IDEAS home Printed from https://ideas.repec.org/a/taf/quantf/v14y2014i3p481-494.html
   My bibliography  Save this article

Multiplicative noise, fast convolution and pricing

Author

Listed:
  • Giacomo Bormetti
  • Sofia Cazzaniga

Abstract

In this work we detail the application of a fast convolution algorithm to compute high-dimensional integrals in the context of multiplicative noise stochastic processes. The algorithm provides a numerical solution to the problem of characterizing conditional probability density functions at arbitrary times, and we apply it successfully to quadratic and piecewise linear diffusion processes. The ability to reproduce statistical features of financial return time series, such as thickness of the tails and scaling properties, makes these processes appealing for option pricing. Since exact analytical results are lacking, we exploit the fast convolution as a numerical method alternative to Monte Carlo simulation both in the objective and risk-neutral settings. In numerical sections we document how fast convolution outperforms Monte Carlo both in speed and efficiency terms.

Suggested Citation

  • Giacomo Bormetti & Sofia Cazzaniga, 2014. "Multiplicative noise, fast convolution and pricing," Quantitative Finance, Taylor & Francis Journals, vol. 14(3), pages 481-494, March.
    • Handle: RePEc:taf:quantf:v:14:y:2014:i:3:p:481-494
    DOI: 10.1080/14697688.2012.729857
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/14697688.2012.729857
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/14697688.2012.729857?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. Carl Chiarella & Nadima El-Hassan, 1997. "Evaluation of Derivative Security Prices in the Heath-Jarrow-Morton Framework as Path Integrals Using Fast Fourier Transform Techniques," Working Paper Series 72, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    2. Lisa Borland, 2007. "A theory of non-Gaussian option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 7(6), pages 701-701.
    3. 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..
    4. Alejandro-Quiñones, Ángel L. & Bassler, Kevin E. & Field, Michael & McCauley, Joseph L. & Nicol, Matthew & Timofeyev, Ilya & Török, Andrew & Gunaratne, Gemunu H., 2006. "A theory of fluctuations in stock prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 363(2), pages 383-392.
    5. J.L. McCauley & G.h. Gunaratne, 2002. "An empirical model of volatility of returns and option pricing," Computing in Economics and Finance 2002 186, Society for Computational Economics.
    6. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    7. Suneal K. Chaudhary, 2007. "A Simple American Option Pricing Method Using The Fast Fourier Transform," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(07), pages 1191-1202.
    8. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    9. Danilo Delpini & Giacomo Bormetti, 2010. "Minimal model of financial stylized facts," Papers 1011.5983, arXiv.org, revised Mar 2011.
    10. G. Bormetti & G. Montagna & N. Moreni & O. Nicrosini, 2006. "Pricing exotic options in a path integral approach," Quantitative Finance, Taylor & Francis Journals, vol. 6(1), pages 55-66.
    11. Michel Vellekoop & Hans Nieuwenhuis, 2007. "On option pricing models in the presence of heavy tails," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 563-573.
    12. Lisa Borland & Jean-Philippe Bouchaud & Jean-Francois Muzy & Gilles Zumbach, 2005. "The Dynamics of Financial Markets -- Mandelbrot's multifractal cascades, and beyond," Papers cond-mat/0501292, arXiv.org.
    13. Eydeland, A, 1994. "A Fast Algorithm for Computing Integrals in Function Spaces: Financial Applications," Computational Economics, Springer;Society for Computational Economics, vol. 7(4), pages 277-285.
    14. G. Bormetti & V. Cazzola & G. Livan & G. Montagna & O. Nicrosini, 2009. "A Generalized Fourier Transform Approach to Risk Measures," Papers 0909.3978, arXiv.org, revised May 2012.
    15. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    16. Montagna, Guido & Nicrosini, Oreste & Moreni, Nicola, 2002. "A path integral way to option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 310(3), pages 450-466.
    17. G. Montagna & O. Nicrosini & N. Moreni, 2002. "A Path Integral Way to Option Pricing," Papers cond-mat/0202143, arXiv.org.
    18. Bouchaud,Jean-Philippe & Potters,Marc, 2003. "Theory of Financial Risk and Derivative Pricing," Cambridge Books, Cambridge University Press, number 9780521819169, October.
    19. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    20. 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.
