Detecting jumps in stochastic volatility jump-diffusion models via the power variation approach

Zhongyu Chen; Juliang Yin; Zhongyu Chen; Juliang Yin

doi:10.3934/math.2025858

AIMS Mathematics

2025, Volume 10, Issue 8: 19189-19216. doi: 10.3934/math.2025858

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Research article

Detecting jumps in stochastic volatility jump-diffusion models via the power variation approach

Zhongyu Chen ,
Juliang Yin ^,

School of Economics and Statistics, Guangzhou University, Guangzhou, Guangdong, China

Received: 06 May 2025 Revised: 16 July 2025 Accepted: 29 July 2025 Published: 22 August 2025
MSC : 62M20, 62P05

It is well-known that pretesting the presence of the jump component in an underlying price process is crucial for modeling this process. In this paper, we propose a consistent test for jump intensity of the conditional Poisson process in a stochastic volatility jump diffusion model. Theoretically, we derive the infill and long-span asymptotic properties of realized power variation under some suitable conditions, and verify the asymptotic size and power of the proposed test. Furthermore, the finite-sample performance of our proposed test is illustrated through simulation analysis, and an application to real price series provides empirical evidence of significant jump intensities.
- high-frequency data,
- jump intensity,
- long time span jump test,
- power variation
Citation: Zhongyu Chen, Juliang Yin. Detecting jumps in stochastic volatility jump-diffusion models via the power variation approach[J]. AIMS Mathematics, 2025, 10(8): 19189-19216. doi: 10.3934/math.2025858

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Abstract

It is well-known that pretesting the presence of the jump component in an underlying price process is crucial for modeling this process. In this paper, we propose a consistent test for jump intensity of the conditional Poisson process in a stochastic volatility jump diffusion model. Theoretically, we derive the infill and long-span asymptotic properties of realized power variation under some suitable conditions, and verify the asymptotic size and power of the proposed test. Furthermore, the finite-sample performance of our proposed test is illustrated through simulation analysis, and an application to real price series provides empirical evidence of significant jump intensities.

