Margrabe Exchange Option

Consider two assets $S_1$ and $S_2$ defined on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{Q})$ . Under the risk-neutral measure $\mathbb{Q}$ associated with the money market account $B_t = e^{rt}$ , the dynamics are of Black-Scholes type:

\frac{dS_i(t)}{S_i(t)} = (r - q_i)\,dt + \sigma_i\,dW_i(t), \quad i = 1,2

where $r$ is constant, $q_i$ is the continuous dividend yield of $S_i$ , $\sigma_i > 0$ , and is a two-dimensional Brownian motion with:

\langle W_1, W_2 \rangle_t = \rho \, t, \quad \rho \in (-1,1)

Consider the Margrabe exchange option with maturity $T$ and payoff:

\big(S_1(T) - S_2(T)\big)^+

Question: Give a closed-form formula for the price $V_0$ of this option at date $0$ .

Margrabe Exchange Option

Answer