Pricing Swaptions with Characteristic Functions under Lévy Dynamics
Summary
The document considers whether characteristic function methods used to price options can be adapted to interest rate derivatives, especially swaptions. It explains the key difficulty: many interest rate models describe short rates or zero coupon bonds, while swaptions depend on the swap rate. The swap rate is not directly modeled in many such frameworks.
The proposed route is to use the swap annuity as numeraire and price under the resulting measure. In the familiar Black setup, the swap rate follows a martingale geometric Brownian motion; the answer suggests instead modeling its logarithm as a Lévy process with an appropriate compensator. This can lead to a characteristic function approach for swaption pricing. The response sketches a modeling idea rather than deriving a pricing integral or giving references, implementation details, or numerical evidence. Its applicability depends on specifying suitable swap rate dynamics and measure assumptions.
Key ideas
- Swaptions depend on swap rates, which many interest rate models do not model directly.
- Using the swap annuity as numeraire defines a measure suited to swaption valuation.
- A Lévy process for the logarithm of the swap rate can extend the Black model setup.
- Characteristic function pricing requires appropriate dynamics and compensation under the chosen measure.
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# Determining swaption prices using the characteristic function
# 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?
## Answer by Kiwiakos (score 2)
https://quant.stackexchange.com/a/17351
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:
- As in Black's model take the annuity as numeraire, and define the relevant measure $Q$
- 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
- 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.)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.