IDEAS home Printed from https://ideas.repec.org/a/taf/quantf/v14y2014i3p409-426.html
   My bibliography  Save this article

Pricing American options written on two underlying assets

Author

Listed:
  • Carl Chiarella
  • Jonathan Ziveyi

Abstract

This paper extends the integral transform approach of McKean [ Ind. Manage. Rev. , 1965, 6 , 32--39] and Chiarella and Ziogas [ J. Econ. Dyn. Control , 2005, 29 , 229--263] to the pricing of American options written on more than one underlying asset under the Black and Scholes [ J. Polit. Econ. , 1973, 81 , 637--659] framework. A bivariate transition density function of the two underlying stochastic processes is derived by solving the associated backward Kolmogorov partial differential equation. Fourier transform techniques are used to transform the partial differential equation to a corresponding ordinary differential equation whose solution can be readily found by using the integrating factor method. An integral expression of the American option written on any two assets is then obtained by applying Duhamel's principle. A numerical algorithm for calculating American spread call option prices is given as an example, with the corresponding early exercise boundaries approximated by linear functions. Numerical results are presented and comparisons made with other alternative approaches.

Suggested Citation

  • Carl Chiarella & Jonathan Ziveyi, 2014. "Pricing American options written on two underlying assets," Quantitative Finance, Taylor & Francis Journals, vol. 14(3), pages 409-426, March.
    • Handle: RePEc:taf:quantf:v:14:y:2014:i:3:p:409-426
    DOI: 10.1080/14697688.2013.810811
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/14697688.2013.810811
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/14697688.2013.810811?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. Jing-Zhi Huang & Marti G. Subrahmanyam & G. George Yu, 1999. "Pricing And Hedging American Options: A Recursive Integration Method," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 8, pages 219-239, World Scientific Publishing Co. Pte. Ltd..
    2. Geltner, David & Riddiough, Timothy & Stojanovic, Srdjan, 1996. "Insights on the Effect of Land Use Choice: The Perpetual Option on the Best of Two Underlying Assets," Journal of Urban Economics, Elsevier, vol. 39(1), pages 20-50, January.
    3. Stulz, ReneM., 1982. "Options on the minimum or the maximum of two risky assets : Analysis and applications," Journal of Financial Economics, Elsevier, vol. 10(2), pages 161-185, July.
    4. Johnson, Herb, 1987. "Options on the Maximum or the Minimum of Several Assets," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(3), pages 277-283, September.
    5. Carl Chiarella & Adam Kucera & Andrew Ziogas, 2004. "A Survey of the Integral Representation of American Option Prices," Research Paper Series 118, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Boyle, Phelim P & Evnine, Jeremy & Gibbs, Stephen, 1989. "Numerical Evaluation of Multivariate Contingent Claims," Review of Financial Studies, Society for Financial Studies, vol. 2(2), pages 241-250.
    7. Chiarella, Carl & Ziogas, Andrew, 2005. "Evaluation of American strangles," Journal of Economic Dynamics and Control, Elsevier, vol. 29(1-2), pages 31-62, January.
    8. Geske, Robert & Johnson, Herb E, 1984. "The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-1524, December.
    9. Mark Broadie & Jérôme Detemple, 1997. "The Valuation of American Options on Multiple Assets," Mathematical Finance, Wiley Blackwell, vol. 7(3), pages 241-286, July.
    10. Carl Chiarella & Boda Kang & Gunter H. Meyer & Andrew Ziogas, 2009. "The Evaluation Of American Option Prices Under Stochastic Volatility And Jump-Diffusion Dynamics Using The Method Of Lines," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 393-425.
    11. Brennan, Michael J & Schwartz, Eduardo S, 1977. "The Valuation of American Put Options," Journal of Finance, American Finance Association, vol. 32(2), pages 449-462, May.
    12. Ibáñez, Alfredo & Zapatero, Fernando, 2004. "Monte Carlo Valuation of American Options through Computation of the Optimal Exercise Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 39(2), pages 253-275, June.
    13. 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.
    14. Peter Carr & Robert Jarrow & Ravi Myneni, 2008. "Alternative Characterizations Of American Put Options," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 5, pages 85-103, World Scientific Publishing Co. Pte. Ltd..
    15. Siim Kallast & Andi Kivinukk, 2003. "Pricing and Hedging American Options Using Approximations by Kim Integral Equations," Review of Finance, Springer, vol. 7(3), pages 361-383.
    16. Elias Tzavalis & Shijun Wang, 2003. "Pricing American Options under Stochastic Volatility: A New Method Using Chebyshev Polynomials to Approximate the Early Exercise Boundary," Working Papers 488, Queen Mary University of London, School of Economics and Finance.
