In the no-arbitrage pricing the log return of the stock price does not have expected return $0$ but $r$, the risk free rate. This is strongly related to the pricing of forward contracts. There you could follow the steps to see that in the arbitrage free world the spot price grows with the risk-free rate in expectation.
Thus if you price an option then the probability (in the martingale measure) that the log return is positive is greater than $1/2$ if there are positive interest rates. If you calculate with a dividend yield then this yield is substracted from the risk-free rate.
All the things that I have said hold for the log-return. If you take the exponential:
S_0 \exp( X_t ) = S_0 (1 + X_t + \frac12 X_t^2 + \cdots)
where $X_t$ is the log return process, and take the expectation then you get the terms $E[X_t] = r t$ for $E[X_t^2]/2 = t \sigma^2/2$ considering terms up to $2nd$ order.
In the Bachelier model, where the stock price is modelled as arithmetic Brownian motion, there you don't have this $\sigma^2$ term.