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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. 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.
    2. Frederik Herzberg, 2013. "First steps towards an equilibrium theory for Lévy financial markets," Annals of Finance, Springer, vol. 9(3), pages 543-572, August.
    3. 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.
    4. Bretó, Carles, 2014. "On idiosyncratic stochasticity of financial leverage effects," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 20-26.
    5. 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.
    6. 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.
    7. 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.
    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. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    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. Carles Bret'o, 2013. "On idiosyncratic stochasticity of financial leverage effects," Papers 1312.5496, arXiv.org.
    19. 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.

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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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