Let S1 = (S1t) and S2 = (S2t) be the price processes of two assets. Consider the option to exchange (at zero cost) one unit of asset 2 for one unit of asset 1 at some prespecified date T. The payoff is thus max (S1T − S2T, 0). The assets have no dividends before time T.

(a) Argue that the time t value of this option can be written as Vt = S2t EQ2 t



max S1T S2T

− 1, 0 , where Q2 is the risk-adjusted probability measure associated with asset 2.

Suppose that S1 and S2 are both geometric Brownian motions so that we may write their joint dynamics as dS1t = S1t [μ1 dt + σ1 dz1t], dS2t = S2t



μ2 dt + ρσ2 dz1t +

'

1 − ρ2σ2 dz2t



.

(b) Find the dynamics of S1t/S2t under the probability measure Q2.

(c) Use the two previous questions and your knowledge of lognormal random variables to show that Vt = S1tN(d1) − S2tN(d2), (13.31)

where d1 = ln(S1t/S2t)

v +

1 2

v, d2 = d1 − v, v =

'

(σ 2 1 + σ 2 2 − 2ρσ1σ2)(T − t).

This formula was first given by Margrabe (1978).

(d) Give pricing formulas (in terms of S1t and S2t) for an option with payoff max(S1T, S2T) and an option with payoff min(S1T, S2T).

(e) What happens to the pricing formula (13.31) if asset 2 is a zero-coupon bond maturing at time T with a payment of K? And if, furthermore, interest rates are constant, what then?