References

[1]	P. Tankov, R. Cont, Financial modelling with jump processes, 2 Eds., New York: Taylor & Francis, 2015.
[2]	E. Platen, N. Bruti-Liberati, Stochastic differential equations with jumps, In: Numerical solution of stochastic differential equations with jumps in finance, Berlin, Heidelberg: Springer, 2010, 1–60. https://doi.org/10.1007/978-3-642-13694-8_1
[3]	V. Todorov, Variance risk-premium dynamics: the role of jumps, Rev. Financ. Stud., 23 (2010), 345–383. https://doi.org/10.1093/rfs/hhp035 doi: 10.1093/rfs/hhp035
[4]	T. Bollerslev, V. Todorov, Tails, fears, and risk premia, J. Finance, 66 (2011), 2165–2211. https://doi.org/10.1111/j.1540-6261.2011.01695.x
[5]	J. Liu, F. A. Longstaff, J. Pan, Dynamic asset allocation with event risk, J. Finance, 58 (2003), 231–259. https://doi.org/10.1111/1540-6261.00523 doi: 10.1111/1540-6261.00523
[6]	Y. Aït-Sahalia, T. R. Hurd, Portfolio choice in markets with contagion, J. Financ. Econom., 14 (2016), 1–28. https://doi.org/10.1093/jjfinec/nbv024 doi: 10.1093/jjfinec/nbv024
[7]	G. Bakshi, C. Cao, Z. W. Chen, Empirical performance of alternative option pricing models, J. Finance, 52 (1997), 2003–2049. https://doi.org/10.1111/j.1540-6261.1997.tb02749.x doi: 10.1111/j.1540-6261.1997.tb02749.x
[8]	T. G. Andersen, N. Fusari, V. Todorov, The risk premia embedded in index options, J. Financ. Econ., 117 (2015), 558–584. https://doi.org/10.1016/j.jfineco.2015.06.005 doi: 10.1016/j.jfineco.2015.06.005
[9]	Y. Aït-Sahalia, J. Jacod, Testing for jumps in a discretely observed process, Ann. Stat., 37 (2009), 184–222. https://doi.org/10.1214/07-AOS568 doi: 10.1214/07-AOS568
[10]	M. Podolskij, D. Ziggel, New tests for jumps in semimartingale models, Stat. Inference Stoch. Process, 13 (2010), 15–41. https://doi.org/10.1007/s11203-009-9037-8 doi: 10.1007/s11203-009-9037-8
[11]	O. E. Barndorff-Nielsen, S. E. Graversen, J. Jacod, M. Podolskij, N. Shephard, A central limit theorem for realised power and bipower variations of continuous semimartingales, In: From stochastic calculus to mathematical finance, Berlin, Heidelberg: Springer, 2006, 33–68. https://doi.org/10.1007/978-3-540-30788-4_3
[12]	O. E. Barndorff-Nielsen, N. Shephard, Econometrics of testing for jumps in financial economics using bipower variation, J. Financ. Econom., 4 (2006), 1–30. https://doi.org/10.1093/jjfinec/nbi022 doi: 10.1093/jjfinec/nbi022
[13]	F. Corsi, D. Pirino, R. Reno, Threshold bipower variation and the impact of jumps on volatility forecasting, J. Econom., 159 (2010), 276–288. https://doi.org/10.1016/j.jeconom.2010.07.008 doi: 10.1016/j.jeconom.2010.07.008
[14]	T. G. Andersen, D. Dobrev, E. Schaumburg, Jump-robust volatility estimation using nearest neighbor truncation, J. Econom., 169 (2012), 75–93. https://doi.org/10.1016/j.jeconom.2012.01.011 doi: 10.1016/j.jeconom.2012.01.011
[15]	G. J. Jiang, R. C. A. Oomen, Testing for jumps when asset prices are observed with noise–a "swap variance" approach, J. Econom., 144 (2008), 352–370. https://doi.org/10.1016/j.jeconom.2008.04.009 doi: 10.1016/j.jeconom.2008.04.009
[16]	A. M. Dumitru, G. Urga, Identifying jumps in financial assets: a comparison between nonparametric jump tests, J. Bus. Econ. Stat., 30 (2012), 242–255. https://doi.org/10.1080/07350015.2012.663250 doi: 10.1080/07350015.2012.663250
[17]	T. G. Andersen, T. Bollerslev, D. Dobrev, No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and i.i.d. noise: theory and testable distributional implications, J. Econom., 138 (2007), 125–180. https://doi.org/10.1016/j.jeconom.2006.05.018 doi: 10.1016/j.jeconom.2006.05.018
[18]	S. S. Lee, P. A. Mykland, Jumps in financial markets: a new nonparametric test and jump dynamics, Rev. Financ. Stud., 21 (2008), 2535–2563. https://doi.org/10.1093/rfs/hhm056 doi: 10.1093/rfs/hhm056
[19]	Y. Xue, R. Gencay, S. Fagan, Jump detection with wavelets for high-frequency financial time series, Quant. Finance, 14 (2014), 1427–1444. https://doi.org/10.1080/14697688.2013.830320 doi: 10.1080/14697688.2013.830320
[20]	E. W. Sun, T. Meinl, A new wavelet-based denoising algorithm for high-frequency financial data mining, Eur. J. Oper. Res., 217 (2012), 589–599. https://doi.org/10.1016/j.ejor.2011.09.049 doi: 10.1016/j.ejor.2011.09.049
[21]	Y. T. Chen, W. N. Lai, E. W. Sun, Jump detection and noise separation by a singular wavelet method for predictive analytics of high-frequency data, Comput. Econ., 54 (2019), 809–844. https://doi.org/10.1007/s10614-019-09881-3 doi: 10.1007/s10614-019-09881-3
[22]	W. Z. Chen, G. Zhang, Jump detection using deep learning: with applications to financial time series data, Comput. Econom., 2025, 1–16. https://doi.org/10.1007/s10614-025-10949-6
[23]	V. Corradi, M. J. Silvapulle, N. R. Swanson, Testing for jumps and jump intensity path dependence, J. Econ., 204 (2018), 248–267. https://doi.org/10.1016/j.jeconom.2018.02.004 doi: 10.1016/j.jeconom.2018.02.004
[24]	J. Jacod, P. Protter, Discretization of processes, Berlin, Heidelberg: Springer, 2012. https://doi.org/10.1007/978-3-642-24127-7
[25]	X. Huang, G. Tauchen, The relative contribution of jumps to total price variance, J. Financ. Econom., 3 (2005), 456–499. https://doi.org/10.1093/jjfinec/nbi025 doi: 10.1093/jjfinec/nbi025
