6. Risk-neutral pricing and Itos lemma

(a) A derivative contract has the following payo at maturity:

fT = erT ln ST ,

where r is a constant interest rate. The underlying asset is described by the following process (under the real world probability),

dS = Sdt + Sdz ,

where, , and are constants, and z is a Wiener process. Show that, when r = 0

2

the price at time zero of the derivative contract is, f0 = lnS0 1

2 T .

(b) Optional. Apply Itos lemma to the following functions:

(i) X = z2

(ii) Y = t2 + ez

where, z is Wiener process, and t represents time.