I have a time series of daily data that I want to calibrate GBM parameters $\mu$ and $\sigma$ to. Using the discretized solution
$$ S_{t_{i+1}} = S_{t_i}\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)\Delta t + \sigma \sqrt{\Delta{}}Z_{i+1}\right), $$ calibrating the parameters $\mu$ and $\sigma$ to a given time series with $n$ values turns out to be simply computing
$$ \sigma = \frac{std(R)}{\sqrt{\Delta t}}, \qquad \mu = \frac{\mathbb{E}[R]}{t} + \frac{\sigma^2}{2}, $$
where $R$ is a vector of log returns with components $R_{i+1} = \log S_{t_{i+1}} / S_{t_i}$, $1 \leq i \leq n-1$. The term $std(R)$ denotes the standard deviation of $R$.
Now, the time step $\Delta t = t_{i+1} - t_i$ is supposed to be the length of time between values in the series. Recall the closed-form solution to a GBM evaluated at "final" time $T$ is $$ S_T = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)T + \sigma W(T)\right). $$ So, if I have a time series history of daily prices spanning exactly one year (say 28 Oct 2013 - 28 Oct 2014), what should $T$ and $\Delta t$ be? In addition, $n=253$ in my series, even though the dates cover 365 days.
Some results: using natural gas futures prices with dates given above.
$T = 1$ and $\Delta t = 1/365$, I get $\sigma = 0.32$ and $\mu = 0.07$.
$T = 1$ and $\Delta t = 1/253$, I get $\sigma = 0.27$ and $\mu = 0.05$.
$T = 365$ and $\Delta t = 1$, I get $\sigma = 0.02$ and $\mu = 0.0002$.
$T = 253$ and $\Delta t = 1$, I get $\sigma = 0.02$ and $\mu = 0.0002$ (same as before).
The first two seem more reasonable for my time series. Any thoughts?