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?