The present value of a risk neutral GBM implies no "drift". A generic GBM is not necessarily a martingale...
The median of a generic GBM with drift alpha and volatility sigma is \( S(0)e^{(\alpha-\frac{1}{2}\sigma^2)t} \)
This is because \( log(S_t)=log(S(0))+(\alpha-\frac{1}{2} \sigma^2)t+\sigma W_t \)
Since the median of \( W_t \) is zero, the median of \( log(S_t) = log(S_0)+(\alpha-\frac{1}{2} \sigma^2)t \) , so the median of \( S_t \) is \( S(0)e^{(\alpha-\frac{1}{2}\sigma^2)t} \)
Since the beta is one, the stock is perfectly correlated with the market (iirc) and thus the question is whether the market is more likely to be above where it is currently at in a year...which I would think is the greater possibility, at least in normal economic times
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