JUMPS IN FINANCIAL MARKET: TESTS AND IMPLICATIONS

Ito’s semimartingales are widely applied in modeling asset prices in financial markets. They are a rich class of stochastic processes including diffusion and jump diffusion processes, Lévy processes, etc. In recent literature, there has seen a rapidly increasing interest in semimartingales with jumps which are known as infrequent large changes in asset prices. And with the greater availability of high-frequency data nowadays and the growth in computing speed, there are many advancements in theories of estimating volatility as well as the nonparametric method of testing jumps. In this article, we start with an overview of various nonparametric methods for estimating integrated volatility and then classify various nonparametric jump test based on high-frequency financial data and explain the underlying logic of each jump test. Furthermore, we introduce the theory of market microstructure noise as well as the prevailing method to alleviate the effect and then propose a modified version of jump test based on the method of pre-averaging that is more robust to the market microstructure noise. Finally, we give the empirical resultabout various realized measures and the empirical evidence of the market microstructure noise in A-share market and then perform pre-averaging based tests to identify and study the jump components in A-share market. Concurrently, we have constructed a series of factors based on jump detection theories and presented the empirical results of the factor RPJV and RSJ to determine whether factors associated with stock price jumps possess the ability to predict future returns and their asset pricing power.

[1] MERTON R C. Option pricing when underlying stock returns are discontinuous[J/OL]. Journalof Financial Economics, 1976, 3(1): 125-144. https://www.sciencedirect.com/science/article/pii/0304405X76900222. DOI: https://doi.org/10.1016/0304-405X(76)90022-2.
[2] ENGLE R F. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance ofUnited Kingdom Inflation[J/OL]. Econometrica, 1982, 50(4): 987-1007
[2024-03-28]. http://www.jstor.org/stable/1912773.
[3] BOLLERSLEV T. Generalized autoregressive conditional heteroskedasticity[J/OL]. Journal ofEconometrics, 1986, 31: 307-327. DOI: 10.1016/0304-4076(86)90063-1.
[4] NELSON D B. CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEWAPPROACH[J/OL]. Econometrica, 1991, 59: 347-370. https://api.semanticscholar.org/CorpusID:153810026.
[5] ANDERSEN T G, BOLLERSLEV T, DIEBOLD F X, et al. The distribution of realized stockreturn volatility[J/OL]. Journal of Financial Economics, 2001, 61: 43-76. DOI: 10.1016/S0304-405X(01)00055-1.
[6] AïT-SAHALIA Y. Disentangling diffusion from jumps[J/OL]. Journal of Financial Economics,2004, 74: 487-528. DOI: 10.1016/J.JFINECO.2003.09.005.
[7] BARNDORFF-NIELSEN O E, SHEPHARD N. Power and Bipower Variation with StochasticVolatility and Jumps[J/OL]. Journal of Financial Econometrics, 2004, 2: 1-37. https://academic.oup.com/jfec/article/2/1/1/960705. DOI: 10.1093/JJFINEC/NBH001.
[8] BARNDORFF-NIELSEN O E, SHEPHARD N. Econometrics of Testing for Jumps in FinancialEconomics Using Bipower Variation[J/OL]. Journal of Financial Econometrics, 2006, 4: 1-30.https://academic.oup.com/jfec/article/4/1/1/833043. DOI: 10.1093/JJFINEC/NBI022.
[9] AïT-SAHALIA Y, JACOD J. Estimating the degree of activity of jumps in high frequency data[J/OL]. Annals of Statistics, 2009, 37: 2202-2244. https://projecteuclid.org/journals/annals-of-statistics/volume-37/issue-5A/Estimating-the-degree-of-activity-of-jumps-in-high-frequency/10.1214/08-AOS640.fullhttps://projecteuclid.org/journals/annals-of-statistics/volume-37/issue-5A/Estimating-the-degree-of-activity-of-jumps-in-high-frequency/10.1214/08-AOS640.short.
