IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-01669082.html
   My bibliography  Save this paper

A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data

Author

Listed:
  • Antoine Lejay

    (TOSCA - TO Simulate and CAlibrate stochastic models - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en Automatique - IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique, IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique)

  • Paolo Pigato

    (WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin] - FVB - Forschungsverbund Berlin e.V. (FVB))

Abstract

In financial markets, low prices are generally associated with high volatilities and vice-versa, this well known stylized fact usually being referred to as leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts of leverage and mean-reversion effects in the dynamics of the prices. This model exhibits a regime switch in the dynamics accordingly to a certain threshold. It can be seen as a continuous-time version of the Self-Exciting Threshold Autoregressive (SETAR) model. We propose an estimation procedure for the volatility and drift coefficients as well as for the threshold level. Parameters estimated on the daily prices of 348 stocks of NYSE and S&P 500, on different time windows, show consistent empirical evidence for leverage effects. Mean-reversion effects are also detected, most markedly in crisis periods.

Suggested Citation

  • Antoine Lejay & Paolo Pigato, 2019. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Post-Print hal-01669082, HAL.
    • Handle: RePEc:hal:journl:hal-01669082
    DOI: 10.1142/S0219024919500171
    Note: View the original document on HAL open archive server: https://inria.hal.science/hal-01669082v5
    as

    Download full text from publisher

    File URL: https://inria.hal.science/hal-01669082v5/document
    Download Restriction: no
    File URL: https://libkey.io/10.1142/S0219024919500171?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
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Andrew W. Lo, A. Craig MacKinlay, 1988. "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test," The Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 41-66.
    2. Andrew Ang & Allan Timmermann, 2012. "Regime Changes and Financial Markets," Annual Review of Financial Economics, Annual Reviews, vol. 4(1), pages 313-337, October.
    3. Fei Su & Kung-Sik Chan, 2017. "Testing for Threshold Diffusion," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(2), pages 218-227, April.
    4. 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.
    5. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    6. Antoine Lejay & Paolo Pigato, 2020. "Maximum likelihood drift estimation for a threshold diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 609-637, September.
    7. Su, Fei & Chan, Kung-Sik, 2015. "Quasi-likelihood estimation of a threshold diffusion process," Journal of Econometrics, Elsevier, vol. 189(2), pages 473-484.
    8. Hens, Thorsten & Steude, Sven C., 2009. "The leverage effect without leverage," Finance Research Letters, Elsevier, vol. 6(2), pages 83-94, June.
    9. Antoine Lejay & Paolo Pigato, 2017. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Working Papers hal-01669082, HAL.
    10. Salhi, Khaled & Deaconu, Madalina & Lejay, Antoine & Champagnat, Nicolas & Navet, Nicolas, 2016. "Regime switching model for financial data: Empirical risk analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 148-157.
    11. Tong, Howell, 2015. "Threshold models in time series analysis—Some reflections," Journal of Econometrics, Elsevier, vol. 189(2), pages 485-491.
    12. Marc Decamps & Marc Goovaerts & Wim Schoutens, 2006. "Self Exciting Threshold Interest Rates Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(07), pages 1093-1122.
    13. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
    14. Spierdijk, Laura & Bikker, Jacob A. & van den Hoek, Pieter, 2012. "Mean reversion in international stock markets: An empirical analysis of the 20th century," Journal of International Money and Finance, Elsevier, vol. 31(2), pages 228-249.
    15. Decamps, Marc & De Schepper, Ann & Goovaerts, Marc, 2004. "Applications of δ-function perturbation to the pricing of derivative securities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 342(3), pages 677-692.
    16. Chan, K. S. & Stramer, O., 1998. "Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 33-44, August.
    17. 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.
    18. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    19. Luis H. R. Alvarez E. & Paavo Salminen, 2017. "Timing in the presence of directional predictability: optimal stopping of skew Brownian motion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 377-400, October.
    20. Rabemananjara, R & Zakoian, J M, 1993. "Threshold Arch Models and Asymmetries in Volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(1), pages 31-49, Jan.-Marc.
    21. Siu, Tak Kuen, 2016. "A self-exciting threshold jump–diffusion model for option valuation," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 168-193.
    22. So, Mike K P & Li, W K & Lam, K, 2002. "A Threshold Stochastic Volatility Model," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 21(7), pages 473-500, November.
    23. Antoine Lejay & Paolo Pigato, 2017. "Data and methods for A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data [Données et méthodes pour "A threshold model for local volatilit," Working Papers hal-01668975, HAL.
    24. Michael Monoyios & Lucio Sarno, 2002. "Mean reversion in stock index futures markets: A nonlinear analysis," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 22(4), pages 285-314, April.
    25. Rossello, Damiano, 2012. "Arbitrage in skew Brownian motion models," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 50-56.
    26. Pradeep K. Yadav & Peter F. Pope & Krishna Paudyal, 1994. "Threshold Autoregressive Modeling In Finance: The Price Differences Of Equivalent Assets1," Mathematical Finance, Wiley Blackwell, vol. 4(2), pages 205-221, April.
    27. Bong‐Gyu Jang & Changki Kim & Kyeong Tae Kim & Seungkyu Lee & Dong‐Hoon Shin, 2015. "Psychological Barriers and Option Pricing," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 35(1), pages 52-74, January.
    28. Poterba, James M. & Summers, Lawrence H., 1988. "Mean reversion in stock prices : Evidence and Implications," Journal of Financial Economics, Elsevier, vol. 22(1), pages 27-59, October.
    29. Fei Su & Kung-Sik Chan, 2016. "Option Pricing with Threshold Diffusion Processes," North American Actuarial Journal, Taylor & Francis Journals, vol. 20(2), pages 133-141, April.
    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. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    2. Antoine Lejay & Paolo Pigato, 2017. "Data and methods for A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data [Données et méthodes pour "A threshold model for local volatilit," Working Papers hal-01668975, HAL.
    3. Antoine Lejay & Paolo Pigato, 2020. "Maximum likelihood drift estimation for a threshold diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 609-637, September.
    4. Dingwen Zhang, 2024. "Determining the Number and Values of Thresholds for Multi-regime Threshold Ornstein–Uhlenbeck Processes," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3581-3626, November.
    5. Manuel L. Esquível & Nadezhda P. Krasii & Pedro P. Mota & Victoria V. Shamraeva, 2023. "Coupled Price–Volume Equity Models with Auto-Induced Regime Switching," Risks, MDPI, vol. 11(11), pages 1-20, November.
    6. Héctor Araya & Meryem Slaoui & Soledad Torres, 2022. "Bayesian inference for fractional Oscillating Brownian motion," Computational Statistics, Springer, vol. 37(2), pages 887-907, April.
    7. Andrey Itkin & Alexander Lipton & Dmitry Muravey, 2021. "Multilayer heat equations and their solutions via oscillating integral transforms," Papers 2112.00949, arXiv.org, revised Dec 2021.

