October 2010 Q5 (iii)

[baidshryans]

Could you explain how the S1^(2) term has been expanded?

Shouldn't there be a factor of half ?

 

[Steve Hales]

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?