Let A denote the event S1T > S2T.

(a) Show that, for i = 1,2, E[MTSiT1A] = Si0 probSi

(A).

Conclude that the value at date 0 of an option to exchange asset 2 for asset 1 at date T is S10 probS1 (A)− S20 probS2 (A).

(b) Define Y = S2/S1. Show that dlogY = −1 2

σ2 dt + σ dB∗ , where B∗ is a Brownian motion under the probability measure probS1 .

Use this fact and the fact that A is the event logYT < 0 to show that probS1 (A) = N(d1), where d1 = log(S10/S20) + 1 2σ2T

σ

√T .

(c) Define Z = S1/S2. Show that dlogZ = −1 2

σ2 dt +σ dB∗ ,

where B∗ is a Brownian motion under the probability measure probS2 .
Use this fact and the fact that A is the event logZT > 0 to show that probS2 (A) = N(d2), where d2 = d1 − σ
√
T .