IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1608.02028.html
   My bibliography  Save this paper

Explicit Heston Solutions and Stochastic Approximation for Path-dependent Option Pricing

Author

Listed:
  • Michael A. Kouritzin

Abstract

New simulation approaches to evaluating path-dependent options without matrix inversion issues nor Euler bias are evaluated. They employ three main contributions: Stochastic approximation replaces regression in the LSM algorithm; Explicit weak solutions to stochastic differential equations are developed and applied to Heston model simulation; and Importance sampling expands these explicit solutions. The approach complements Heston (1993) and Broadie and Kaya (2006) by handling the case of path-dependence in the option's execution strategy. Numeric comparison against standard Monte Carlo methods demonstrate up to two orders of magnitude speed improvement. The general ideas will extend beyond the important Heston setting.

Suggested Citation

  • Michael A. Kouritzin, 2016. "Explicit Heston Solutions and Stochastic Approximation for Path-dependent Option Pricing," Papers 1608.02028, arXiv.org, revised Apr 2018.
  • Handle: RePEc:arx:papers:1608.02028
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1608.02028
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Carriere, Jacques F., 1996. "Valuation of the early-exercise price for options using simulations and nonparametric regression," Insurance: Mathematics and Economics, Elsevier, vol. 19(1), pages 19-30, December.
    3. Alexander Van Haastrecht & Antoon Pelsser, 2010. "Efficient, Almost Exact Simulation Of The Heston Stochastic Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(01), pages 1-43.
    4. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    5. 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..
    6. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. "Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-1632, December.
    7. Yu, Jun, 2005. "On leverage in a stochastic volatility model," Journal of Econometrics, Elsevier, vol. 127(2), pages 165-178, August.
    8. Michael A. Kouritzin & Yong Zeng, 2005. "Bayesian Model Selection Via Filtering For A Class Of Micro-Movement Models Of Asset Price," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 97-121.
    9. Drost, Feike C. & Werker, Bas J. M., 1996. "Closing the GARCH gap: Continuous time GARCH modeling," Journal of Econometrics, Elsevier, vol. 74(1), pages 31-57, September.
    10. 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.
    11. Boyle, Phelim P., 1977. "Options: A Monte Carlo approach," Journal of Financial Economics, Elsevier, vol. 4(3), pages 323-338, May.
    12. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    13. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
    14. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    15. Boyle, Phelim & Broadie, Mark & Glasserman, Paul, 1997. "Monte Carlo methods for security pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1267-1321, June.
    16. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    17. Panagiota Daskalopoulos & Paul M. N. Feehan, 2011. "Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance," Papers 1109.1075, arXiv.org.
    18. 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.
    19. Wilmott,Paul & Howison,Sam & Dewynne,Jeff, 1995. "The Mathematics of Financial Derivatives," Cambridge Books, Cambridge University Press, number 9780521497893, November.
    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. Philip Protter & Emmanuelle Clément & Damien Lamberton, 2002. "An analysis of a least squares regression method for American option pricing," Finance and Stochastics, Springer, vol. 6(4), pages 449-471.
    22. Schwartz, Eduardo S., 1977. "The valuation of warrants: Implementing a new approach," Journal of Financial Economics, Elsevier, vol. 4(1), pages 79-93, January.
    23. 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. Dashti Moghaddam, M. & Serota, R.A., 2021. "Combined multiplicative–Heston model for stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 561(C).
    2. Iro Ren'e Kouarfate & Michael A. Kouritzin & Anne MacKay, 2020. "Explicit solution simulation method for the 3/2 model," Papers 2009.09058, arXiv.org, revised Jan 2021.
    3. Kouritzin, Michael A. & MacKay, Anne, 2018. "VIX-linked fees for GMWBs via explicit solution simulation methods," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 1-17.
    4. Michael A. Kouritzin & Anne MacKay, 2019. "Branching Particle Pricers with Heston Examples," Papers 1907.00219, arXiv.org, revised Nov 2019.
    5. Michael A. Kouritzin & Anne Mackay, 2020. "Branching Particle Pricers With Heston Examples," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-29, February.
    6. Michael A. Kouritzin & Anne MacKay, 2017. "VIX-linked fees for GMWBs via Explicit Solution Simulation Methods," Papers 1708.06886, arXiv.org, revised Apr 2018.
    7. Chih-Chen Hsu & Chung-Gee Lin & Tsung-Jung Kuo, 2020. "Pricing of Arithmetic Asian Options under Stochastic Volatility Dynamics: Overcoming the Risks of High-Frequency Trading," Mathematics, MDPI, vol. 8(12), pages 1-16, December.

