Recently we had an interesting discussion in one of our training sessions regarding exchange options. The option is a pretty simple vanilla structure and the solution in closed form (Black-Scholes type) is trivial. In that respect there is nothing interesting about the exchange option. However, like many basket or multi-asset products an exchange option is a correlation play and it is interesting to see how correlation impacts the price of the product. Secondly, an exchange option can be used in managing pension liabilities vis-a-vis assets of a pension fund and that transformation is also pretty unique. We shall discuss that in a subsequent article.

But let's see what is an exchange option and how does the correlation impact its value.

Say an investor is betting on the relative performance of two stock indices in a year's time and in his opinion the return from the first stock index (say, FTSE) will outperform the return from the second stock index (say, DAX). In that case he can buy a one year exchange option where the payoff of this option can be written as:

Where, the return on the index could be defined as

The above is an option to exchange the return from DAX for the return from FTSE. An exchange option is a spread product, i.e. the payoff is a spread between two assets. In general the payoff of an exchange option can be written as:

In the above formula, and are any two constants and and are two assets. The correlation between the two assets enters through the volatility of the spread between them. An exchange option is a spread product by definition and the formula for the volatility of a spread, say between two equity indices (or two yields) is given by:

In the above formula, is the volatility of the first asset (equity index or the yield), is the volatility of the second asset (again the second equity index or the yield) and is the instantaneous correlation between the two asset returns.

From the above formula we see that a holder of an exchange option is essentially betting that the correlation between the two assets will drop from the present level (the level at which the holder buys the option). If the correlation becomes more negative, i.e. if the two assets move apart from each other and in opposite directions, then the exchange option will gain in value. The opposite will happen if the two assets get more aligned with each other and the correlation between them increases.

Closed form solutions for exchange options exist (Margrabe's formula) and you can check out the impact of the correlation by using that formula. Say, a stock is trading at 100 and a second stock is also trading at 50. And both of them belong to the same industry. The first One has a dividend yield of 1.5% and the second stock has a dividend yield of 3.15%. The volatilities are 15% and 10% respectively and the correlation between the two stocks is currently at 0.5. If the investor believes that these two stocks will move away from each other, i.e the correlation between them will decrease then he'll want to buy an exchange option whereby at maturity he exchanges asset two and acquires asset one. And if the correlation indeed drops from the present level of 0.5 and moves into negative territory then the option would be worth more and he will make money. The valuation is pretty straightforward.

We use the formula for the spread volatility as shown above and Margrabe's formula as shown below:

We use the above parameters for both the assets and a maturity of one year. The graph below depicts the variation of option values with respect to the correlation:

The above makes sense intuitively as well. Why would an investor buy an option to exchange one asset with another? He will do so if he believes that one asset will be worth much more than the other one at maturity. In the above example, the only way that the first stock will be worth more than the second stock will be if it goes up and the second one goes down, or does not go up by as much from their present levels. This can only happen if the correlation between the two stocks starts to drop from the present level of 0.5.

Of course, the other scenario is also possible: a falling correlation can make the second stock go up and the first one go down, thereby lowering the value of the exchange option for the investor. But the necessary condition for stock one to go up and stock two to go down or not to go up by as much - something which the investor (buyer of the option) is betting on - will have to be decreasing correlation.

We would like to ask our readers the following questions:

**Questions: **

- Is the above a correlation play? And if so, then is the investor long the correlation or short the correlation?

- How will the seller of the above option (presumably the bank) hedge itself?

- How can the above product be decomposed into a "best-of" or "worst-of" product?

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