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Time is expressed in fractions of year in the GBM formula. Therefore, $T=1$ year and $\Delta t = 1/m$. Considered that you have $253$ observations, I would use $m = 253$, so the second option as Drew suggested. In general, using 253 or 365 days in a year depends on how you consider reality: do you think that when markets are closed (i.e. weekends) the price ...

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The second one will be the best estimate. Also, a smaller timestep usually corresponds to a smaller bias. But I agree, the answer is not obvious. You should be careful about increasing $T$ though, because for negative drifts there is a threshold value ($2\mu + \sigma^2 < 0$) beyond which the variance of the price process stops increasing. It's an ...

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