IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v236y2025icp411-425.html

Valuation of European call options for the Scott’s stochastic volatility model: An explicit finite difference scheme

Author

Listed:
  • Marín-Sánchez, Freddy H.
  • Barrera, Alejandro Pinilla
  • Zambrano, Cristhian Montoya
  • Hurtado, Santiago Medina

Abstract

This paper addresses the construction of an explicit finite difference scheme for options valuation when the underlying asset is described by the stochastic volatility model proposed by Scott (1987). A numerical analysis of the scheme is conducted to ensure positivity, monotonicity, stability, consistency, and convergence conditions. Several numerical experiments are presented for the valuation of European call options, and the results obtained from the explicit finite difference scheme in 2D are compared with Monte Carlo simulations in order to show the feasibility of our results.

Suggested Citation

  • Marín-Sánchez, Freddy H. & Barrera, Alejandro Pinilla & Zambrano, Cristhian Montoya & Hurtado, Santiago Medina, 2025. "Valuation of European call options for the Scott’s stochastic volatility model: An explicit finite difference scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 236(C), pages 411-425.
    • Handle: RePEc:eee:matcom:v:236:y:2025:i:c:p:411-425
    DOI: 10.1016/j.matcom.2025.03.033
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475425001193
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2025.03.033?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Rubinstein, Mark, 1985. "Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978," Journal of Finance, American Finance Association, vol. 40(2), pages 455-480, June.
    2. Hull, John & White, Alan, 1990. "Valuing Derivative Securities Using the Explicit Finite Difference Method," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(1), pages 87-100, March.
    3. Bhattacharya, Mihir, 1980. "Empirical Properties of the Black-Scholes Formula Under Ideal Conditions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 15(5), pages 1081-1105, December.
    4. 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.
    5. Clement, E. & Gourieroux, C. & Monfort, A., 2000. "Econometric specification of the risk neutral valuation model," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 117-143.
    6. 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.
    7. Jorion, Philippe, 1995. "Predicting Volatility in the Foreign Exchange Market," Journal of Finance, American Finance Association, vol. 50(2), pages 507-528, June.
    8. Geske, Robert, 1979. "The valuation of compound options," Journal of Financial Economics, Elsevier, vol. 7(1), pages 63-81, March.
    9. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    10. Fred Espen Benth & Jūratė Šaltytė-Benth, 2004. "The Normal Inverse Gaussian Distribution And Spot Price Modelling In Energy Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(02), pages 177-192.
    11. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    12. Stephen J. Taylor, 1994. "Modeling Stochastic Volatility: A Review And Comparative Study," Mathematical Finance, Wiley Blackwell, vol. 4(2), pages 183-204, April.
    13. Johnson, Herb & Shanno, David, 1987. "Option Pricing when the Variance Is Changing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(2), pages 143-151, June.
    14. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    15. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    16. 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.
    17. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    18. Rubinstein, Mark, 1983. "Displaced Diffusion Option Pricing," Journal of Finance, American Finance Association, vol. 38(1), pages 213-217, March.
    19. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    20. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    21. 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.
    22. 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)

    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. 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.
    2. 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, March.
    3. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    4. Mondher Bellalah, 2009. "Derivatives, Risk Management & Value," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 7175, March.
    5. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    6. David S. Bates, 1995. "Testing Option Pricing Models," NBER Working Papers 5129, National Bureau of Economic Research, Inc.
    7. David Heath & Simon Hurst & Eckhard Platen, 1999. "Modelling the Stochastic Dynamics of Volatility for Equity Indices," Research Paper Series 7, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. Dimitris Bertsimas & Leonid Kogan & Andrew W. Lo, 2001. "Hedging Derivative Securities and Incomplete Markets: An (epsilon)-Arbitrage Approach," Operations Research, INFORMS, vol. 49(3), pages 372-397, June.
    9. Bertsimas, Dimitris. & Kogan, Leonid, 1974- & Lo, Andrew W., 1997. "Pricing and hedging derivative securities in incomplete markets : an e-arbitrage approach," Working papers WP 3973-97., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    10. Jonathan Ziveyi, 2011. "The Evaluation of Early Exercise Exotic Options," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2011, January-A.
    11. Christoffersen, Peter & Jacobs, Kris & Chang, Bo Young, 2013. "Forecasting with Option-Implied Information," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 581-656, Elsevier.
    12. Darsinos, T. & Satchell, S.E., 2001. "Bayesian Forecasting of Options Prices: A Natural Framework for Pooling Historical and Implied Volatiltiy Information," Cambridge Working Papers in Economics 0116, Faculty of Economics, University of Cambridge.
    13. 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.
    14. Duan, Jin-Chuan & Zhang, Hua, 2001. "Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1989-2014, November.
    15. Dimitris Bertsimas & Leonid Kogan & Andrew W. Lo, 1997. "Pricing and Hedging Derivative Securities in Incomplete Markets: An E-Aritrage Model," NBER Working Papers 6250, National Bureau of Economic Research, Inc.
    16. René Garcia & Richard Luger & Eric Renault, 2000. "Asymmetric Smiles, Leverage Effects and Structural Parameters," Working Papers 2000-57, Center for Research in Economics and Statistics.
    17. 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.
    18. Guidolin, Massimo & Timmermann, Allan, 2003. "Option prices under Bayesian learning: implied volatility dynamics and predictive densities," Journal of Economic Dynamics and Control, Elsevier, vol. 27(5), pages 717-769, March.
    19. Christian Gourieroux & Razvan Sufana, 2004. "Derivative Pricing with Multivariate Stochastic Volatility : Application to Credit Risk," Working Papers 2004-31, Center for Research in Economics and Statistics.
    20. 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.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    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:eee:matcom:v:236:y:2025:i:c:p:411-425. 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.