    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. Antoine Lejay & Paolo Pigato, 2017. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Working Papers hal-01669082, HAL.
    2. Antoine Lejay & Paolo Pigato, 2020. "Maximum likelihood drift estimation for a threshold diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 609-637, September.
    3. Dingwen Zhang, 2024. "Determining the Number and Values of Thresholds for Multi-regime Threshold Ornstein–Uhlenbeck Processes," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3581-3626, November.
    4. Adam Zaremba & Jacob Koby Shemer, 2018. "Price-Based Investment Strategies," Springer Books, Springer, number 978-3-319-91530-2, January.
    5. Lux, Thomas & Morales-Arias, Leonardo & Sattarhoff, Cristina, 2011. "A Markov-switching multifractal approach to forecasting realized volatility," Kiel Working Papers 1737, Kiel Institute for the World Economy (IfW Kiel).
    6. Linton, Oliver & Whang, Yoon-Jae & Yen, Yu-Min, 2016. "A nonparametric test of a strong leverage hypothesis," Journal of Econometrics, Elsevier, vol. 194(1), pages 153-186.
    7. Zhi Liu, 2022. "Testing for the Presence of the Leverage Effect without Estimation," Mathematics, MDPI, vol. 10(14), pages 1-16, July.
    8. Akarim, Yasemin Deniz & Sevim, Serafettin, 2013. "The impact of mean reversion model on portfolio investment strategies: Empirical evidence from emerging markets," Economic Modelling, Elsevier, vol. 31(C), pages 453-459.
    9. Shively, Philip A., 2007. "Asymmetric temporary and permanent stock-price innovations," Journal of Empirical Finance, Elsevier, vol. 14(1), pages 120-130, January.
    10. Marcus Alexander Ong, 2015. "An information theoretic analysis of stock returns, volatility and trading volumes," Applied Economics, Taylor & Francis Journals, vol. 47(36), pages 3891-3906, August.
    11. 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.
    12. Tim Bollerslev & Robert J. Hodrick, 1992. "Financial Market Efficiency Tests," NBER Working Papers 4108, National Bureau of Economic Research, Inc.
    13. McAleer, Michael & Medeiros, Marcelo C., 2008. "A multiple regime smooth transition Heterogeneous Autoregressive model for long memory and asymmetries," Journal of Econometrics, Elsevier, vol. 147(1), pages 104-119, November.
    14. Claudeci Da Silva & Hugo Agudelo Murillo & Joaquim Miguel Couto, 2014. "Early Warning Systems: Análise De Ummodelo Probit De Contágio De Crise Dos Estados Unidos Para O Brasil(2000-2010)," Anais do XL Encontro Nacional de Economia [Proceedings of the 40th Brazilian Economics Meeting] 110, ANPEC - Associação Nacional dos Centros de Pós-Graduação em Economia [Brazilian Association of Graduate Programs in Economics].
    15. Heiko Rachinger & Edward M. H. Lin & Henghsiu Tsai, 2024. "A bootstrap test for threshold effects in a diffusion process," Computational Statistics, Springer, vol. 39(5), pages 2859-2872, July.
    16. Cathy W. S. Chen & Mike K. P. So & Ming-Tien Chen, 2005. "A Bayesian threshold nonlinearity test for financial time series," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 24(1), pages 61-75.
    17. Sandrine Jacob Leal, 2015. "Fundamentalists, chartists and asset pricing anomalies," Quantitative Finance, Taylor & Francis Journals, vol. 15(11), pages 1837-1850, November.
    18. Dangxing Chen, 2019. "Does the leverage effect affect the return distribution?," Papers 1909.08662, arXiv.org, revised Sep 2019.
    19. Ashok Chanabasangouda Patil & Shailesh Rastogi, 2019. "Time-Varying Price–Volume Relationship and Adaptive Market Efficiency: A Survey of the Empirical Literature," JRFM, MDPI, vol. 12(2), pages 1-18, June.
    20. Sebastien Valeyre & Denis Grebenkov & Sofiane Aboura & Qian Liu, 2013. "The reactive volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 13(11), pages 1697-1706, November.

    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:hal:journl:hal-01669082. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.