IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v80y2009i2p245-255.html
   My bibliography  Save this article

Polynomial chaos for simulating random volatilities

Author

Listed:
  • Pulch, Roland
  • van Emmerich, Cathrin

Abstract

In financial mathematics, the fair price of options can be achieved by solutions of parabolic differential equations. The volatility usually enters the model as a constant parameter. However, since this constant has to be estimated with respect to the underlying market, it makes sense to replace the volatility by an according random variable. Consequently, a differential equation with stochastic input occurs, whose solution determines the fair price in the refined model. Corresponding expected values and variances can be computed approximately via a Monte Carlo method. Alternatively, the generalised polynomial chaos yields an efficient approach for calculating the required data. Based on a parabolic equation modelling the fair price of Asian options, the technique is developed and corresponding numerical simulations are presented.

Suggested Citation

  • Pulch, Roland & van Emmerich, Cathrin, 2009. "Polynomial chaos for simulating random volatilities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(2), pages 245-255.
    • Handle: RePEc:eee:matcom:v:80:y:2009:i:2:p:245-255
    DOI: 10.1016/j.matcom.2009.05.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475409001645
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2009.05.008?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. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    2. 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..
    3. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    5. 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.
    6. 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. Pulch, Roland, 2011. "Modelling and simulation of autonomous oscillators with random parameters," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(6), pages 1128-1143.
    2. Kathrin Hellmuth & Christian Klingenberg, 2022. "Computing Black Scholes with Uncertain Volatility-A Machine Learning Approach," Papers 2202.07378, arXiv.org.
    3. Lin, Y.-T. & Shih, Y.-T. & Chien, C.-S. & Sheng, Q., 2021. "A note on stochastic polynomial chaos expansions for uncertain volatility and Asian option pricing," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    4. Ledermann, Daniel & Alexander, Carol, 2012. "Further properties of random orthogonal matrix simulation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 83(C), pages 56-79.

    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. Charles J. Corrado & Tie Su, 1996. "Skewness And Kurtosis In S&P 500 Index Returns Implied By Option Prices," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 19(2), pages 175-192, June.
    2. Stentoft, Lars, 2011. "American option pricing with discrete and continuous time models: An empirical comparison," Journal of Empirical Finance, Elsevier, vol. 18(5), pages 880-902.
    3. 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).
    4. Gonçalo Faria & João Correia-da-Silva, 2014. "A closed-form solution for options with ambiguity about stochastic volatility," Review of Derivatives Research, Springer, vol. 17(2), pages 125-159, July.
    5. Damien Ackerer & Damir Filipović, 2020. "Option pricing with orthogonal polynomial expansions," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 47-84, January.
    6. Hu, May & Park, Jason, 2019. "Valuation of collateralized debt obligations: An equilibrium model," Economic Modelling, Elsevier, vol. 82(C), pages 119-135.
    7. Yijuan Liang & Xiuchuan Xu, 2019. "Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities," Sustainability, MDPI, vol. 11(3), pages 1-21, February.
    8. Rombouts, Jeroen V.K. & Stentoft, Lars, 2015. "Option pricing with asymmetric heteroskedastic normal mixture models," International Journal of Forecasting, Elsevier, vol. 31(3), pages 635-650.
    9. 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.
    10. Alexander Lipton & Artur Sepp, 2022. "Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility," Papers 2202.07849, arXiv.org.
    11. Huang, Yu Chuan & Chen, Shing Chun, 2002. "Warrants pricing: Stochastic volatility vs. Black-Scholes," Pacific-Basin Finance Journal, Elsevier, vol. 10(4), pages 393-409, September.
    12. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    13. Mondher Bellalah, 2009. "Derivatives, Risk Management & Value," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 7175, January.
    14. 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.
    15. Badescu Alex & Kulperger Reg & Lazar Emese, 2008. "Option Valuation with Normal Mixture GARCH Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 12(2), pages 1-42, May.
    16. Robert F. Engle & Joshua V. Rosenberg, 1995. "GARCH Gamma," NBER Working Papers 5128, National Bureau of Economic Research, Inc.
    17. 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..
    18. 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.
    19. Yacine Ait-Sahalia & Robert Kimmel, 2004. "Maximum Likelihood Estimation of Stochastic Volatility Models," NBER Working Papers 10579, National Bureau of Economic Research, Inc.
    20. Damien Ackerer & Damir Filipovic, 2017. "Option Pricing with Orthogonal Polynomial Expansions," Papers 1711.09193, arXiv.org, revised May 2019.

    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:matcom:v:80:y:2009:i:2:p:245-255. 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/mathematics-and-computers-in-simulation/ .

    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.