Pricing Exchange Options with the Margrabe Formula
Summary
The document considers an option that pays the positive difference between two asset values at expiry. Rather than extending a basic binomial tree to track combinations of movements in both assets, the answer points to the Margrabe formula under Black–Scholes assumptions. It defines the effective volatility of the asset ratio from each asset’s volatility and their correlation, then gives the option value using the starting asset prices, dividend rates, expiry, and normal cumulative probabilities.
This offers a direct pricing expression for a European exchange option, avoiding explicit enumeration of joint binomial paths. Its applicability depends on the model assumptions, including the specified volatilities, correlation, and dividend yields. The document does not explain how to estimate these inputs or how to adapt the result for early exercise or departures from the assumed price dynamics.
Key ideas
- An exchange option pays the positive difference between two asset values at expiry.
- The Margrabe formula provides a direct valuation under the two-asset Black–Scholes framework.
- The formula combines the assets’ volatilities and correlation into an effective volatility.
- Dividend rates and the assets’ initial values also enter the price.
- The result does not address input estimation, early exercise, or model departures.
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Full text
# How to modify binomial tree to incorporate one more asset?
# How to modify binomial tree to incorporate one more asset?
I wonder, what would happen if we use the binomial tree to price exchange option, an option to exchange one asset for another at the expiry date. Payoff is $\max(S_1-S_2,0)$
For instance, I have two assets whose payoff are the following: $\begin{bmatrix}1.1&0.9\\1.1&1.1\\0.9&1.1\end{bmatrix}$, and $S_0^1=100$ and $S_0^2=95$, risk free rate is $R_f=4\%$ and $T=6$
How to deal with the one more dimension? My intuition is to use the binomial tree as usual. However, it is difficult to tell how many possible paths lead to a payoff, given that I have listed all possible combination of payoffs of asset 1 and 2.
## Answer by Ezy (score 3, accepted)
https://quant.stackexchange.com/a/43876
Under Black-Scholes assumption for the 2 assets $S_1$ and $S_2$ with volatilities $\sigma_{1,2}$ and correlation $\rho$ the value of this option has an explicit expression which is the Margrabe formula
To quote the result explicitly
Introducing $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 - 2 \sigma_1\sigma_2\rho}$, Margrabe's formula states that the fair price for the option at time 0 and expiry $T$ is:
$$e^{-q_1 T}S_1(0) N(d_1) - e^{-q_2 T}S_2(0) N(d_2)$$
where $q_1,q_2$ are the expected dividend rates of the prices $S_1,S_2$ under the appropriate risk-neutral measure, $N$ denotes the cumulative distribution function for a normal distribution,
$$d_1 = (\ln (S_1(0)/S_2(0)) + (q_2 - q_1 + \sigma^2/2)T)/ \sigma\sqrt{T}$$, $$d_2 = d_1 - \sigma\sqrt{T}$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.