## Exchange options

Exchange options grant the holder the right to exchange one risky asset to another risky asset at maturity. It is easy to see that simple options are special forms of exchange options where one of the risky assets is a constant amount of money (the strike price).

The pricing formula of an exchange option was first derived by *Margrabe, 1978*. The model assumptions, the pricing principles, and the resultant formula of `Margrabe`

are very similar to (more precisely, the generalization of) those of Black, Scholes, and Merton. Now we will show how to determine the value of an exchange option.

Let's denote the spot prices of the two risky assets at time *t* by *S _{1t}* and

*S*. We assume that these prices under the risk neutral probability measure (

_{2t}*Q*) follow geometric Brownian motion with drifts equal to the risk-free rate (

*r*), shown as and .

Here, *W _{1}* and

*W*are standard Wiener processes under

_{2}*Q*, with correlation

*ρ*. You may observe that here, the assets have no yield (for example, stocks...