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Option pricing in the presence of natural boundaries and a quadratic diffusion term (*)

Author

Listed:
  • Sven Rady

    (Graduate School of Business, Stanford University, Stanford, CA 94305-5015, USA)

Abstract

This paper uses a probabilistic change-of-numeraire technique to compute closed-form prices of European options to exchange one asset against another when the relative price of the underlying assets follows a diffusion process with natural boundaries and a quadratic diffusion coefficient. The paper shows in particular how to interpret the option price formula in terms of exercise probabilities which are calculated under the martingale measures associated with two specific numeraire portfolios. An application to the pricing of bond options and certain interest rate derivatives illustrates the main results.

Suggested Citation

  • Sven Rady, 1997. "Option pricing in the presence of natural boundaries and a quadratic diffusion term (*)," Finance and Stochastics, Springer, vol. 1(4), pages 331-344.
    • Handle: RePEc:spr:finsto:v:1:y:1997:i:4:p:331-344
    Note: received: January 1996; final version received: December 1996
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    References listed on IDEAS

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    Cited by:

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    2. Alexander Lipton & Andrey Gal & Andris Lasis, 2013. "Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: survey and new results," Papers 1312.5693, arXiv.org.
    3. Sanjay K. Nawalkha & Xiaoyang Zhuo, 2022. "A Theory of Equivalent Expectation Measures for Contingent Claim Returns," Journal of Finance, American Finance Association, vol. 77(5), pages 2853-2906, October.
    4. Alexander Lipton & Andrey Gal & Andris Lasis, 2014. "Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1899-1922, November.
    5. U Hou Lok & Yuh-Dauh Lyuu, 2022. "A Valid and Efficient Trinomial Tree for General Local-Volatility Models," Computational Economics, Springer;Society for Computational Economics, vol. 60(3), pages 817-832, October.
    6. Antoine Jacquier & Martin Keller-Ressel, 2015. "Implied volatility in strict local martingale models," Papers 1508.04351, arXiv.org.
    7. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    8. Peter Carr, 2017. "Bounded Brownian Motion," Risks, MDPI, vol. 5(4), pages 1-11, November.
    9. Martin HERDEGEN & Martin SCHWEIZER, 2015. "Economics-Based Financial Bubbles (and Why They Imply Strict Local Martingales)," Swiss Finance Institute Research Paper Series 15-05, Swiss Finance Institute.
    10. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "Why are quadratic normal volatility models analytically tractable?," Papers 1202.6187, arXiv.org, revised Mar 2013.
    11. Zühlsdorff, Christian, 2002. "The Pricing of Derivatives on Assets with Quadratic Volatility," Bonn Econ Discussion Papers 5/2002, University of Bonn, Bonn Graduate School of Economics (BGSE).
    12. Hui, Cho-Hoi & Lo, Chi-Fai & Liu, Chi-Hei, 2022. "Exchange rate dynamics with crash risk and interventions," International Review of Economics & Finance, Elsevier, vol. 79(C), pages 18-37.
    13. Martin Herdegen & Martin Schweizer, 2016. "Strong Bubbles And Strict Local Martingales," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-44, June.
    14. Antje Mahayni, 2003. "Effectiveness of Hedging Strategies under Model Misspecification and Trading Restrictions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 6(05), pages 521-552.
    15. Erik Schlögl, 2001. "Arbitrage-Free Interpolation in Models of Market Observable Interest Rates," Research Paper Series 71, Quantitative Finance Research Centre, University of Technology, Sydney.
    16. Samuel Drapeau & Yunbo Zhang, 2019. "Pricing and Hedging Performance on Pegged FX Markets Based on a Regime Switching Model," Papers 1910.08344, arXiv.org, revised May 2020.
    17. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.
    18. Christian Zuhlsdorff, 2001. "The pricing of derivatives on assets with quadratic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 8(4), pages 235-262.

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    More about this item

    Keywords

    Option pricing; bond options; change-of-numeraire technique; diffusion process; quadratic diffusion terms;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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