Interest Rate Swaptions: A Review and Derivation of Swaption Pricing Formulae

Nicholas Burgess ()
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Nicholas Burgess: University of Reading

Journal of Economics and Financial Analysis, 2018, vol. 2, issue 2, 87-103

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; European Swaption Pricing; Martingale Representation Theorem; Radon-Nikodym Derivative; Generalized Black-Scholes Model. (search for similar items in EconPapers)
JEL-codes: C02 C20 E43 E47 E49 G15 (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:trp:01jefa:jefa0019

DOI: 10.1991/jefa.v2i2.a19

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