[10] LEE S S, MYKLAND P A. Jumps in Financial Markets: A New Nonparametric Test and JumpDynamics[J/OL]. The Review of Financial Studies, 2008, 21: 2535-2563. https://academic.oup.com/rfs/article/21/6/2535/1574138. DOI: 10.1093/RFS/HHM056.
[11] JIANG G J, OOMEN R C. Testing for jumps when asset prices are observed with noise–a “swapvariance” approach[J/OL]. Journal of Econometrics, 2008, 144: 352-370. DOI: 10.1016/J.JECONOM.2008.04.009.
[12] AïT-SAHALIA Y, JACOD J, LI J. Testing for jumps in noisy high frequency data[J/OL]. Journalof Econometrics, 2012, 168: 207-222. DOI: 10.1016/J.JECONOM.2011.12.004.
[13] LI J. Robust Estimation and Inference for Jumps in Noisy High Frequency Data: A Localto-Continuity Theory for the Pre-Averaging Method[J/OL]. Econometrica, 2013, 81: 1673-1693. https://onlinelibrary.wiley.com/doi/full/10.3982/ECTA10534https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA10534https://onlinelibrary.wiley.com/doi/10.3982/ECTA10534.
[14] Aı¨TAı¨T-SAHALIA Y, MYKLAND P A, ZHANG L. How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise[J/OL]. The Review of FinancialStudies, 2005, 18. DOI: 10.1093/rfs/hhi016.
[15] XIU D. Quasi-maximum likelihood estimation of volatility with high frequency data[J/OL].Journal of Econometrics, 2010, 159: 235-250. DOI: 10.1016/J.JECONOM.2010.07.002.
[16] SHEPHARD N, XIU D. Econometric analysis of multivariate realised QML: Estimation of thecovariation of equity prices under asynchronous trading[J/OL]. Journal of Econometrics, 2017,201: 19-42. DOI: 10.1016/j.jeconom.2017.04.003.
[17] ZHANG L, MYKLAND P A, AïT-SAHALIA Y. A tale of two time scales: Determining integratedvolatility with noisy high-frequency data[J/OL]. Journal of the American StatisticalAssociation, 2005, 100: 1394-1411. DOI: 10.1198/016214505000000169.
[18] ZHANG L. Efficient estimation of stochastic volatility using noisy observations: a multi-scaleapproach[J/OL]. Bernoulli, 2006, 12: 1019-1043. https://projecteuclid.org/journals/bernoulli/volume-12/issue-6/Efficient-estimation-of-stochastic-volatility-using-noisy-observations--a/10.3150/bj/1165269149.fullhttps://projecteuclid.org/journals/bernoulli/volume-12/issue-6/Efficient-estimation-of-stochastic-volatility-using-noisy-observations--a/10.3150/bj/1165269149.short. DOI: 10.3150/BJ/1165269149.
[19] BARNDORFF-NIELSEN O, HANSEN P, LUNDE A, et al. Designing realized Kernels tomeasure the ex post variation of equity prices in the presence of noise[J/OL]. Econometrica,2008, 76: 1481-1536. DOI: 10.3982/ECTA6495.
[20] AïT-SAHALIA Y, JACOD J. Is Brownian motion necessary to model high-frequency data?[J/OL]. https://doi.org/10.1214/09-AOS749, 2010, 38: 3093-3128. https://projecteuclid.org/journals/annals-of-statistics/volume-38/issue-5/Is-Brownian-motion-necessary-to-model-high-frequency-data/10.1214/09-AOS749.fullhttps://projecteuclid.org/journals/annals-of-statistics/volume-38/issue-5/Is-Brownian-motion-necessary-to-model-high-frequency-data/10.1214/09-AOS749.short.
[21] JACOD J, LI Y, ZHENG X. Statistical Properties of Microstructure Noise[J/OL]. Econometrica,2017, 85: 1133-1174. https://onlinelibrary.wiley.com/doi/full/10.3982/ECTA13085https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA13085https://onlinelibrary.wiley.com/doi/10.3982/ECTA13085.
