In the Geometric Brownian Motion model for a commodity spot price with convenience yield(or for a stock paying continuous dividend rate ), we have:

dST/St=()dt+dWt

whereWis a (real world) Brownian motion. We know that forT > t,

E [STSt] = e()(Tt)

(a) Find the conditional variance var(STSt).

(b)Find the probability thatSincreases by at least % between timetandT. Does the answer depend on the valueSt , and why (or not)?

(c)Now suppose we have a Schwarz one-factor (i.e. exponential Ornstein-Uhlenbeck) model instead:St=eYt, whereYis an Ornstein-Uhlenbeck process:

dYt=Ytdt+dWt

Find E [STSt]

(d) In the Schwarz model, find the probability thatSincreases by at least % between timetandT.