    17. Siim Kallast & Andi Kivinukk, 2003. "Pricing and Hedging American Options Using Approximations by Kim Integral Equations," Review of Finance, European Finance Association, vol. 7(3), pages 361-383.
    18. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    19. Ju, Nengjiu, 1998. "Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function," Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 627-646.
    20. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14, April.
    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. Sha Lin & Song‐Ping Zhu, 2022. "Pricing callable–puttable convertible bonds with an integral equation approach," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(10), pages 1856-1911, October.
    2. Len Patrick Dominic M. Garces & Gerald H. L. Cheang, 2021. "A Numerical Approach to Pricing Exchange Options under Stochastic Volatility and Jump-Diffusion Dynamics," Papers 2106.07362, arXiv.org.
    3. Len Patrick Dominic M. Garces & Gerald H. L. Cheang, 2020. "A Put-Call Transformation of the Exchange Option Problem under Stochastic Volatility and Jump Diffusion Dynamics," Papers 2002.10194, arXiv.org.
    4. Shen, Yang & Sherris, Michael & Ziveyi, Jonathan, 2016. "Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 127-137.
    5. Jérôme Detemple, 2014. "Optimal Exercise for Derivative Securities," Annual Review of Financial Economics, Annual Reviews, vol. 6(1), pages 459-487, 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. Jonathan Ziveyi, 2011. "The Evaluation of Early Exercise Exotic Options," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2011.
    2. Jonathan Ziveyi, 2011. "The Evaluation of Early Exercise Exotic Options," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 12, July-Dece.
    3. Mark Broadie & Jérôme Detemple, 1996. "Recent Advances in Numerical Methods for Pricing Derivative Securities," CIRANO Working Papers 96s-17, CIRANO.
    4. Zhongkai Liu & Tao Pang, 2016. "An efficient grid lattice algorithm for pricing American-style options," International Journal of Financial Markets and Derivatives, Inderscience Enterprises Ltd, vol. 5(1), pages 36-55.
    5. Chockalingam, Arun & Muthuraman, Kumar, 2015. "An approximate moving boundary method for American option pricing," European Journal of Operational Research, Elsevier, vol. 240(2), pages 431-438.
    6. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    7. 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.
    8. Fabozzi, Frank J. & Paletta, Tommaso & Stanescu, Silvia & Tunaru, Radu, 2016. "An improved method for pricing and hedging long dated American options," European Journal of Operational Research, Elsevier, vol. 254(2), pages 656-666.
    9. Manuel Moreno & Javier Navas, 2003. "On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives," Review of Derivatives Research, Springer, vol. 6(2), pages 107-128, May.
    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. Minqiang Li, 2010. "A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes," Review of Derivatives Research, Springer, vol. 13(2), pages 177-217, July.
    12. Muthuraman, Kumar, 2008. "A moving boundary approach to American option pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 32(11), pages 3520-3537, November.
    13. Chiarella, Carl & Ziogas, Andrew, 2005. "Evaluation of American strangles," Journal of Economic Dynamics and Control, Elsevier, vol. 29(1-2), pages 31-62, January.
    14. Ruas, João Pedro & Dias, José Carlos & Vidal Nunes, João Pedro, 2013. "Pricing and static hedging of American-style options under the jump to default extended CEV model," Journal of Banking & Finance, Elsevier, vol. 37(11), pages 4059-4072.
    15. Song-Ping Zhu, 2006. "An exact and explicit solution for the valuation of American put options," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 229-242.
    16. Detemple, Jérôme & Emmerling, Thomas, 2009. "American chooser options," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 128-153, January.
    17. 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.
    18. Pressacco, Flavio & Gaudenzi, Marcellino & Zanette, Antonino & Ziani, Laura, 2008. "New insights on testing the efficiency of methods of pricing and hedging American options," European Journal of Operational Research, Elsevier, vol. 185(1), pages 235-254, February.
    19. Jérôme Detemple, 2014. "Optimal Exercise for Derivative Securities," Annual Review of Financial Economics, Annual Reviews, vol. 6(1), pages 459-487, December.
    20. In oon Kim & Bong-Gyu Jang & Kyeong Tae Kim, 2013. "A simple iterative method for the valuation of American options," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 885-895, May.

    More about this item

    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:taf:quantf:v:14:y:2014:i:3:p:409-426. 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: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RQUF20 .

    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.