IDEAS home Printed from https://ideas.repec.org/p/uts/rpaper/256.html
   My bibliography  Save this paper

An Analysis of American Options Under Heston Stochastic Volatility and Jump-Diffusion Dynamics

Author

Listed:

Abstract

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square root process as used by Heston (1993), and by a Poisson jump process as introduced by Merton (1976). Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalises in an intuitive way the structure of the solution to the corresponding European option pricing problem in the case of a call option and constant interest rates obtained by Scott (1997).

Suggested Citation

  • Gerald Cheang & Carl Chiarella & Andrew Ziogas, 2009. "An Analysis of American Options Under Heston Stochastic Volatility and Jump-Diffusion Dynamics," Research Paper Series 256, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:256
    as

    Download full text from publisher

    File URL: http://www.qfrc.uts.edu.au/research//sites/default/files/qfr-archive-03/QFR-rp256.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    2. Carl Chiarella & Andrew Ziogas, 2009. "American Call Options Under Jump-Diffusion Processes - A Fourier Transform Approach," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(1), pages 37-79.
    3. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
    4. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    7. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    8. Louis O. Scott, 1997. "Pricing Stock Options in a Jump‐Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier Inversion Methods," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 413-426, October.
    9. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    10. Andrew Ziogas & Carl Chiarella, 2005. "Pricing American Options under Stochastic Volatility," Computing in Economics and Finance 2005 77, Society for Computational Economics.
    11. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    12. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    13. 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.
    14. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    15. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    16. Chandrasekhar Reddy Gukhal, 2001. "Analytical Valuation of American Options on Jump‐Diffusion Processes," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 97-115, January.
    17. 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.
    18. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    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. Fernanda D'Ippoliti & Enrico Moretto & Sara Pasquali & Barbara Trivellato, 2010. "Exact Pricing With Stochastic Volatility And Jumps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(06), pages 901-929.
    2. 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.
    3. Carl Chiarella & Jonathan Ziveyi, 2011. "Two Stochastic Volatility Processes - American Option Pricing," Research Paper Series 292, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Gerald H. L. Cheang & Carl Chiarella & Andrew Ziogas, 2013. "The representation of American options prices under stochastic volatility and jump-diffusion dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 13(2), pages 241-253, January.

    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. Gerald H. L. Cheang & Carl Chiarella & Andrew Ziogas, 2013. "The representation of American options prices under stochastic volatility and jump-diffusion dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 13(2), pages 241-253, January.
    2. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    3. Eckhard Platen & Hardy Hulley, 2008. "Hedging for the Long Run," Research Paper Series 214, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Blessing Taruvinga & Boda Kang & Christina Sklibosios Nikitopoulos, 2018. "Pricing American Options with Jumps in Asset and Volatility," Research Paper Series 394, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. 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.
    6. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 19, July-Dece.
    7. Gerald H. L. Cheang & Len Patrick Dominic M. Garces, 2020. "Representation of Exchange Option Prices under Stochastic Volatility Jump-Diffusion Dynamics," Papers 2002.10202, arXiv.org.
    8. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2009.
    9. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    10. Cheng Few Lee & Yibing Chen & John Lee, 2020. "Alternative Methods to Derive Option Pricing Models: Review and Comparison," World Scientific Book Chapters, in: Cheng Few Lee & John C Lee (ed.), HANDBOOK OF FINANCIAL ECONOMETRICS, MATHEMATICS, STATISTICS, AND MACHINE LEARNING, chapter 102, pages 3573-3617, World Scientific Publishing Co. Pte. Ltd..
    11. R. Merino & J. Pospíšil & T. Sobotka & J. Vives, 2018. "Decomposition Formula For Jump Diffusion Models," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-36, December.
    12. Raul Merino & Jan Posp'iv{s}il & Tom'av{s} Sobotka & Josep Vives, 2019. "Decomposition formula for jump diffusion models," Papers 1906.06930, arXiv.org.
    13. Chen, An-Sing & Leung, Mark T., 2005. "Modeling time series information into option prices: An empirical evaluation of statistical projection and GARCH option pricing model," Journal of Banking & Finance, Elsevier, vol. 29(12), pages 2947-2969, December.
    14. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 2000. "Pricing and hedging long-term options," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 277-318.
    15. Bin Xie & Weiping Li & Nan Liang, 2021. "Pricing S&P 500 Index Options with L\'evy Jumps," Papers 2111.10033, arXiv.org, revised Nov 2021.
    16. Chan, Tat Lung (Ron), 2019. "Efficient computation of european option prices and their sensitivities with the complex fourier series method," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    17. Belssing Taruvinga, 2019. "Solving Selected Problems on American Option Pricing with the Method of Lines," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2019.
    18. Carl Chiarella & Xue-Zhong He & Christina Sklibosios Nikitopoulos, 2015. "Derivative Security Pricing," Dynamic Modeling and Econometrics in Economics and Finance, Springer, edition 127, number 978-3-662-45906-5, July-Dece.
    19. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    20. Mussa Juma & Min Cherng Lee & Seong Tah Chin & Kian Wah Liew, 2017. "Evaluation of variable annuity guarantees with the effect of jumps in the asset price process," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1326218-132, January.

    More about this item

    Keywords

    American options; stochastic volatility; jump-diffusion processes; Volterra integral equations; free boundary problem; method of lines;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:uts:rpaper:256. 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: Duncan Ford (email available below). General contact details of provider: https://edirc.repec.org/data/qfutsau.html .

    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.