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Bond, futures and option evaluation in the quadratic interest rate model

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  • Farshid Jamshidian

Abstract

This paper develops the quadratic interest-rate model of Beaglehole and Tenney in detail. For the quadratic model as well as the multifactor Cox-Ingersoll-Ross square-root model, explicit pricing formulae in terms of one-dimensional integrals of elementary functions are given for bond options, bond exchange options, caps, options on bond futures and forward contracts, and futures delivery options. For the quadratic model, certain forward and transport equations are found that explicitly determine the dynamics of the term structure in terms of initial yield and volatility curves. These option-pricing formulae are thus determined in term of the initial curves. Some shortcomings of the model are identified. New formulae for some distributions and their truncated moments are also derived.

Suggested Citation

  • Farshid Jamshidian, 1996. "Bond, futures and option evaluation in the quadratic interest rate model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(2), pages 93-115.
    • Handle: RePEc:taf:apmtfi:v:3:y:1996:i:2:p:93-115
    DOI: 10.1080/13504869600000005
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    Cited by:

    1. Nicole El Karoui & Caroline Hillairet & Mohamed Mrad, 2022. "Ramsey rule with forward/backward utility for long-term yield curves modeling," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 375-414, June.
    2. Leippold, Markus & Wu, Liuren, 2002. "Asset Pricing under the Quadratic Class," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 37(2), pages 271-295, June.
    3. Andrea Macrina, 2012. "Heat Kernel Framework for Asset Pricing in Finite Time," Papers 1211.0856, arXiv.org, revised Sep 2013.
    4. Björk, Tomas & Landen, Camilla, 2000. "On the Term Structure of Futures and Forward Prices," SSE/EFI Working Paper Series in Economics and Finance 0417, Stockholm School of Economics, revised 20 Dec 2000.
    5. Gibson, Rajna & Lhabitant, Francois-Serge & Talay, Denis, 2010. "Modeling the Term Structure of Interest Rates: A Review of the Literature," Foundations and Trends(R) in Finance, now publishers, vol. 5(1–2), pages 1-156, December.
    6. Peng Cheng & Olivier Scaillet, 2002. "Linear-Quadratic Jump-Diffusion Modeling with Application to Stochastic Volatility," FAME Research Paper Series rp67, International Center for Financial Asset Management and Engineering.
    7. Satoshi Yamashita & Toshinao Yoshiba, 2011. "Analytical Solution for the Loss Distribution of a Collateralized Loan under a Quadratic Gaussian Default Intensity Process," IMES Discussion Paper Series 11-E-20, Institute for Monetary and Economic Studies, Bank of Japan.
    8. Markus Leippold & Zvi Wiener, 2005. "Efficient Calibration of Trinomial Trees for One-Factor Short Rate Models," Review of Derivatives Research, Springer, vol. 7(3), pages 213-239, October.
    9. Enlin Pan & Liuren Wu, 2006. "Taking Positive Interest Rates Seriously," World Scientific Book Chapters, in: Cheng-Few Lee (ed.), Advances In Quantitative Analysis Of Finance And Accounting, chapter 14, pages 327-356, World Scientific Publishing Co. Pte. Ltd..
    10. Frédéric Godin & Ramin Eghbalzadeh & Patrice Gaillardetz, 2023. "Pricing swaptions and zero-coupon futures options under the discrete-time arbitrage-free Nelson–Siegel model," Review of Derivatives Research, Springer, vol. 26(2), pages 171-206, October.
    11. Gaspar, Raquel M., 2004. "General Quadratic Term Structures of Bond, Futures and Forward Prices," SSE/EFI Working Paper Series in Economics and Finance 559, Stockholm School of Economics.
    12. Christa Cuchiero & Claudio Fontana & Alessandro Gnoatto, 2019. "Affine multiple yield curve models," Mathematical Finance, Wiley Blackwell, vol. 29(2), pages 568-611, April.
    13. Niels Rom-Poulsen, 2007. "Semi-analytical MBS Pricing," The Journal of Real Estate Finance and Economics, Springer, vol. 34(4), pages 463-498, May.
    14. Rafael Mendoza-Arriaga & Vadim Linetsky, 2014. "Time-changed CIR default intensities with two-sided mean-reverting jumps," Papers 1403.5402, arXiv.org.
    15. Markus Leippold & Liuren Wu, 2003. "Design and Estimation of Quadratic Term Structure Models," Review of Finance, European Finance Association, vol. 7(1), pages 47-73.
    16. Ben-Ameur, Hatem & Breton, Michele & Karoui, Lotfi & L'Ecuyer, Pierre, 2007. "A dynamic programming approach for pricing options embedded in bonds," Journal of Economic Dynamics and Control, Elsevier, vol. 31(7), pages 2212-2233, July.
    17. Hans-Peter Bermin, 2012. "Bonds and Options in Exponentially Affine Bond Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(6), pages 513-534, December.
    18. Si Cheng & Michael R. Tehranchi, 2015. "Polynomial term structure models," Papers 1504.03238, arXiv.org, revised Dec 2020.
    19. Tomoaki Shouda, 2005. "Dynamical analysis of corporate bonds based on the yield spread term-quality surface," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 12(4), pages 307-332, December.
    20. Satoshi Yamashita & Toshinao Yoshiba, 2013. "A collateralized loan's loss under a quadratic Gaussian default intensity process," Quantitative Finance, Taylor & Francis Journals, vol. 13(12), pages 1935-1946, December.
    21. Rogers, L. C. G. & Stummer, Wolfgang, 2000. "Consistent fitting of one-factor models to interest rate data," Insurance: Mathematics and Economics, Elsevier, vol. 27(1), pages 45-63, August.
    22. Dijkstra, Theo K. & Yao, Yong, 2002. "Moment generating function approach to pricing interest rate and foreign exchange rate claims," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 163-178, October.
    23. Li, Chenxu & Wu, Linjia, 2019. "Exact simulation of the Ornstein–Uhlenbeck driven stochastic volatility model," European Journal of Operational Research, Elsevier, vol. 275(2), pages 768-779.
    24. Duffie, Darrell, 2003. "Intertemporal asset pricing theory," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 11, pages 639-742, Elsevier.

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