IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v09y2006i07ns0219024906003962.html
   My bibliography  Save this article

A New Analytical Approximation Formula For The Optimal Exercise Boundary Of American Put Options

Author

Listed:
  • SONG-PING ZHU

    (School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia)

Abstract

In this paper, a new analytical formula as an approximation to the value of American put options and their optimal exercise boundary is presented. A transform is first introduced to better deal with the terminal condition and, most importantly, the optimal exercise price which is an unknown moving boundary and the key reason that valuing American options is much harder than valuing its European counterparts. The pseudo-steady-state approximation is then used in the performance of the Laplace transform, to convert the systems of partial differential equations to systems of ordinary differential equations in the Laplace space. A simple and elegant formula is found for the optimal exercise boundary as well as the option price of the American put with constant interest rate and volatility. Other hedge parameters as the derivatives of this solution are also presented.

Suggested Citation

  • Song-Ping Zhu, 2006. "A New Analytical Approximation Formula For The Optimal Exercise Boundary Of American Put Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(07), pages 1141-1177.
    • Handle: RePEc:wsi:ijtafx:v:09:y:2006:i:07:n:s0219024906003962
    DOI: 10.1142/S0219024906003962
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024906003962
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219024906003962?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Wilmott,Paul & Howison,Sam & Dewynne,Jeff, 1995. "The Mathematics of Financial Derivatives," Cambridge Books, Cambridge University Press, number 9780521497893, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Maria do Rosário Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Pricing American Call Option by the Black-Scholes Equation with a Nonlinear Volatility Function," Working Papers REM 2017/18, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    2. Lu, Xiaoping & Putri, Endah R.M., 2020. "A semi-analytic valuation of American options under a two-state regime-switching economy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    3. Jose Cruz & Daniel Sevcovic, 2020. "On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models," Papers 2003.03851, arXiv.org.
    4. Maria do Rosario Grossinho & Yaser Kord Faghan & Daniel Sevcovic, 2016. "Pricing Perpetual Put Options by the Black-Scholes Equation with a Nonlinear Volatility Function," Papers 1611.00885, arXiv.org, revised Nov 2017.
    5. Jean-Philippe Aguilar & Cyril Coste & Hagen Kleinert & Jan Korbel, 2016. "Distributional Mellin calculus in $\mathbb{C}^n$, with applications to option pricing," Papers 1611.03239, arXiv.org, revised Nov 2016.
    6. Tomas Bokes, 2010. "A unified approach to determining the early exercise boundary position at expiry for American style of general class of derivatives," Papers 1012.0348, arXiv.org, revised Mar 2011.
    7. Anna Clevenhaus & Matthias Ehrhardt & Michael Günther & Daniel Ševčovič, 2020. "Pricing American Options with a Non-Constant Penalty Parameter," JRFM, MDPI, vol. 13(6), pages 1-7, June.
    8. Song-Ping Zhu & Xin-Jiang He & XiaoPing Lu, 2018. "A new integral equation formulation for American put options," Quantitative Finance, Taylor & Francis Journals, vol. 18(3), pages 483-490, March.
    9. Zafar Ahmad & Reilly Browne & Rezaul Chowdhury & Rathish Das & Yushen Huang & Yimin Zhu, 2023. "Fast American Option Pricing using Nonlinear Stencils," Papers 2303.02317, arXiv.org, revised Oct 2023.
    10. Minqiang Li, 2010. "Analytical approximations for the critical stock prices of American options: a performance comparison," Review of Derivatives Research, Springer, vol. 13(1), pages 75-99, April.
    11. Lu, Xiaoping & Yan, Dong & Zhu, Song-Ping, 2022. "Optimal exercise of American puts with transaction costs under utility maximization," Applied Mathematics and Computation, Elsevier, vol. 415(C).
    12. Denis Veliu & Roberto De Marchis & Mario Marino & Antonio Luciano Martire, 2022. "An Alternative Numerical Scheme to Approximate the Early Exercise Boundary of American Options," Mathematics, MDPI, vol. 11(1), pages 1-12, December.
    13. Song-Ping Zhu & Nhat-Tan Le & Wen-Ting Chen & Xiaoping Lu, 2015. "Pricing Parisian down-and-in options," Papers 1511.01564, arXiv.org.
    14. Maria do Rosário Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Pricing Perpetual Put Options by the Black-Scholes Equation with a Nonlinear Volatility Function," Working Papers REM 2017/19, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    15. Zhiqiang Zhou & Hongying Wu, 2018. "Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries," Discrete Dynamics in Nature and Society, Hindawi, vol. 2018, pages 1-12, September.
    16. Zhu, Song-Ping & Chen, Wen-Ting, 2013. "An inverse finite element method for pricing American options," Journal of Economic Dynamics and Control, Elsevier, vol. 37(1), pages 231-250.
    17. Maria do Rosario Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function," Papers 1707.00358, arXiv.org, revised Jun 2018.
    18. Maria do Rosário Grossinho & Yaser Kord Faghan & Daniel Ševčovič, 2017. "Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 24(4), pages 291-308, December.

