Exchange Option: Pricing the Right to Swap One Stock for Another
An exchange option gives its holder the right, at maturity $T$, to exchange one share of stock 1 for one share of stock 2; its payoff is $\max(S_{2,T} - S_{1,T},\,0)$. Both stocks pay no dividends and follow geometric Brownian motions, $$\frac{dS_1}{S_1} = \mu_1\,dt + \sigma_1\,dW_1, \qquad \frac{dS_2}{S_2} = \mu_2\,dt + \sigma_2\,dW_2, \qquad dW_1\,dW_2 = \rho\,dt,$$ and the risk-free rate is $r$. Find the price of the exchange option today (time $0$), when the stock prices are $S_1$ and $S_2$.
Hints
- Factor the payoff: $\max(S_{2,T} - S_{1,T}, 0) = S_{1,T}\max(S_{2,T}/S_{1,T} - 1, 0)$. Measured in units of stock 1, this is a plain call on the ratio $S_2/S_1$ with strike $1$.
- Use stock 1 as the numeraire. Under the measure associated with $S_1$, the ratio $S_2/S_1$ is a martingale (drift zero) and is lognormal with volatility $\sigma_s = \sqrt{\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2}$.
- Apply Black-Scholes with spot $S_2/S_1$, strike $1$, rate $0$ and volatility $\sigma_s$, then multiply by $S_1$ to convert back to dollars.
Worked Solution
How to Think About It: The payoff is homogeneous of degree one in $(S_1, S_2)$, so the problem has one fewer dimension than it looks. Choose stock 1 as the numeraire: prices measured in shares of stock 1 are martingales under the corresponding measure, and the payoff in those units is a vanilla call on the ratio $S_2/S_1$ with strike $1$. Black-Scholes with zero rate does the rest.
Quick Estimate: With $S_1 = S_2 = 100$, $\sigma_1 = 20\%$, $\sigma_2 = 30\%$, $\rho = 0.5$, $T = 1$: $\sigma_s = \sqrt{0.04 - 0.06 + 0.09} = \sqrt{0.07} = 0.265$. The option is at the money in ratio terms, so the at-the-money approximation gives $0.4\,\sigma_s S\sqrt{T} = 0.4 \times 0.265 \times 100 = 10.6$. The exact value is $10.52$.
Formal Solution:
*Step 1 -- Dynamics of the ratio.* Let $X = S_2/S_1$. By Ito's lemma, $d\ln X = d\ln S_2 - d\ln S_1$, and since $\ln S_i$ has diffusion $\sigma_i\,dW_i$, $$\text{Var}(d\ln X) = (\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2)\,dt \equiv \sigma_s^2\,dt.$$ So $X$ is a geometric Brownian motion with volatility $\sigma_s = \sqrt{\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2}$.
*Step 2 -- Change of numeraire.* Take $S_1$ as the numeraire. Under the associated measure $Q^{1}$, every traded asset divided by $S_1$ is a martingale; in particular $X = S_2/S_1$ has zero drift, and the option's value satisfies $$\frac{V_0}{S_1} = E^{Q^1}\!\left[\frac{\max(S_{2,T} - S_{1,T}, 0)}{S_{1,T}}\right] = E^{Q^1}\!\left[\max(X_T - 1, 0)\right].$$ (Equivalently: the payoff is homogeneous of degree one, so $V(S_1, S_2, t) = S_1\,g(S_2/S_1, t)$ and $g$ satisfies the Black-Scholes PDE with rate $0$ and volatility $\sigma_s$.)
*Step 3 -- Black-Scholes on the ratio.* $E^{Q^1}[\max(X_T - 1, 0)]$ is the undiscounted Black-Scholes price of a call with spot $X_0 = S_2/S_1$, strike $1$, rate $0$ and volatility $\sigma_s$: $$E^{Q^1}[\max(X_T - 1,0)] = X_0\,N(d_1) - N(d_2), \qquad d_1 = \frac{\ln(S_2/S_1) + \tfrac12\sigma_s^2 T}{\sigma_s\sqrt{T}}, \quad d_2 = d_1 - \sigma_s\sqrt{T}.$$
*Step 4 -- Convert back to dollars.* Multiply by $S_1$: $$V_0 = S_2\,N(d_1) - S_1\,N(d_2).$$ Neither $r$ nor the real-world drifts $\mu_i$ appear: both assets are traded, so their risk-neutral drifts are $r$ and cancel in the ratio.
*Step 5 -- Example and sanity checks.* $S_1 = S_2 = 100$, $\sigma_1 = 0.2$, $\sigma_2 = 0.3$, $\rho = 0.5$, $T = 1$: $\sigma_s = 0.2646$, $d_1 = 0.1323$, $d_2 = -0.1323$, $V_0 = 100\,[N(0.1323) - N(-0.1323)] = 10.52$. If $\rho \to 1$ and $\sigma_1 = \sigma_2$, then $\sigma_s \to 0$ and the option is worth $\max(S_2 - S_1, 0)$ today (the stocks move in lockstep). If $S_1$ is replaced by a constant $K$ (so $\sigma_1 = 0$ and $S_1$ is a bond worth $Ke^{-rT}$), the formula collapses to Black-Scholes.
Answer: $V_0 = S_2\,N(d_1) - S_1\,N(d_2)$ with $\sigma_s = \sqrt{\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2}$, $d_1 = \dfrac{\ln(S_2/S_1) + \sigma_s^2 T/2}{\sigma_s\sqrt{T}}$, $d_2 = d_1 - \sigma_s\sqrt{T}$ (Margrabe's formula). For $S_1 = S_2 = 100$, $\sigma_1 = 20\%$, $\sigma_2 = 30\%$, $\rho = 0.5$, $T = 1$ the price is $10.52$.
Intuition
An exchange option is an ordinary call once you change the unit of account: measured in shares of stock 1, it pays $\max(S_2/S_1 - 1, 0)$, a call on the ratio with strike $1$. The ratio of two lognormals is lognormal with volatility given by the variance of the difference of their log returns, $\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2$, and under the stock-1 numeraire it has no drift, so the interest rate disappears. This is Margrabe's formula, and the same numeraire trick prices spread options, best-of and worst-of baskets, quanto and FX cross options; higher correlation makes the two stocks move together, shrinks $\sigma_s$, and makes the right to swap them cheaper.