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

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  • Nicholas Burgess

    (University of Reading)

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.

Suggested Citation

  • Nicholas Burgess, 2018. "Interest Rate Swaptions: A Review and Derivation of Swaption Pricing Formulae," Journal of Economics and Financial Analysis, Tripal Publishing House, vol. 2(2), pages 87-103.
    • Handle: RePEc:trp:01jefa:jefa0019
    DOI: http://dx.doi.org/10.1991/jefa.v2i2.a19
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    References listed on IDEAS

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    1. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    2. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
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    More about this item

    Keywords

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

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
    • E47 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Forecasting and Simulation: Models and Applications
    • E49 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Other
    • G15 - Financial Economics - - General Financial Markets - - - International Financial Markets

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