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Stochastic volatility and stochastic leverage

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  • Almut Veraart
  • Luitgard Veraart

Abstract

This paper proposes the new concept of stochastic leverage in stochastic volatility models. Stochastic leverage refers to a stochastic process which replaces the classical constant correlation parameter between the asset return and the stochastic volatility process. We provide a systematic treatment of stochastic leverage and propose to model the stochastic leverage effect explicitly, e.g. by means of a linear transformation of a Jacobi process. Such models are both analytically tractable and allow for a direct economic interpretation. In particular, we propose two new stochastic volatility models which allow for a stochastic leverage effect: the generalised Heston model and the generalised Barndorff-Nielsen & Shephard model. We investigate the impact of a stochastic leverage effect in the risk neutral world by focusing on implied volatilities generated by option prices derived from our new models. Furthermore, we give a detailed account on statistical properties of the new models.
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Suggested Citation

  • Almut Veraart & Luitgard Veraart, 2012. "Stochastic volatility and stochastic leverage," Annals of Finance, Springer, vol. 8(2), pages 205-233, May.
    • Handle: RePEc:kap:annfin:v:8:y:2012:i:2:p:205-233
    DOI: 10.1007/s10436-010-0157-3
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    Cited by:

    1. Frederik Herzberg, 2013. "First steps towards an equilibrium theory for Lévy financial markets," Annals of Finance, Springer, vol. 9(3), pages 543-572, August.
    2. René Aïd & Luciano Campi & Nicolas Langrené & Huyên Pham, 2012. "A probabilistic numerical method for optimal multiple switching problems in high dimension," Working Papers hal-00747229, HAL.
    3. 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.
    4. Hoang Nguyen & Trong-Nghia Nguyen & Minh-Ngoc Tran, 2023. "A dynamic leverage stochastic volatility model," Applied Economics Letters, Taylor & Francis Journals, vol. 30(1), pages 97-102, January.
    5. Almut Veraart, 2011. "How precise is the finite sample approximation of the asymptotic distribution of realised variation measures in the presence of jumps?," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 95(3), pages 253-291, September.
    6. Curato, Imma Valentina, 2019. "Estimation of the stochastic leverage effect using the Fourier transform method," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3207-3238.
    7. Ting, Sai Hung Marten & Ewald, Christian-Oliver & Wang, Wen-Kai, 2013. "On the investment–uncertainty relationship in a real option model with stochastic volatility," Mathematical Social Sciences, Elsevier, vol. 66(1), pages 22-32.
    8. Imma Valentina Curato & Simona Sanfelici, 2019. "Stochastic leverage effect in high-frequency data: a Fourier based analysis," Papers 1910.06660, arXiv.org, revised Mar 2021.
    9. Yacine Aït-Sahalia & Jianqing Fan & Roger J. A. Laeven & Christina Dan Wang & Xiye Yang, 2017. "Estimation of the Continuous and Discontinuous Leverage Effects," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1744-1758, October.
    10. Aboura, Sofiane & Chevallier, Julien, 2018. "Tail risk and the return-volatility relation," Research in International Business and Finance, Elsevier, vol. 46(C), pages 16-29.
    11. Aïd, René & Campi, Luciano & Langrené, Nicolas & Pham, Huyên, 2014. "A probabilistic numerical method for optimal multiple switching problems in high dimension," LSE Research Online Documents on Economics 63011, London School of Economics and Political Science, LSE Library.
    12. Giacomo Toscano & Maria Cristina Recchioni, 2022. "Bias-optimal vol-of-vol estimation: the role of window overlapping," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 137-185, June.
    13. Ren'e Aid & Luciano Campi & Nicolas Langren'e & Huy^en Pham, 2012. "A probabilistic numerical method for optimal multiple switching problem and application to investments in electricity generation," Papers 1210.8175, arXiv.org.
    14. Curato, Imma Valentina & Sanfelici, Simona, 2022. "Stochastic leverage effect in high-frequency data: a Fourier based analysis," Econometrics and Statistics, Elsevier, vol. 23(C), pages 53-82.
    15. Carles Bret'o, 2013. "On idiosyncratic stochasticity of financial leverage effects," Papers 1312.5496, arXiv.org.
    16. Marie Roy de Chaumaray, 2018. "Moderate deviations for parameters estimation in a geometrically ergodic Heston process," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 553-567, October.
    17. Ole E. Barndorff-Nielsen & Almut E. D. Veraart, 2009. "Stochastic volatility of volatility in continuous time," CREATES Research Papers 2009-25, Department of Economics and Business Economics, Aarhus University.
    18. Bretó, Carles, 2014. "On idiosyncratic stochasticity of financial leverage effects," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 20-26.
    19. Alexander Schnurr, 2015. "An Ordinal Pattern Approach to Detect and to Model Leverage Effects and Dependence Structures Between Financial Time Series," Papers 1502.07321, arXiv.org.

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    More about this item

    Keywords

    Stochastic volatility; Volatility of volatility; Stochastic correlation; Leverage effect; Jacobi process; Ornstein–Uhlenbeck process; Square root diffusion; Lévy process; Heston model; Barndorff-Nielsen & Shephard model; C1; C5; G0; G1;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets

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