Hi.
Here we're working under the set of risk-neutral probabilities, or equivalent martginale measure (EMM) Q. By definition, this is the set of probabilities that ensure that the expected rate of (continuously compounded) return/drift of the underlying risky share is equal to the risk-free force of interest, r.
So, under Q, the stochastic differential equation for S(t) must be:
dS(t) = S(t).[r.dt + sigma.dZ(t)~]
where Z(t)~ is a standard Brownian motion under the Q probabilities.
(This is what you have already stated in part (i) of the question.)
So, if we apply Ito's Lemma to the function:
G(S(t) = ln S(t)
as per the example on p22 of Chapter 7 of the Course Notes, then we can see that the corresponding formula for S(t) itself is:
S(t) = S(0). exp[(r - sigma/2)t + sigma.Z(t)~]
{Note that whenever you have a formula of this type, if you ignore the -sigma/2 term, then whatever else is in the coefficient on t (ie r here), tells you what the drift of S(t) is. The -sigma/2 term appears because of the 2nd-order term in Ito's Lemma.}
Also, if we log the equation for S(t), then we can show that lnS(t) has a Normal distribution, as shown on p34 of the ASET, and hence that S(t) itself has a logNormal distribution.
Hope this helps.
Graham
