# Determining swaption prices using the characteristic function

There exist multiple techniques to determine call option prices that make use of the characteristic function. These techniques boil down to some integral expression of the option price in terms of the characteristic function. A popular approach uses the Fast Fourier Transform, see Carr and Madan, 1999.

Most literature out there revolves around applying these techniques to equity models, such as the Heston model. I'm interested in applying it to the interest-rate models, such as Hull-White.

Now, the results obtained by Carr and Madan are not as relevant in the the IR world, since here the most liquid options are Swaptions. These are like Call Options on the Swap Rate. But most IR models do not model the Swap Rate directly.

I suppose therefore that what we need is an expression for the characteristic function with respect to the Swap Rate. But I haven't found any literature on this; the few papers I found focus on the characteristic function of the (integrated) short rate (i.e. Zero Coupons Bonds). I suppose you can apply these to Interest Rate Caps, but not simply to Swaptions.

Are there any good references that apply the characteristic-function approach to the interest rate world? Does there exist an expression for the price of a Swaption in terms of a characteristic function?

## 1 Answer

Pricing via characteristic functions arises naturally in models that involve Levy processes. Therefore I can see how Black's formula for swaptions can be generalized for Levy dynamics:

1. As in Black's model take the annuity as numeraire, and define the relevant measure $Q$
2. Black assumes that under this measure the swap rate is martingale GBM, that is to say $$dS = \sigma S dW \text{, or that }d\log S= -\sigma^2/2+\sigma dW$$ You can assume instead that log-swap is a Levy process with the appropriate compensator
3. This should lead to swaption pricing via characteristic function

(As a matter of fact, you could calculate Black's formula via cf if you really wanted to.)