Margrabe's Formula for Exchange Options via Change of Numeraire

Scenario

An exchange option grants the holder the right to exchange asset 2 for asset 1 at time $T$, delivering payoff $\max(S_1(T) - S_2(T), 0)$. Both assets follow correlated geometric Brownian motions under risk-neutral measure $\mathbb{Q}$: $dS_1 = r S_1 dt + \sigma_1 S_1 dW_1, \quad dS_2 = r S_2 dt + \sigma_2 S_2 dW_2, \quad d\langle W_1, W_2 \rangle = \rho dt$

Your Task

1. By selecting asset 2 ($S_2$) as the numeraire, define the relative price process $Z_t = \frac{S_1(t)}{S_2(t)}$.
2. Use Girsanov's theorem to prove that $Z_t$ is a martingale under the numeraire measure $\mathbb{Q}^{S_2}$ with effective volatility $\hat{\sigma} = \sqrt{\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2}$.
3. Derive Margrabe's closed-form pricing formula: $V(0) = S_1(0) \Phi(d_1) - S_2(0) \Phi(d_2), \quad d_1 = \frac{\ln(S_1(0)/S_2(0)) + \frac{1}{2}\hat{\sigma}^2 T}{\hat{\sigma}\sqrt{T}}, \quad d_2 = d_1 - \hat{\sigma}\sqrt{T}$
4. Compute the exchange option value for $S_1 = 100, S_2 = 100, \sigma_1 = 0.20, \sigma_2 = 0.30, \rho = 0.50, T = 1.0$, and verify that the price is completely independent of the risk-free rate $r$.

Constraints

• Correlated geometric Brownian motions, continuous trading.

Examples When $\sigma_1 = \sigma_2$ and $\rho = 1$, relative volatility $\hat{\sigma} = 0$, and the option is worth $\max(S_1 - S_2, 0)$.