Journal of Economics and Financial Analysis, 2 (2), pp. 87-103, [2018]

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



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.


Interest Rate Swaps; European Swaption Pricing; Martingale Representation Theorem; Radon-Nikodym Derivative; Generalized Black-Scholes Model.

JEL Classification

C02, C20, E43, E47, E49, G15.

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Baxter, M., and Rennie, A. (1966). Textbook: Financial Calculus – An Introduction to Derivatives Pricing. Cambridge University Press.

Black, F., and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.

Black, F. (1976). The Pricing of Commodity Contracts. Journal of Financial Economics, 3(1-2), 167-179.

Burgess, N. (2014). Martingale Measures & Change of Measure Explained. Available at SSRN: or

Burgess, N. (2017a). How to Price Swaps in Your Head - An Interest Rate Swap & Asset Swap Primer. Available at SSRN: or

Burgess, N. (2017b). A Review of the Generalized Black-Scholes Formula & It’s Application to Different Underlying Assets. Available at SSRN: or

Derman, E., and Taleb, N. (2005). The Illusion of Dynamic Delta Replication. Quantitative Finance, 5(4), 323-326.

Hull, J. (2011). Textbook: Options, Futures and Other Derivatives. 8th ed., Pearson Education Limited

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