    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. Peter Buchen & Otto Konstandatos, 2005. "A New Method Of Pricing Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 245-259, April.
    2. George Chang, 2018. "Examining the Efficiency of American Put Option Pricing by Monte Carlo Methods with Variance Reduction," International Journal of Economics and Finance, Canadian Center of Science and Education, vol. 10(2), pages 10-13, February.
    3. Jiao Li, 2016. "Trading VIX Futures under Mean Reversion with Regime Switching," Papers 1605.07945, arXiv.org, revised Jun 2016.
    4. Moon, Yongma & Baran, Mesut, 2018. "Economic analysis of a residential PV system from the timing perspective: A real option model," Renewable Energy, Elsevier, vol. 125(C), pages 783-795.
    5. Zaheer Imdad & Tusheng Zhang, 2014. "Pricing European options in a delay model with jumps," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(04), pages 1-13.
    6. San-Lin Chung & Mark Shackleton, 2005. "On the use and improvement of Hull and White's control variate technique," Applied Financial Economics, Taylor & Francis Journals, vol. 15(16), pages 1171-1179.
    7. Tarn Driffield & Peter C. Smith, 2007. "A Real Options Approach to Watchful Waiting: Theory and an Illustration," Medical Decision Making, , vol. 27(2), pages 178-188, March.
    8. Xueping Wu & Jin Zhang, 1999. "Options on the minimum or the maximum of two average prices," Review of Derivatives Research, Springer, vol. 3(2), pages 183-204, May.
    9. Peter Buchen & Otto Konstandatos, 2009. "A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(6), pages 497-515.
    10. Kenji Hamatani & Masao Fukushima, 2011. "Pricing American options with uncertain volatility through stochastic linear complementarity models," Computational Optimization and Applications, Springer, vol. 50(2), pages 263-286, October.
    11. Moriconi, L., 2007. "Delta hedged option valuation with underlying non-Gaussian returns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 343-350.
    12. Masatoshi Miyake & Hiroshi Inoue & Satoru Takahashi, 2011. "Option Pricing For Weighted Average Of Asset Prices," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 28(05), pages 651-672.
    13. Ulm, Eric R., 2014. "Analytic solution for ratchet guaranteed minimum death benefit options under a variety of mortality laws," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 14-23.
    14. Xiaolin Luo & Pavel V. Shevchenko, 2014. "Fast and simple method for pricing exotic options using Gauss–Hermite quadrature on a cubic spline interpolation," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(04), pages 1-31.
    15. Conrad, Jon M. & Kotani, Koji, 2005. "When to drill? Trigger prices for the Arctic National Wildlife Refuge," Resource and Energy Economics, Elsevier, vol. 27(4), pages 273-286, November.
    16. Philippe Jacquinot & Nikolay Sukhomlin, 2010. "A direct formulation of implied volatility in the Black-Scholes model," Post-Print hal-02533014, HAL.
    17. Gao, Pei-wang, 2009. "Options strategies with the risk adjustment," European Journal of Operational Research, Elsevier, vol. 192(3), pages 975-980, February.
    18. Saphores, Jean-Daniel & Gravel, Éric & Bernard, Jean-Thomas, 2003. "Environmental Impact Assessment and Investment under Uncertainty. An Application to Power Grid Interconnection," Cahiers de recherche 0303, GREEN.
    19. L. Ingber & J.K. Wilson, 1999. "Volatility of volatility of financial markets," Lester Ingber Papers 99vv, Lester Ingber.
    20. Alexander Buryak & Ivan Guo, 2014. "New analytic approach to address Put - Call parity violation due to discrete dividends," Papers 1407.7328, arXiv.org.

    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:wsi:ijtafx:v:09:y:2006:i:07:n:s0219024906003962. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    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.