4. Suppose that S1 and S2 follow geometric Brownian motion and pay continuous proportional dividends at the rates δ1 and δ2. Use the martingale argument to show that the value of a claim paying S1(T ) if S1(T) > KS2(T ) is S1(t)e

−δ1(T−t)N ln(S1(t)/KS2(t)) + (δ2 − δ1+ 0.5σ2)(T − t)

σ

√

T − t



where σ2 = σ2 1

+ σ2 2

− 2ρ1, 2σ1σ2 and δ1 and δ2 are the dividend yields on the two stocks.