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Do price and volatility jump together?


  • Jean Jacod
  • Viktor Todorov


We consider a process $X_t$, which is observed on a finite time interval $[0,T]$, at discrete times $0,\Delta_n,2\Delta_n,\ldots.$ This process is an It\^{o} semimartingale with stochastic volatility $\sigma_t^2$. Assuming that $X$ has jumps on $[0,T]$, we derive tests to decide whether the volatility process has jumps occurring simultaneously with the jumps of $X_t$. There are two different families of tests for the two possible null hypotheses (common jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh $\Delta_n$ goes to $0$. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use on S&P 500 index data.

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  • Jean Jacod & Viktor Todorov, 2010. "Do price and volatility jump together?," Papers 1010.4990,
  • Handle: RePEc:arx:papers:1010.4990

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    1. S. P. Sethi & N. A. Derzko & J. P. Lehoczky, 1991. "A Stochastic Extension of the Miller-Modigliani Framework," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 57-76.
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    Cited by:

    1. Torben G. Andersen & Nicola Fusari & Viktor Todorov, 2015. "Parametric Inference and Dynamic State Recovery From Option Panels," Econometrica, Econometric Society, vol. 83(3), pages 1081-1145, May.
    2. Mykland, Per A. & Zhang, Lan, 2016. "Between data cleaning and inference: Pre-averaging and robust estimators of the efficient price," Journal of Econometrics, Elsevier, vol. 194(2), pages 242-262.
    3. Worapree Maneesoonthorn & Catherine S. Forbes & Gael M. Martin, 2013. "Inference on Self-Exciting Jumps in Prices and Volatility using High Frequency Measures," Monash Econometrics and Business Statistics Working Papers 28/13, Monash University, Department of Econometrics and Business Statistics.
    4. repec:eee:econom:v:202:y:2018:i:1:p:18-44 is not listed on IDEAS
    5. Caporin, Massimiliano & Rossi, Eduardo & Santucci de Magistris, Paolo, 2017. "Chasing volatility," Journal of Econometrics, Elsevier, vol. 198(1), pages 122-145.
    6. Thomas Kokholm & Martin Stisen, 2015. "Joint pricing of VIX and SPX options with stochastic volatility and jump models," Journal of Risk Finance, Emerald Group Publishing, vol. 16(1), pages 27-48, January.
    7. Andras Fulop & Junye Li & Jun Yu, 2012. "Investigating Impacts of Self-Exciting Jumps in Returns and Volatility: A Bayesian Learning Approach," Global COE Hi-Stat Discussion Paper Series gd12-264, Institute of Economic Research, Hitotsubashi University.
    8. Isao Ishida & Michael McAleer & Kosuke Oya, 2011. "Estimating the leverage parameter of continuous-time stochastic volatility models using high frequency S&P 500 and VIX," Managerial Finance, Emerald Group Publishing, vol. 37(11), pages 1048-1067, September.
    9. Aït-Sahalia, Yacine & Xiu, Dacheng, 2016. "Increased correlation among asset classes: Are volatility or jumps to blame, or both?," Journal of Econometrics, Elsevier, vol. 194(2), pages 205-219.
    10. Christian Palmes & Jeannette H. C. Woerner, 2016. "The Gumbel test and jumps in the volatility process," Statistical Inference for Stochastic Processes, Springer, vol. 19(2), pages 235-258, July.
    11. repec:eee:econom:v:201:y:2017:i:2:p:417-432 is not listed on IDEAS
    12. Cecilia Mancini & Vanessa Mattiussi & Roberto Renò, 2015. "Spot volatility estimation using delta sequences," Finance and Stochastics, Springer, vol. 19(2), pages 261-293, April.
    13. 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.
    14. Bregantini, Daniele, 2013. "Moment-based estimation of stochastic volatility," Journal of Banking & Finance, Elsevier, vol. 37(12), pages 4755-4764.
    15. Worapree Maneesoonthorn & Catherine S. Forbes & Gael M. Martin, 2016. "Inference on Self-Exciting Jumps in Prices and Volatility using High Frequency Measures," Monash Econometrics and Business Statistics Working Papers 8/16, Monash University, Department of Econometrics and Business Statistics.
    16. Markus Bibinger & Lars Winkelmann, 2014. "Common price and volatility jumps in noisy high-frequency data," SFB 649 Discussion Papers SFB649DP2014-037, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    17. Markus Bibinger & Moritz Jirak & Mathias Vetter, 2015. "Nonparametric change-point analysis of volatility," SFB 649 Discussion Papers SFB649DP2015-008, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    18. Chan, Kam Fong & Bowman, Robert G. & Neely, Christopher J., 2017. "Systematic cojumps, market component portfolios and scheduled macroeconomic announcements," Journal of Empirical Finance, Elsevier, vol. 43(C), pages 43-58.
    19. Behme, Anita & Chong, Carsten & Klüppelberg, Claudia, 2015. "Superposition of COGARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1426-1469.
    20. Pun, Chi Seng & Chung, Shing Fung & Wong, Hoi Ying, 2015. "Variance swap with mean reversion, multifactor stochastic volatility and jumps," European Journal of Operational Research, Elsevier, vol. 245(2), pages 571-580.
    21. Chiarella, Carl & He, Xue-Zhong & Huang, Weihong & Zheng, Huanhuan, 2012. "Estimating behavioural heterogeneity under regime switching," Journal of Economic Behavior & Organization, Elsevier, vol. 83(3), pages 446-460.
    22. Ishida, I. & McAleer, M.J. & Oya, K., 2011. "Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 VIX," Econometric Institute Research Papers EI 2011-10, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    23. Bandi, F.M. & Renò, R., 2016. "Price and volatility co-jumps," Journal of Financial Economics, Elsevier, vol. 119(1), pages 107-146.
    24. Karl Friedrich Hofmann & Thorsten Schulz, 2016. "A General Ornstein–Uhlenbeck Stochastic Volatility Model With Lévy Jumps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(08), pages 1-23, December.
    25. Jacod, Jean & Klüppelberg, Claudia & Müller, Gernot, 2017. "Testing for non-correlation between price and volatility jumps," Journal of Econometrics, Elsevier, vol. 197(2), pages 284-297.

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