Is Swaption an Exchange Option?
October 30, 2006
European swaptions are universally valued using Black's closed form model, which itself is derived from the closed form Black-Scholes model for pricing options. What is a swaption? A swaption is an option to enter into an interest rate swap. In a receiver swaption, or the put swaption, the buyer enters into a forward swap to receive fixed and pay floating interest rates. In a payer swaption, or the call swaption, the buyer enters into a forward swap to pay fixed and receive floating interest rates.
In the usual Black's model for the swaption the strike price is , the fixed rate. Thus the payoff of a payer swaption becomes:
Where, L is the floating rate and K is the fixed rate. But the option has a maturity and at maturity both the fixed rate and the floating rate will have some value that is not known today.
If at maturity the value of the fixed rate is say, and the value of the floating rate is then the payoff function for a payer swaption should actually be:
In the above payoff function, , the value of the fixed interest rate is used as a strike price rather than simply, which is the fixed interest rate. Why is the difference between the two payoff function?
This is because the interest rates will move during the life of a swaption, which is a trivial yet an important observation. And if interest rates move during the life of the swaption then the value of the fixed rate as well as the floating rate, both will change. Therefore, the payoff of the swaption should, strictly speaking, not be in the form of (1) but rather in the form of (2) above.
Therefore, the payoff equation (2) is not in the usual form of a Black or the Black-Scholes model but rather it is in the form of an exchange option. In Black's model there is one stochastic underlying and a fixed strike (as shown in equation (1)) but in equation (2) the payoff of the swaption has two stochastic underlying, i.e. the fixed and the floating rates.
Some practitioners have therefore advocated that closed form exchange option valuation model of Margrabe should be used to value a swaption.
However, as it so happens, almost all traders and quants price the swaption using Black's closed form model or even if a Monte Carlo simulation model is used, the payoff function that is taken is that of (1) and not (2). This is made possible by manipulating the values and in such a way as to fit the Black's payoff function. But it would be interesting to see if the values of swaption would vary greatly - from the conventional Black's model - if Margrabe's exchange option closed form solution is used.
Reference: Interested readers may check out Richard Flavell's Swaps & Other Derivatives for more on this.
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