IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i19p4201-d1255656.html
   My bibliography  Save this article

From Constant to Rough: A Survey of Continuous Volatility Modeling

Author

Listed:
  • Giulia Di Nunno

    (Department of Mathematics, University of Oslo, 0851 Oslo, Norway
    Department of Business and Management Science, NHH Norwegian School of Economics, 5045 Bergen, Norway)

  • Kęstutis Kubilius

    (Faculty of Mathematics and Informatics, Vilnius University, LT-03225 Vilnius, Lithuania)

  • Yuliya Mishura

    (Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
    Division of Mathematics and Physics, Mälardalen University, 722 20 Västerås, Sweden)

  • Anton Yurchenko-Tytarenko

    (Department of Mathematics, University of Oslo, 0851 Oslo, Norway)

Abstract

In this paper, we present a comprehensive survey of continuous stochastic volatility models, discussing their historical development and the key stylized facts that have driven the field. Special attention is dedicated to fractional and rough methods: without advocating for either roughness or long memory, we outline the motivation behind them and characterize some landmark models. In addition, we briefly touch on the problem of VIX modeling and recent advances in the SPX-VIX joint calibration puzzle.

Suggested Citation

  • Giulia Di Nunno & Kęstutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From Constant to Rough: A Survey of Continuous Volatility Modeling," Mathematics, MDPI, vol. 11(19), pages 1-35, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4201-:d:1255656
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/19/4201/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/19/4201/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. George Skiadopoulos, 2001. "Volatility Smile Consistent Option Models: A Survey," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(03), pages 403-437.
    2. Jean-Pierre Fouque & George Papanicolaou & Ronnie Sircar & Knut Solna, 2004. "Maturity cycles in implied volatility," Finance and Stochastics, Springer, vol. 8(4), pages 451-477, November.
    3. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    4. 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).
    5. Cheung, Yin-Wong & Ng, Lilian K, 1992. "Stock Price Dynamics and Firm Size: An Empirical Investigation," Journal of Finance, American Finance Association, vol. 47(5), pages 1985-1997, December.
    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. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    8. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    9. Aït-Sahalia, Yacine & Fan, Jianqing & Li, Yingying, 2013. "The leverage effect puzzle: Disentangling sources of bias at high frequency," Journal of Financial Economics, Elsevier, vol. 109(1), pages 224-249.
    10. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    11. A. James Boness, 1964. "Elements of a Theory of Stock-Option Value," Journal of Political Economy, University of Chicago Press, vol. 72(2), pages 163-163.
    12. 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.
    13. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-778, April.
    14. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    15. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    16. Das, Sanjiv Ranjan & Sundaram, Rangarajan K., 1999. "Of Smiles and Smirks: A Term Structure Perspective," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(2), pages 211-239, June.
    17. 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.
    18. Latane, Henry A & Rendleman, Richard J, Jr, 1976. "Standard Deviations of Stock Price Ratios Implied in Option Prices," Journal of Finance, American Finance Association, vol. 31(2), pages 369-381, May.
    19. Christie, Andrew A., 1982. "The stochastic behavior of common stock variances : Value, leverage and interest rate effects," Journal of Financial Economics, Elsevier, vol. 10(4), pages 407-432, December.
    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. 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..
    22. Duffee, Gregory R., 1995. "Stock returns and volatility A firm-level analysis," Journal of Financial Economics, Elsevier, vol. 37(3), pages 399-420, March.
    23. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    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. Orimar Sauri, 2024. "Asymptotic Error Distribution of the Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise," Papers 2402.08513, arXiv.org.
    2. Giulia Di Nunno & Anton Yurchenko-Tytarenko, 2023. "Power law in Sandwiched Volterra Volatility model," Papers 2311.01228, arXiv.org.
    3. Mishari Al-Foraih & Jan Posp'iv{s}il & Josep Vives, 2023. "Computation of Greeks under rough Volterra stochastic volatility models using the Malliavin calculus approach," Papers 2312.00405, arXiv.org.

    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. Giulia Di Nunno & Kk{e}stutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From constant to rough: A survey of continuous volatility modeling," Papers 2309.01033, arXiv.org, revised Sep 2023.
    2. Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    3. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    4. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    5. Choi, Jaewon & Richardson, Matthew, 2016. "The volatility of a firm's assets and the leverage effect," Journal of Financial Economics, Elsevier, vol. 121(2), pages 254-277.
    6. 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, January.
    7. Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, University Library of Munich, Germany.
    8. Bollerslev, Tim & Engle, Robert F. & Nelson, Daniel B., 1986. "Arch models," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 49, pages 2959-3038, Elsevier.
    9. Chuang, Wen-I & Huang, Teng-Ching & Lin, Bing-Huei, 2013. "Predicting volatility using the Markov-switching multifractal model: Evidence from S&P 100 index and equity options," The North American Journal of Economics and Finance, Elsevier, vol. 25(C), pages 168-187.
    10. Christoffersen, Peter & Heston, Steve & Jacobs, Kris, 2006. "Option valuation with conditional skewness," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 253-284.
    11. Meddahi, N., 2001. "An Eigenfunction Approach for Volatility Modeling," Cahiers de recherche 2001-29, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    12. Font, Begoña, 1998. "Modelización de series temporales financieras. Una recopilación," DES - Documentos de Trabajo. Estadística y Econometría. DS 3664, Universidad Carlos III de Madrid. Departamento de Estadística.
    13. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
    14. Yueh-Neng Lin & Ken Hung, 2008. "Is Volatility Priced?," Annals of Economics and Finance, Society for AEF, vol. 9(1), pages 39-75, May.
    15. René Garcia & Eric Ghysels & Eric Renault, 2004. "The Econometrics of Option Pricing," CIRANO Working Papers 2004s-04, CIRANO.
    16. Asai, Manabu & McAleer, Michael, 2015. "Leverage and feedback effects on multifactor Wishart stochastic volatility for option pricing," Journal of Econometrics, Elsevier, vol. 187(2), pages 436-446.
    17. Kaehler, Jürgen, 1991. "Modelling and forecasting exchange-rate volatility with ARCH-type models," ZEW Discussion Papers 91-02, ZEW - Leibniz Centre for European Economic Research.
    18. Robert F. Engle & Emil N. Siriwardane, 2018. "Structural GARCH: The Volatility-Leverage Connection," The Review of Financial Studies, Society for Financial Studies, vol. 31(2), pages 449-492.
    19. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat & Wang, Yintian, 2008. "Option valuation with long-run and short-run volatility components," Journal of Financial Economics, Elsevier, vol. 90(3), pages 272-297, December.
    20. Jin Zhang & Yi Xiang, 2008. "The implied volatility smirk," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 263-284.

    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:gam:jmathe:v:11:y:2023:i:19:p:4201-:d:1255656. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.