Pricing a Two-Asset Exchange Option with the Margrabe Formula

Summary

The note identifies a payoff equal to the positive difference between two asset prices at maturity as an exchange option. This is equivalent to a call on one asset priced in units of the other, with a zero strike. For a maturity of three years in the stated problem, the suggested route is to derive each asset’s forward price from its price dynamics and dividend yield, then determine both volatilities and the correlation between their returns.

Those inputs allow the use of the Margrabe formula to value the option. The document gives a method outline rather than carrying out the calculation, so it does not provide a numerical price or show intermediate formulas. Applying the approach requires the assets’ volatility and correlation information as well as their forward values; dividend yields alone are insufficient. The answer assumes the provided price equations supply the needed dynamics and parameters.

Key ideas

• A payoff based on the positive difference between two asset prices is an exchange option.
• The exchange-option payoff can be viewed as a zero-strike call on one asset relative to the other.
• Pricing requires both assets’ forward values, volatilities, and return correlation.
• The Margrabe formula provides the valuation method once those inputs are known.

Tags

# How to calculate the price of an asset using Black-Scholes equation?


# How to calculate the price of an asset using Black-Scholes equation?












I'm trying to solve this problem given:

> The dividend yield for asset 1 (asset 2) is 0.05 (0.03), and it is also given the time zero stock prices, and both assets' Black-Scholes equation.

I need to find the time zero price of an asset which pays:

$[max(S_{3}^{1}-S_{3}^{2},0)]$ at T = 3.

Just wanted to get a hint on how to approach the problem?

Thanks!

## Answer by ZRH (score 3)

https://quant.stackexchange.com/a/44225

Basically what you are after is an exchange option on the two assets, i.e., a zero-strike call option on the price difference between `asset1` and `asset2`. Having the equations of motion for both asset prices, you are in a position to work out:

i) the forward prices for both assets at $T=3$;

ii) both assets' volatilities $\sigma_1$ and $\sigma_2$; AND

iii) the correlation coefficient $\rho_{1,2}$.

Having all of this, you can apply the Margrabe formula Margrabe's formula on Wikipedia

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.