IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v389y2010i24p5736-5748.html
   My bibliography  Save this article

Pricing European options with a log Student’s t-distribution: A Gosset formula

Author

Listed:
  • Cassidy, Daniel T.
  • Hamp, Michael J.
  • Ouyed, Rachid

Abstract

The distributions of returns for stocks are not well described by a normal probability density function (pdf). Student’s t-distributions, which have fat tails, are known to fit the distributions of the returns. We present pricing of European call or put options using a log Student’s t-distribution, which we call a Gosset approach in honour of W.S. Gosset, the author behind the nom de plume Student. The approach that we present can be used to price European options using other distributions and yields the Black–Scholes formula for returns described by a normal pdf.

Suggested Citation

  • Cassidy, Daniel T. & Hamp, Michael J. & Ouyed, Rachid, 2010. "Pricing European options with a log Student’s t-distribution: A Gosset formula," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(24), pages 5736-5748.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:24:p:5736-5748
    DOI: 10.1016/j.physa.2010.08.037
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437110007405
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2010.08.037?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. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    2. Pinn, Klaus, 2000. "Minimal variance hedging of options with student-t underlying," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 276(3), pages 581-595.
    3. Blattberg, Robert C & Gonedes, Nicholas J, 1974. "A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices," The Journal of Business, University of Chicago Press, vol. 47(2), pages 244-280, April.
    4. 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.
    5. McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2007. "Martingale option pricing," MPRA Paper 2151, University Library of Munich, Germany.
    6. Frantisek Slanina, 2010. "A contribution to the systematics of stochastic volatility models," Papers 1009.2696, arXiv.org.
    7. Jean-Philippe Bouchaud & Didier Sornette, 1994. "The Black-Scholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes," Science & Finance (CFM) working paper archive 500040, Science & Finance, Capital Fund Management.
    8. Lax, Melvin & Cai, Wei & Xu, Min, 2006. "Random Processes in Physics and Finance," OUP Catalogue, Oxford University Press, number 9780198567769.
    9. McCauley, J.L. & Gunaratne, G.H. & Bassler, K.E., 2007. "Martingale option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 351-356.
    10. J. L. McCauley & G. H. Gunaratne & K. E. Bassler, 2006. "Martingale Option Pricing," Papers physics/0606011, arXiv.org, revised Feb 2007.
    11. Vicente, Renato & de Toledo, Charles M. & Leite, Vitor B.P. & Caticha, Nestor, 2006. "Underlying dynamics of typical fluctuations of an emerging market price index: The Heston model from minutes to months," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(1), pages 272-288.
    12. 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.
    13. Lim, G.C. & Martin, G.M. & Martin, V.L., 2006. "Pricing currency options in the presence of time-varying volatility and non-normalities," Journal of Multinational Financial Management, Elsevier, vol. 16(3), pages 291-314, July.
    14. Slanina, František, 2010. "A contribution to the systematics of stochastic volatility models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3230-3239.
    15. 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.
    16. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    17. Silva, A. Christian & Prange, Richard E. & Yakovenko, Victor M., 2004. "Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 227-235.
    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. Ahmadi-Javid, Amir & Fallah-Tafti, Malihe, 2019. "Portfolio optimization with entropic value-at-risk," European Journal of Operational Research, Elsevier, vol. 279(1), pages 225-241.
    2. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    3. Claudia Yeap & Simon S Kwok & S T Boris Choy, 2018. "A Flexible Generalized Hyperbolic Option Pricing Model and Its Special Cases," Journal of Financial Econometrics, Oxford University Press, vol. 16(3), pages 425-460.
    4. Yeap, Claudia & Kwok, Simon S. & Choy, S. T. Boris, 2016. "A Flexible Generalised Hyperbolic Option Pricing Model and its Special Cases," Working Papers 2016-14, University of Sydney, School of Economics.
    5. Federica De Domenico & Giacomo Livan & Guido Montagna & Oreste Nicrosini, 2023. "Modeling and Simulation of Financial Returns under Non-Gaussian Distributions," Papers 2302.02769, arXiv.org.
    6. Wang, Xiao-Tian & Li, Zhe & Zhuang, Le, 2017. "European option pricing under the Student’s t noise with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 848-858.
    7. Lasko Basnarkov & Viktor Stojkoski & Zoran Utkovski & Ljupco Kocarev, 2019. "Option Pricing With Heavy-Tailed Distributions Of Logarithmic Returns," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-35, November.
    8. Till Massing, 2019. "What is the best Lévy model for stock indices? A comparative study with a view to time consistency," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 33(3), pages 277-344, September.
    9. Rui Li & Saralees Nadarajah, 2020. "A review of Student’s t distribution and its generalizations," Empirical Economics, Springer, vol. 58(3), pages 1461-1490, March.
    10. Till Massing, 2018. "Simulation of Student–Lévy processes using series representations," Computational Statistics, Springer, vol. 33(4), pages 1649-1685, December.
    11. De Domenico, Federica & Livan, Giacomo & Montagna, Guido & Nicrosini, Oreste, 2023. "Modeling and simulation of financial returns under non-Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    12. Daniel T. Cassidy & Michael J. Hamp & Rachid Ouyed, 2013. "Log Student’s t -distribution-based option sensitivities: Greeks for the Gosset formulae," Quantitative Finance, Taylor & Francis Journals, vol. 13(8), pages 1289-1302, July.
    13. Massing, Till & Ramos, Arturo, 2021. "Student’s t mixture models for stock indices. A comparative study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 580(C).
    14. Cassidy, Daniel T., 2011. "Describing n-day returns with Student’s t-distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(15), pages 2794-2802.
    15. Abootaleb Shirvani & Stefan Mittnik & W. Brent Lindquist & Svetlozar T. Rachev, 2021. "Bitcoin Volatility and Intrinsic Time Using Double Subordinated Levy Processes," Papers 2109.15051, arXiv.org, revised Aug 2023.
    16. Li, Hao & Melnikov, Alexander, 2014. "Polynomial extensions of distributions and their applications in actuarial and financial modeling," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 250-260.
    17. López Martín, María del Mar & García, Catalina García & García Pérez, José, 2012. "Treatment of kurtosis in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(5), pages 2032-2045.

