IDEAS home Printed from https://ideas.repec.org/p/arx/papers/math-0612691.html
   My bibliography  Save this paper

Option pricing with log-stable L\'{e}vy processes

Author

Listed:
  • Przemys{l}aw Repetowicz
  • Peter Richmond

Abstract

We model the logarithm of the price (log-price) of a financial asset as a random variable obtained by projecting an operator stable random vector with a scaling index matrix $\underline{\underline{E}}$ onto a non-random vector. The scaling index $\underline{\underline{E}}$ models prices of the individual financial assets (stocks, mutual funds, etc.). We find the functional form of the characteristic function of real powers of the price returns and we compute the expectation value of these real powers and we speculate on the utility of these results for statistical inference. Finally we consider a portfolio composed of an asset and an option on that asset. We derive the characteristic function of the deviation of the portfolio, \mbox{${\mathfrak D}_t^{({\mathfrak t})}$}, defined as a temporal change of the portfolio diminished by the the compound interest earned. We derive pseudo-differential equations for the option as a function of the log-stock-price and time and we find exact closed-form solutions to that equation. These results were not known before. Finally we discuss how our solutions correspond to other approximate results known from literature,in particular to the well known Black & Scholes equation.

Suggested Citation

  • Przemys{l}aw Repetowicz & Peter Richmond, 2006. "Option pricing with log-stable L\'{e}vy processes," Papers math/0612691, arXiv.org.
    • Handle: RePEc:arx:papers:math/0612691
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/math/0612691
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Boyarchenko Svetlana & Levendorskii Sergei Z, 2006. "General Option Exercise Rules, with Applications to Embedded Options and Monopolistic Expansion," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 6(1), pages 1-53, June.
    2. Alvaro Cartea & Sam Howison, 2002. "Distinguished Limits of Levy-Stable Processes, and Applications to Option Pricing," OFRC Working Papers Series 2002mf04, Oxford Financial Research Centre.
    3. 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..
    4. J. Huston McCulloch, 2003. "The Risk-Neutral Measure and Option Pricing under Log-Stable Uncertainty," Working Papers 03-07, Ohio State University, Department of Economics.
    5. P. Gopikrishnan & M. Meyer & L.A.N. Amaral & H.E. Stanley, 1998. "Inverse cubic law for the distribution of stock price variations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 3(2), pages 139-140, July.
    6. S. R. Hurst & Eckhard Platen & S. T. Rachev, 1999. "Option pricing for a logstable asset price model," Published Paper Series 1999-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    Full references (including those not matched with items on IDEAS)

    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. Lombardi, Marco J. & Calzolari, Giorgio, 2009. "Indirect estimation of [alpha]-stable stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2298-2308, April.
    2. Derksen, M. & Kleijn, B. & de Vilder, R., 2022. "Heavy tailed distributions in closing auctions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 593(C).
    3. Andria, Joseph & di Tollo, Giacomo & Kalda, Jaan, 2022. "The predictive power of power-laws: An empirical time-arrow based investigation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    4. Pablo Su'arez-Garc'ia & David G'omez-Ullate, 2012. "Scaling, stability and distribution of the high-frequency returns of the IBEX35 index," Papers 1208.0317, arXiv.org.
    5. Taisei Kaizoji, 2005. "Comparison of volatility distributions in the periods of booms and stagnations: an empirical study on stock price indices," Papers physics/0506114, arXiv.org.
    6. Gabaix, Xavier & Gopikrishnan, Parameswaran & Plerou, Vasiliki & Eugene Stanley, H., 2008. "Quantifying and understanding the economics of large financial movements," Journal of Economic Dynamics and Control, Elsevier, vol. 32(1), pages 303-319, January.
    7. Philipp Weber & Bernd Rosenow, 2006. "Large stock price changes: volume or liquidity?," Quantitative Finance, Taylor & Francis Journals, vol. 6(1), pages 7-14.
    8. Jean-Philippe Aguilar & Cyril Coste & Jan Korbel, 2016. "Non-Gaussian analytic option pricing: a closed formula for the L\'evy-stable model," Papers 1609.00987, arXiv.org, revised Nov 2017.
    9. Michele Leonardo Bianchi & Svetlozar T. Rachev & Frank J. Fabozzi, 2018. "Calibrating the Italian Smile with Time-Varying Volatility and Heavy-Tailed Models," Computational Economics, Springer;Society for Computational Economics, vol. 51(3), pages 339-378, March.
    10. Stanisław Drożdż & Jarosław Kwapień & Rafał Rak, 2011. "Characteristics of Financial Fluctuations," Ekonomia journal, Faculty of Economic Sciences, University of Warsaw, vol. 25.
    11. Anirban Chakraborti & Ioane Muni Toke & Marco Patriarca & Frederic Abergel, 2011. "Econophysics review: I. Empirical facts," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 991-1012.
    12. Sabiou M. Inoua & Vernon L. Smith, 2022. "Perishable goods versus re-tradable assets: A theoretical reappraisal of a fundamental dichotomy," Chapters, in: Sascha Füllbrunn & Ernan Haruvy (ed.), Handbook of Experimental Finance, chapter 15, pages 162-171, Edward Elgar Publishing.
    13. Sabiou M. Inoua, 2020. "News-Driven Expectations and Volatility Clustering," JRFM, MDPI, vol. 13(1), pages 1-14, January.
    14. Yoshio Miyahara & Alexander Novikov, 2001. "Geometric Lévy Process Pricing Model," Research Paper Series 66, Quantitative Finance Research Centre, University of Technology, Sydney.
    15. Stanley, H.E. & Gopikrishnan, P. & Plerou, V. & Amaral, L.A.N., 2000. "Quantifying fluctuations in economic systems by adapting methods of statistical physics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 339-361.
    16. Marcin Wk{a}torek & Stanis{l}aw Dro.zd.z & Jaros{l}aw Kwapie'n & Ludovico Minati & Pawe{l} O'swik{e}cimka & Marek Stanuszek, 2020. "Multiscale characteristics of the emerging global cryptocurrency market," Papers 2010.15403, arXiv.org, revised Mar 2021.
    17. Climent Hernández José Antonio & Venegas Martínez Francisco, 2013. "Valuación de opciones sobre subyacentes con rendimientos a-estables," Contaduría y Administración, Accounting and Management, vol. 58(4), pages 119-150, octubre-d.
    18. Alvaro Cartea & Sam Howison, 2009. "Option pricing with Levy-Stable processes generated by Levy-Stable integrated variance," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 397-409.
    19. Tetsuya Takaishi, 2016. "Dynamical cross-correlation of multiple time series Ising model," Evolutionary and Institutional Economics Review, Springer, vol. 13(2), pages 455-468, December.
    20. Troy Tassier, 2013. "Handbook of Research on Complexity, by J. Barkley Rosser, Jr. and Edward Elgar," Eastern Economic Journal, Palgrave Macmillan;Eastern Economic Association, vol. 39(1), pages 132-133.

    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:arx:papers:math/0612691. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.