    21. McCauley, Joseph L. & Gunaratne, Gemunu H., 2003. "An empirical model of volatility of returns and option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 329(1), pages 178-198.
    22. Xavier Gabaix & Parameswaran Gopikrishnan & Vasiliki Plerou & H. Eugene Stanley, 2003. "A theory of power-law distributions in financial market fluctuations," Nature, Nature, vol. 423(6937), pages 267-270, May.
    23. Bormetti, Giacomo & Cisana, Enrica & Montagna, Guido & Nicrosini, Oreste, 2007. "A non-Gaussian approach to risk measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 532-542.
    24. 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..
    25. Lisa Borland & Jean-Philippe Bouchaud & Jean-Francois Muzy & Gilles Zumbach, 2005. "The Dynamics of Financial Markets -- Mandelbrot's multifractal cascades, and beyond," Science & Finance (CFM) working paper archive 500061, Science & Finance, Capital Fund Management.
    26. G. Bormetti & G. Montagna & N. Moreni & O. Nicrosini, 2004. "Pricing Exotic Options in a Path Integral Approach," Papers cond-mat/0407321, arXiv.org, revised May 2006.
    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. Giacomo Bormetti & Sofia Cazzaniga, 2011. "Multiplicative noise, fast convolution, and pricing," Papers 1107.1451, arXiv.org.
    2. Jovanovic, Franck & Schinckus, Christophe, 2017. "Econophysics and Financial Economics: An Emerging Dialogue," OUP Catalogue, Oxford University Press, number 9780190205034.
    3. 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.
    4. Shi, Leilei, 2006. "Does security transaction volume–price behavior resemble a probability wave?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 419-436.
    5. Michele Vodret & Iacopo Mastromatteo & Bence Tóth & Michael Benzaquen, 2023. "Microfounding GARCH models and beyond: a Kyle-inspired model with adaptive agents," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 18(3), pages 599-625, July.
    6. 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).
    7. Axel A. Araneda & Marcelo J. Villena, 2018. "Computing the CEV option pricing formula using the semiclassical approximation of path integral," Papers 1803.10376, arXiv.org.
    8. 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.
    9. Antonio Doria, Francisco, 2011. "J.B. Rosser Jr. , Handbook of Research on Complexity, Edward Elgar, Cheltenham, UK--Northampton, MA, USA (2009) 436 + viii pp., index, ISBN 978 1 84542 089 5 (cased)," Journal of Economic Behavior & Organization, Elsevier, vol. 78(1-2), pages 196-204, April.
    10. 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).
    11. Alexander Subbotin & Thierry Chauveau & Kateryna Shapovalova, 2009. "Volatility Models: from GARCH to Multi-Horizon Cascades," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00390636, HAL.
    12. Chuo Chang, 2020. "Dynamic correlations and distributions of stock returns on China's stock markets," Journal of Applied Finance & Banking, SCIENPRESS Ltd, vol. 10(1), pages 1-6.
    13. Subbotin, Alexandre, 2009. "Volatility Models: from Conditional Heteroscedasticity to Cascades at Multiple Horizons," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 15(3), pages 94-138.
    14. Leopoldo S'anchez-Cant'u & Carlos Arturo Soto-Campos & Andriy Kryvko, 2016. "Evidence of Self-Organization in Time Series of Capital Markets," Papers 1604.03996, arXiv.org, revised Mar 2017.
    15. 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.
    16. Bucsa, G. & Jovanovic, F. & Schinckus, C., 2011. "A unified model for price return distributions used in econophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3435-3443.
    17. Borkowski, Bolesław & Krawiec, Monika & Shachmurove, Yochanan, 2013. "Impact of volatility estimation method on theoretical option values," Global Finance Journal, Elsevier, vol. 24(2), pages 119-128.
    18. Eom, Cheoljun & Kaizoji, Taisei & Scalas, Enrico, 2019. "Fat tails in financial return distributions revisited: Evidence from the Korean stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    19. Boleslaw Borkowski & Monika Krawiec & Yochanan Shachmurove, 2013. "Modeling and Estimating Volatility of Options on Standard & Poor’s 500 Index," PIER Working Paper Archive 13-015, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    20. Shu-Heng Chen & Sai-Ping Li, 2011. "Econophysics: Bridges over a Turbulent Current," Papers 1107.5373, arXiv.org.

    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:481-494. 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.