# Interest Rate Swaptions - A Review & Derivation of Swaption Pricing Formulae

Journal of Economics and Financial Analysis, Vol:2, No:2 (2018) 87-103

15 Pages Posted: 5 Jan 2018 Last revised: 16 Feb 2018

See all articles by Nicholas Burgess

## Nicholas Burgess

University of Oxford, Said Business School

Date Written: February 4, 2018

### Abstract

In this paper we outline the European interest rate swaption pricing formula from first principles using the Martingale Representation Theorem and the annuity measure. This leads to an expression that allows us to apply the generalized Black-Scholes result. We show that a swaption pricing formula is nothing more than the Black-76 formula scaled by the underlying swap annuity factor.

Firstly we review the Martingale Representation Theorem for pricing options, which allows us to price options under a numeraire of our choice. We also highlight and consider European call and put option pricing payoffs. Next we discuss how to evaluate and price an interest swap, which is the swaption underlying instrument. We proceed to examine how to price interest rate swaptions using the martingale representation theorem with the annuity measure to simplify the calculation. Finally applying the Radon-Nikodym derivative to change measure from the annuity measure to the savings account measure we arrive at the swaption pricing formula expressed in terms of the Black-76 formula. We also provide a full derivation of the generalized Black-Scholes formula for completeness.

Keywords: Interest Rate Swaps, Swaptions, European Swaption Pricing, Cash/Physical Settlement, Martingale Representation Theorem, Annuity Martingale Measure, Radon-Nikodym Derivative, Change of Measure, Generalized Black-Scholes, Black-76 Model

JEL Classification: A30, A31, A33, C02, C20, C50, C60, C65, G15

Suggested Citation

Burgess, Nicholas, Interest Rate Swaptions - A Review & Derivation of Swaption Pricing Formulae (February 4, 2018). Journal of Economics and Financial Analysis, Vol:2, No:2 (2018) 87-103 , Available at SSRN: https://ssrn.com/abstract=3095336 or http://dx.doi.org/10.2139/ssrn.3095336