In the Black-Scholes setting of the question we know that \[ dS_{t}=S_{t}(r dt + \sigma dB_{t}) \] to which the solution is: \[ S_t=S_0 \exp\left\{\left( r - \frac{\sigma^2}{2}\right)t+\sigma B_t \right\}. \] You can use this to find an expression for \(S_{1}^{2}\), and then you'll probably want to use the MGF of the standard normal distribution to help find its expectation. Remember that \[ E\left[ e^{2\sigma B_1}\right]=e^{\frac{1}{2}(4\sigma^2)}=e^{2\sigma^2} \] which might be where your factor of a half is missing?
