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

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  • Antoine Lejay

    (TOSCA, IECL)

  • Paolo Pigato

    (WIAS)

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 leverageeffects. Mean-reversion effects are also detected, most markedly in crisis periods.

Suggested Citation

  • Antoine Lejay & Paolo Pigato, 2017. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Papers 1712.08329, arXiv.org, revised Feb 2019.
    • Handle: RePEc:arx:papers:1712.08329
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    1. 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.
    2. 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.
    3. Andrew Ang & Allan Timmermann, 2012. "Regime Changes and Financial Markets," Annual Review of Financial Economics, Annual Reviews, vol. 4(1), pages 313-337, October.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    9. 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.
    10. Su, Fei & Chan, Kung-Sik, 2015. "Quasi-likelihood estimation of a threshold diffusion process," Journal of Econometrics, Elsevier, vol. 189(2), pages 473-484.
    11. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    12. Hens, Thorsten & Steude, Sven C., 2009. "The leverage effect without leverage," Finance Research Letters, Elsevier, vol. 6(2), pages 83-94, June.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. Tong, Howell, 2015. "Threshold models in time series analysis—Some reflections," Journal of Econometrics, Elsevier, vol. 189(2), pages 485-491.
    22. 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.
    23. 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.
    24. Rossello, Damiano, 2012. "Arbitrage in skew Brownian motion models," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 50-56.
    25. 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.
    26. 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.
    27. 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.
    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.
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    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. Héctor Araya & Meryem Slaoui & Soledad Torres, 2022. "Bayesian inference for fractional Oscillating Brownian motion," Computational Statistics, Springer, vol. 37(2), pages 887-907, April.
    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. 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.
    5. 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.

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