Following on our previous article, let's talk a bit more about the exchange option. An exchange option is the option to exchange one asset for the other. The two assets could be two yield curves or any two equity indices. The holder (buyer) of an exchange option has the option to exchange one asset for the other at the maturity.

Let's look at the payoff of an exchange option. The generalized payoff is given by:

........................(1)

In the above payoff, equation and are constants. It is an interesting payoff. Why? First, because the strike price of the option is not obvious. Both and are underlying assets, so what is the strike price of the option? Secondly, it is an interesting payoff because the dimensionality of the above can be reduced from three to two and thereby the above payoff can be tied to the payoff of another exotic option, namely, the ratio option.

In the above payoff equation if we do the following transformation then:

..........(2)

From the above transformed payoff we see that the ratio of becomes the strike price of the option. is the ratio of two assets. In the first equation the payoff is a function of three variables, , and the time variable. But in the second equation the payoff is a function of only two variables and the time variable because the function that needs to be evaluated for option payoff is now .

The above is an example of what we call "similarity reduction". A third, very interesting feature of an exchange option comes to light if we examine the value of the hedged portfolio in the context of delta hedging. Interested readers are referred to Wilmott for an analysis.

* Reference: The above is based on Paul Wilmott (2006). *

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