    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. Lasko Basnarkov & Viktor Stojkoski & Zoran Utkovski & Ljupco Kocarev, 2019. "Option Pricing With Heavy-Tailed Distributions Of Logarithmic Returns," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-35, November.
    2. Daniel T. Cassidy & Michael J. Hamp & Rachid Ouyed, 2010. "Student's t-Distribution Based Option Sensitivities: Greeks for the Gosset Formulae," Papers 1003.1344, arXiv.org, revised Jul 2010.
    3. Daniel T. Cassidy & Michael J. Hamp & Rachid Ouyed, 2013. "Log Student’s t -distribution-based option sensitivities: Greeks for the Gosset formulae," Quantitative Finance, Taylor & Francis Journals, vol. 13(8), pages 1289-1302, July.
    4. Jovanovic, Franck & Schinckus, Christophe, 2017. "Econophysics and Financial Economics: An Emerging Dialogue," OUP Catalogue, Oxford University Press, number 9780190205034.
    5. Cassidy, Daniel T., 2011. "Describing n-day returns with Student’s t-distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(15), pages 2794-2802.
    6. Wang, Xiao-Tian & Li, Zhe & Zhuang, Le, 2017. "European option pricing under the Student’s t noise with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 848-858.
    7. David Edelman & Thomas Gillespie, 2000. "The Stochastically Subordinated Poisson Normal Process for Modelling Financial Assets," Annals of Operations Research, Springer, vol. 100(1), pages 133-164, December.
    8. Yeap, Claudia & Kwok, Simon S. & Choy, S. T. Boris, 2016. "A Flexible Generalised Hyperbolic Option Pricing Model and its Special Cases," Working Papers 2016-14, University of Sydney, School of Economics.
    9. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
    10. Zhu, Song-Ping & Lian, Guang-Hua, 2015. "Pricing forward-start variance swaps with stochastic volatility," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 920-933.
    11. Gilles Daniel & Nathan Joseph & David Bree, 2005. "Stochastic volatility and the goodness-of-fit of the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 5(2), pages 199-211.
    12. Göncü, Ahmet & Karahan, Mehmet Oğuz & Kuzubaş, Tolga Umut, 2016. "A comparative goodness-of-fit analysis of distributions of some Lévy processes and Heston model to stock index returns," The North American Journal of Economics and Finance, Elsevier, vol. 36(C), pages 69-83.
    13. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    14. Liu, Chang & Chang, Chuo, 2021. "Combination of transition probability distribution and stable Lorentz distribution in stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    15. Didier SORNETTE, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based Models," Swiss Finance Institute Research Paper Series 14-25, Swiss Finance Institute.
    16. Stephane Crepey, 2004. "Delta-hedging vega risk?," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 559-579.
    17. Muneer Shaik & S. Maheswaran, 2019. "Robust Volatility Estimation with and Without the Drift Parameter," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 17(1), pages 57-91, March.
    18. 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.
    19. Sosa-Correa, William O. & Ramos, Antônio M.T. & Vasconcelos, Giovani L., 2018. "Investigation of non-Gaussian effects in the Brazilian option market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 525-539.
    20. Chang, Carolyn W. & S.K. Chang, Jack & Lim, Kian-Guan, 1998. "Information-time option pricing: theory and empirical evidence," Journal of Financial Economics, Elsevier, vol. 48(2), pages 211-242, May.

    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:eee:phsmap:v:389:y:2010:i:24:p:5736-5748. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.