[26]	M. M. Cheng, N. R. Swanson, Fixed and long time span jump tests: new monte carlo and empirical evidence, Econometrics, 7 (2019), 1–32. https://doi.org/10.3390/econometrics7010013 doi: 10.3390/econometrics7010013
[27]	P. Brémaud, Point processes and queues: martingale dynamics, Springer, 1981.
[28]	I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, New York: Springer, 1998. https://doi.org/10.1007/978-1-4612-0949-2
[29]	X. Mao, Stochastic differential equations and applications, 2 Eds., Elsevier, 2007.
[30]	Z. F. Fu, Z. H. Li, Stochastic equations of non-negative processes with jumps, Stoch. Proc. Appl., 120 (2010), 306–330. https://doi.org/10.1016/j.spa.2009.11.005 doi: 10.1016/j.spa.2009.11.005
[31]	F. J. Mhlanga, L. Rundora, On the global positivity solutions of non-homogeneous stochastic differential equations, Front. Appl. Math. Stat., 8 (2022), 847896. https://doi.org/10.3389/fams.2022.847896 doi: 10.3389/fams.2022.847896
[32]	A. Friedman, Stochastic differential equations and applications; Vol.1, New York: Academic Press, 1975.
[33]	J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, 2 Eds., Berlin, Heidelberg: Springer, 2003. https://doi.org/10.1007/978-3-662-05265-5
[34]	M. Uchida, N. Yoshida, Estimation for misspecified ergodic diffusion processes from discrete observations, ESAIM Probab. Stat., 15 (2011), 270–290. https://doi.org/10.1051/ps/2010001 doi: 10.1051/ps/2010001
[35]	Y. P. Song, Z. Y. Lin, Empirical likelihood inference for the second-order jump-diffusion model, Stat. Probab. Lett., 83 (2013), 184–195. https://doi.org/10.1016/j.spl.2012.09.005 doi: 10.1016/j.spl.2012.09.005
[36]	N. M. Jakobsen, M. Sørensen, Estimating functions for jump-diffusions, Stoch. Proc. Appl., 129 (2019), 3282–3318. https://doi.org/10.1016/j.spa.2018.09.006 doi: 10.1016/j.spa.2018.09.006
[37]	W. Fang, Adaptive timestepping for SDEs with non-globally Lipschitz drift, PhD thesis, University of Oxford, 2019.
[38]	C. Amorino, A. Gloter, Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function, Stat. Inference Stoch. Process, 24 (2021), 61–148. https://doi.org/10.1007/s11203-020-09227-z doi: 10.1007/s11203-020-09227-z
[39]	M. Jeong, J. Y. Park, Asymptotic theory of maximum likelihood estimator for diffusion model, Technical report, 2013.
[40]	L. C. G. Rogers, D. Williams, Diffusions, Markov processes, and martingales, Cambridge University Press, 2000.
[41]	M. B. Alaya, A. Kebaier, Parameter estimation for the square-root diffusions: ergodic and nonergodic cases, Stoch. Models, 28 (2012), 609–634. https://doi.org/10.1080/15326349.2012.726042 doi: 10.1080/15326349.2012.726042
[42]	Y. Aït-Sahalia, J. Jacod, Analyzing the spectrum of asset returns: jump and volatility components in high frequency data, J. Econ. Lit., 50 (2012), 1007–1050. http://dx.doi.org/10.1257/jel.50.4.1007 doi: 10.1257/jel.50.4.1007
[43]	C. Mancini, Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps, Scand. J. Stat., 36 (2009), 270–296. https://doi.org/10.1111/j.1467-9469.2008.00622.x doi: 10.1111/j.1467-9469.2008.00622.x
[44]	S. Kwok, A consistent and robust test for autocorrelated jump occurrences, J. Financ. Econom., 22 (2024), 157–186. https://doi.org/10.1093/jjfinec/nbac031 doi: 10.1093/jjfinec/nbac031
[45]	G. Li, C. Zhang, On the relationship between conditional jump intensity and diffusive volatility, J. Empir. Finance, 37 (2016), 196–213. https://doi.org/10.1016/j.jempfin.2016.04.001 doi: 10.1016/j.jempfin.2016.04.001
[46]	Y. J. Dong, Y. K. Tse, On estimating market microstructure noise variance, Econ. Lett., 150 (2017), 59–62. https://doi.org/10.1016/j.econlet.2016.11.009 doi: 10.1016/j.econlet.2016.11.009
[47]	S. S. Lee, P. A. Mykland, Jumps in equilibrium prices and market microstructure noise, J. Econom., 168 (2012), 396–406. https://doi.org/10.1016/j.jeconom.2012.03.001 doi: 10.1016/j.jeconom.2012.03.001
[48]	L. Zhang, Estimating covariation: Epps effect, microstructure noise, J. Econom., 160 (2011), 33–47. https://doi.org/10.1016/j.jeconom.2010.03.012 doi: 10.1016/j.jeconom.2010.03.012
[49]	L. Zhang, P. A. Mykland, Y. Aït-Sahalia, A tale of two time scales: determining integrated volatility with noisy high-frequency data, J. Am. Stat. Assoc., 100 (2005), 1394–1411.
[50]	J. C. Zhuang, Estimation, diagnostics, and extensions of nonparametric Hawkes processes with kernel functions, Jpn. J. Stat. Data Sci., 3 (2020), 391–412. https://doi.org/10.1007/s42081-019-00060-0 doi: 10.1007/s42081-019-00060-0
[51]	J. Gatheral, T. Jaisson, M. Rosenbaum, Volatility is rough, Quant. Finance, 18 (2018), 933–949. https://doi.org/10.1080/14697688.2017.1393551
[52]	M. Bibinger, M. Sonntag, Testing for jumps in processes with integral fractional part and jump-robust inference on the hurst exponent, Econom. Stat., 2025. https://doi.org/10.1016/j.ecosta.2025.03.004
[53]	M. Podolskij, M. Vetter, Bipower-type estimation in a noisy diffusion setting, Stoch. Proc. Appl., 119 (2009), 2803–2831. https://doi.org/10.1016/j.spa.2009.02.006 doi: 10.1016/j.spa.2009.02.006
[54]	L. V. Dung, T. C. Son, On the rate of convergence in the central limit theorem for arrays of random vectors, Stat. Probab. Lett., 158 (2020), 108671. https://doi.org/10.1016/j.spl.2019.108671 doi: 10.1016/j.spl.2019.108671

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