    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. Michael A. Kouritzin, 2018. "Explicit Heston Solutions And Stochastic Approximation For Path-Dependent Option Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(01), pages 1-45, February.
    2. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    3. 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.
    4. Katarzyna Toporek, 2012. "Simple is better. Empirical comparison of American option valuation methods," Ekonomia journal, Faculty of Economic Sciences, University of Warsaw, vol. 29.
    5. 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.
    6. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    7. Lars Stentoft, 2008. "Option Pricing using Realized Volatility," CREATES Research Papers 2008-13, Department of Economics and Business Economics, Aarhus University.
    8. Damien Ackerer & Damir Filipović, 2020. "Option pricing with orthogonal polynomial expansions," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 47-84, January.
    9. Manuel Moreno & Javier Navas, 2003. "On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives," Review of Derivatives Research, Springer, vol. 6(2), pages 107-128, May.
    10. Lars Stentoft, 2013. "American option pricing using simulation with an application to the GARCH model," Chapters, in: Adrian R. Bell & Chris Brooks & Marcel Prokopczuk (ed.), Handbook of Research Methods and Applications in Empirical Finance, chapter 5, pages 114-147, Edward Elgar Publishing.
    11. Chuang-Chang Chang & Jun-Biao Lin & Wei-Che Tsai & Yaw-Huei Wang, 2012. "Using Richardson extrapolation techniques to price American options with alternative stochastic processes," Review of Quantitative Finance and Accounting, Springer, vol. 39(3), pages 383-406, October.
    12. Zhongkai Liu & Tao Pang, 2016. "An efficient grid lattice algorithm for pricing American-style options," International Journal of Financial Markets and Derivatives, Inderscience Enterprises Ltd, vol. 5(1), pages 36-55.
    13. Damien Ackerer & Damir Filipovic, 2017. "Option Pricing with Orthogonal Polynomial Expansions," Papers 1711.09193, arXiv.org, revised May 2019.
    14. Bégin Jean-François & Bédard Mylène & Gaillardetz Patrice, 2015. "Simulating from the Heston model: A gamma approximation scheme," Monte Carlo Methods and Applications, De Gruyter, vol. 21(3), pages 205-231, September.
    15. Ravi Kashyap, 2022. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Annals of Operations Research, Springer, vol. 315(2), pages 1175-1215, August.
    16. Mariano González-Sánchez & Eva M. Ibáñez Jiménez & Ana I. Segovia San Juan, 2022. "Market and model risks: a feasible joint estimate methodology," Risk Management, Palgrave Macmillan, vol. 24(3), pages 187-213, September.
    17. Anatoliy Swishchuk, 2013. "Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8660, July.
    18. Stentoft, Lars, 2005. "Pricing American options when the underlying asset follows GARCH processes," Journal of Empirical Finance, Elsevier, vol. 12(4), pages 576-611, September.
    19. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    20. Li, Chenxu & Ye, Yongxin, 2019. "Pricing and Exercising American Options: an Asymptotic Expansion Approach," Journal of Economic Dynamics and Control, Elsevier, vol. 107(C), pages 1-1.

    More about this item

    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:arx:papers:1608.02028. 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.