[22] LI Z M, LINTON O. A ReMeDI for Microstructure Noise[J/OL]. Econometrica, 2022, 90: 367-389. https://onlinelibrary.wiley.com/doi/full/10.3982/ECTA17505https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA17505https://onlinelibrary.wiley.com/doi/10.3982/ECTA17505.
[23] TAUCHEN G, ZHOU H. Realized jumps on financial markets and predicting credit spreads[J/OL]. Journal of Econometrics, 2011, 160: 102-118. DOI: 10.1016/J.JECONOM.2010.03.023.
[24] JIANG G J, YAO T. Stock Price Jumps and Cross-Sectional Return Predictability[J/OL]. TheJournal of Financial and Quantitative Analysis, 2013, 48(5): 1519-1544
[2024-03-30]. http://www.jstor.org/stable/43303850.
[25] BOLLERSLEV T, LI S Z, ZHAO B. Good Volatility, Bad Volatility, and the Cross Sectionof Stock Returns[J/OL]. Journal of Financial and Quantitative Analysis, 2020, 55: 751-781.https://www.cambridge.org/core/journals/journal-of-financial-and-quantitative-analysis/article/good-volatility-bad-volatility-and-the-cross-section-of-stock-returns/B9203F87AB93D8DEAD51CB692587350E. DOI: 10.1017/S0022109019000097.
[26] JIANG G J, ZHU K X. Information Shocks and Short-Term Market Underreaction[J/OL].Journal of Financial Economics, 2017, 124: 43-64. DOI: 10.1016/J.JFINECO.2016.06.006.
[27] CUI X, SENSOY A, NGUYEN D K, et al. Positive information shocks, investor behaviorand stock price crash risk[J/OL]. Journal of Economic Behavior & Organization, 2022, 197:493-518. DOI: 10.1016/J.JEBO.2022.03.016.
[28] AïT-SAHALIA Y, JACOD J. Analyzing the Spectrum of Asset Returns: Jump and VolatilityComponents in High Frequency Data[J/OL]. Journal of Economic Literature, 2012, 50: 1007-50. DOI: 10.1257/JEL.50.4.1007.
[29] ÇINLAR E. Probability and Stochastics: Vol. 261[M/OL]. Springer New York, 2011. https://link.springer.com/10.1007/978-0-387-87859-1.
[30] GALL J F L. Brownian Motion, Martingales, and Stochastic Calculus: Vol. 274[M/OL].Springer International Publishing, 2016. http://link.springer.com/10.1007/978-3-319-31089-3.
[31] JACOD J, SHIRYAEV A N. Limit Theorems for Stochastic Processes: Vol. 288[M/OL].Springer Berlin Heidelberg, 2003. http://link.springer.com/10.1007/978-3-662-05265-5.
[32] JACOD J, LI Y, ZHENG X. Estimating the integrated volatility with tick observations[J/OL].Journal of Econometrics, 2019, 208: 80-100. DOI: 10.1016/j.jeconom.2018.09.006.
[33] AïT-SAHALIA Y, JACOD J. High-frequency financial econometrics[M/OL]. Princeton UniversityPress, 2014: 1-659. DOI: 10.3390/risks4010005.
[34] AïT-SAHALIA Y, MYKLAND P, ZHANG L. How often to sample a continuous-time processin the presence of market microstructure noise[J/OL]. Review of Financial Studies, 2005, 18:351-416. DOI: 10.1093/rfs/hhi016.
[35] AïT-SAHALIA Y, MYKLAND P. The effects of random and discrete sampling when estimatingcontinuous-time diffusions[J/OL]. Econometrica, 2003, 71: 483-549. DOI: 10.1111/1468-0262.t01-1-00416.
[36] BARNDORFF-NIELSEN O E, SHEPHARD N, WINKEL M. Limit theorems for multipowervariation in the presence of jumps[J/OL]. Stochastic Processes and their Applications, 2006,116: 796-806. DOI: 10.1016/J.SPA.2006.01.007.
[37] FAN Y, FAN J. Testing and detecting jumps based on a discretely observed process[J/OL].Journal of Econometrics, 2011, 164: 331-344. DOI: 10.1016/J.JECONOM.2011.06.014.
