IDEAS home Printed from https://ideas.repec.org/a/trp/01jefa/jefa0019.html
   My bibliography  Save this article

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

Author

Listed:
  • 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: https://dx.doi.org/10.1991/jefa.v2i2.a19
    as

    Download full text from publisher

    File URL: https://ojs.tripaledu.com/jefa/article/download/38/40
    Download Restriction: no
    File URL: https://ojs.tripaledu.com/jefa/article/view/38/44
    Download Restriction: no
    File URL: https://libkey.io/https://dx.doi.org/10.1991/jefa.v2i2.a19?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Turvey, Calum G., 2001. "Random Walks And Fractal Structures In Agricultural Commodity Futures Prices," Working Papers 34151, University of Guelph, Department of Food, Agricultural and Resource Economics.
    2. Bujar Huskaj & Marcus Nossman, 2013. "A Term Structure Model for VIX Futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(5), pages 421-442, May.
    3. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    4. Gustavo Silva Araujo & Ricardo Alves Carmo Ribeiro, 2018. "Is Petrobras Options Market Efficient? A Study Using The Delta-Gamma Neutral Strategy," Anais do XLIV Encontro Nacional de Economia [Proceedings of the 44th Brazilian Economics Meeting] 126, ANPEC - Associação Nacional dos Centros de Pós-Graduação em Economia [Brazilian Association of Graduate Programs in Economics].
    5. Rodriguez, Ricardo J., 2002. "Lognormal option pricing for arbitrary underlying assets: a synthesis," The Quarterly Review of Economics and Finance, Elsevier, vol. 42(3), pages 577-586.
    6. Tabesh, Hamid, 1987. "Hedging price risk to soybean producers with futures and options: a case study," ISU General Staff Papers 1987010108000010306, Iowa State University, Department of Economics.
    7. Christoffersen, Peter & Jacobs, Kris & Chang, Bo Young, 2013. "Forecasting with Option-Implied Information," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 581-656, Elsevier.
    8. Chuong Luong & Nikolai Dokuchaev, 2018. "Forecasting of Realised Volatility with the Random Forests Algorithm," JRFM, MDPI, vol. 11(4), pages 1-15, October.
    9. Samson Assefa, 2007. "Pricing Swaptions and Credit Default Swaptions in the Quadratic Gaussian Factor Model," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 3-2007, January-A.
    10. Robert McDonald & Daniel Siegel, 1981. "Option Pricing When the Underlying Asset is Non-Stored," Discussion Papers 512, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    11. Back, Janis & Prokopczuk, Marcel & Rudolf, Markus, 2013. "Seasonality and the valuation of commodity options," Journal of Banking & Finance, Elsevier, vol. 37(2), pages 273-290.
    12. Andrew Carver & Matthew Ennis, 2011. "The real options content of oil producer stocks," Applied Financial Economics, Taylor & Francis Journals, vol. 21(4), pages 217-231.
    13. Szakmary, Andrew & Ors, Evren & Kyoung Kim, Jin & Davidson, Wallace III, 2003. "The predictive power of implied volatility: Evidence from 35 futures markets," Journal of Banking & Finance, Elsevier, vol. 27(11), pages 2151-2175, November.
    14. Ako Doffou & Jimmy E. Hilliard, 2001. "Pricing Currency Options Under Stochastic Interest Rates And Jump-Diffusion Processes," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 24(4), pages 565-585, December.
    15. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    16. Carey, Alexander, 2006. "Path-conditional forward volatility," MPRA Paper 4964, University Library of Munich, Germany.
    17. Siriopoulos, Costas & Fassas, Athanasios, 2012. "An investor sentiment barometer — Greek Implied Volatility Index (GRIV)," Global Finance Journal, Elsevier, vol. 23(2), pages 77-93.
    18. Dew-Becker, Ian & Giglio, Stefano & Kelly, Bryan, 2021. "Hedging macroeconomic and financial uncertainty and volatility," Journal of Financial Economics, Elsevier, vol. 142(1), pages 23-45.
    19. Daniel Guterding, 2023. "Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities," Risks, MDPI, vol. 11(5), pages 1-24, April.
    20. Paul Handro & Bogdan Dima, 2024. "Analyzing Financial Markets Efficiency: Insights from a Bibliometric and Content Review," Journal of Financial Studies, Institute of Financial Studies, vol. 16(9), pages 119-175, May.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:trp:01jefa:jefa0019. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: David Simon Hall (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.