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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

    (Université de Lorraine, IECL, UMR 7502, Vandœuvre-lés-Nancy F-54600, France2CNRS, IECL, UMR 7502, Vandœuvre-lés-Nancy F-54600, France3Inria, Villers-lés-Nancy F-54600, France)

  • PAOLO PIGATO

    (Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, Berlin 10117, Germany)

Abstract

In financial markets, low prices are generally associated with high volatilities and vice-versa, this well-known stylized fact is usually referred to as the leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts for leverage and mean-reversion effects in the dynamics of the prices. This model exhibits a regime switch in the dynamics according 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 351 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," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    • Handle: RePEc:wsi:ijtafx:v:22:y:2019:i:04:n:s0219024919500171
    DOI: 10.1142/S0219024919500171
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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. 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.

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