[38] LIU Z, KONG X B, JING B Y. Estimating the integrated volatility using high-frequency datawith zero durations[J/OL]. Journal of Econometrics, 2018, 204: 18-32. DOI: 10.1016/J.JECONOM.2017.12.008.50REFERENCES
[39] JACOD J. Limit of Random measures associated with the increments of a Brownian semimartingale[J/OL]. Journal of Financial Econometrics, 2018, 16: 526-569. DOI: 10.1093/jjfinec/nbx021.
[40] BARNDORFF-NIELSEN O E, KINNEBROUK S, SHEPHARD N. Measuring downside risk:realised semivariance[M]. (edited by t. bollerslev, j. russell and m. watson) ed. Oxford UniversityPress, 2010: 117-136.
[41] BOLLERSLEV T. Realized Semi(co)variation: Signs That All Volatilities are Not CreatedEqual[J/OL]. Journal of Financial Econometrics, 2022, 20: 219-252. https://dx.doi.org/10.1093/jjfinec/nbab025. DOI: 10.1093/JJFINEC/NBAB025.
[42] DUONG D, SWANSON N R. Empirical evidence on the importance of aggregation, asymmetry,and jumps for volatility prediction[J/OL]. Journal of Econometrics, 2015, 187: 606-621. DOI:10.1016/J.JECONOM.2015.02.042.
[43] HUANG X, TAUCHEN G. The Relative Contribution of Jumps to Total Price Variance[J/OL].Journal of Financial Econometrics, 2005, 3(4): 456-499. https://doi.org/10.1093/jjfinec/nbi025.
[44] CORSI F, PIRINO D, RENò R. Threshold bipower variation and the impact of jumps on volatilityforecasting[J/OL]. Journal of Econometrics, 2010, 159: 276-288. DOI: 10.1016/J.JECONOM.2010.07.008.
[45] ANDERSEN T G, BOLLERSLEV T, DIEBOLD F X. Roughing It Up: Including Jump Componentsin the Measurement, Modeling, and Forecasting of Return Volatility[J/OL]. The Reviewof Economics and Statistics, 2007, 89: 701-720. https://dx.doi.org/10.1162/rest.89.4.701. DOI:10.1162/REST.89.4.701.
[46] KOLOKOLOV A, RENò R. Efficient Multipowers[J/OL]. Journal of Financial Econometrics,2018, 16: 629-659. https://dx.doi.org/10.1093/jjfinec/nbx018. DOI: 10.1093/JJFINEC/NBX018.
[47] MANCINI C. Non-parametric threshold estimation for models with stochastic diffusion coefficientand jumps[J/OL]. Scandinavian Journal of Statistics, 2009, 36. DOI: 10.1111/j.1467-9469.2008.00622.x.
[48] ANDERSEN T G, DOBREV D, SCHAUMBURG E. Jump-robust volatility estimation usingnearest neighbor truncation[J/OL]. Journal of Econometrics, 2012, 169: 75-93. DOI: 10.1016/J.JECONOM.2012.01.011.
[49] AïT-SAHALIA Y, JACOD J. Testing for jumps in a discretely observed process[J/OL]. Annalsof Statistics, 2009, 37: 184-222. https://projecteuclid.org/journals/annals-of-statistics/volume-37/issue-1/Testing-for-jumps-in-a-discretely-observed-process/10.1214/07-AOS568.fullhttps://projecteuclid.org/journals/annals-of-statistics/volume-37/issue-1/Testing-for-jumps-in-a-discretely-observed-process/10.1214/07-AOS568.short.
[50] JACOD J, MYKLAND P A. Microstructure noise in the continuous case: Approximate efficiencyof the adaptive pre-averaging method[J/OL]. Stochastic Processes and their Applications,2015, 125: 2910-2936. DOI: 10.1016/J.SPA.2015.02.005.
[51] NEUBERGER A. The Log Contract[J/OL]. The Journal of Portfolio Management, 1994, 20:74-80. https://www.pm-research.com/content/iijpormgmt/20/2/74. DOI: 10.3905/JPM.1994.409478.