Incorporating Realized Quarticity into a Realized Stochastic Volatility Model

  • Didit Budi Nugroho
  • Takayuki MorimotoEmail author


This study proposes an extension of the realized stochastic volatility model by incorporating realized quarticity RQ into the volatility process. We employ an efficient Riemann Manifold Hamiltonian Monte Carlo method in a Markov Chain Monte Carlo algorithm to estimate parameters that could not be sampled directly. We investigate the empirical performance of the proposed model using data for six equity indices and 24 individual stocks listed on the Tokyo Stock Exchange. Parameter estimates and two Bayesian model selection criteria reveal evidence supporting RQ-based models that are driven by the maximum value of RQ data. That model consistently outperforms benchmark realized stochastic volatility models in capturing spikes in volatility caused by large RQ values. Including such stylised facts as the asymmetric effect of returns-volatility and heavy tailed returns, our results reveal that the proposed models exhibit a weaker correlation between stock returns and volatility and heavier tails in equity returns.


Asymmetric effect Markov Chain Monte Carlo Realized quarticity Stochastic volatility Student-t distribution 



We are especially grateful to Professor Hiroki Masuda of Kyushu University for his profound insights and numerous valuable suggestion. This study is partly supported by JSPS KAKENHI Grant Number 18K01554.


  1. Abanto-Valle, C. A., Migon, H. S., & Lachos, V. H. (2012). Stochastic volatility in mean models with heavy-tailed distributions. Brazilian Journal of Probability and Statistics, 26(4), 402–422.CrossRefGoogle Scholar
  2. Andersen, T. G., & Bollerslev, T. (1997). Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance, 4, 115–158.CrossRefGoogle Scholar
  3. Andersen, T. G., Bollerslev, T., & Diebold, F. X. (2007). Roughing it up: Including jump components in the measurement, modeling and forecasting of return volatility. Review of Economics and Statistics, 89(4), 701–720.CrossRefGoogle Scholar
  4. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2001). The distribution of realized exchange rate volatility. Journal of the American Statistical Association, 96(453), 42–55.CrossRefGoogle Scholar
  5. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica, 71(2), 579–625.CrossRefGoogle Scholar
  6. Andersen, T. G., Dobrev, D., & Schaumburg, E. (2012). Jump-robust volatility estimation using nearest neighbor truncation. Journal of Econometrics, 169(1), 75–93.CrossRefGoogle Scholar
  7. Asai, M., Chang, C.-L., & McAleer, M. (2017). Realized stochastic volatility with general asymmetry and long memory. Journal of Econometrics, 199(2), 202–212.CrossRefGoogle Scholar
  8. Bandi, F. M., & Russell, J. R. (2008). Microstructure noise, realized variance, and optimal sampling. Review of Economic Studies, 75(2), 339–369.CrossRefGoogle Scholar
  9. Barndorff-Nielsen, O. E., Hansen, P., Lunde, A., & Shephard, N. (2008). Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica, 76(6), 1481–1536.CrossRefGoogle Scholar
  10. Barndorff-Nielsen, O. E., & Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society Series B, 64(2), 253–280.CrossRefGoogle Scholar
  11. Barndorff-Nielsen, O. E., & Shephard, N. (2005). How accurate is the asymptotic approximation to the distribution of realised volatility? In D. W. K. Andrews & J. H. Stock (Eds.), In identification and inference for econometric models: A Festschrift for Thomas J. Rothenberg (pp. 306–331). New York: Cambridge University Press.Google Scholar
  12. Bekierman, J., & Gribisch, B. (2016). Estimating stochastic volatility models using realized measures. Studies in Nonlinear Dynamics & Econometrics, 20(3), 279–300.CrossRefGoogle Scholar
  13. Bekierman, J., & Manner, H. (2018). Forecasting realized variance measures using time-varying coefficient models. International Journal of Forecasting, 34, 276–287.CrossRefGoogle Scholar
  14. Berg, A., Meyer, R., & Yu, J. (2004). Deviance information criterion for comparing stochastic volatility models. Journal of Business and Economic Statistics, 22(1), 107–120.CrossRefGoogle Scholar
  15. Bollerslev, T., Patton, A. J., & Quaedvlieg, R. (2016). Exploiting the errors: A simple approach for improved volatility forecasting. Journal of Econometrics, 192(1), 1–18.CrossRefGoogle Scholar
  16. Broto, C., & Ruiz, E. (2004). Estimation methods for stochastic volatility models: A survey. Journal of Economic Surveys, 18, 613–649.CrossRefGoogle Scholar
  17. Carnero, M. A., Pena, D., & Ruiz, E. (2004). Persistence and kurtosis in GARCH and stochastic volatility models. Journal of Financial Econometrics, 2(2), 319–342.CrossRefGoogle Scholar
  18. Chen, M. H., & Shao, Q. M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69–92.Google Scholar
  19. Chevallier, J., & Sévi, B. (2011). On the realized volatility of the ECX \({\text{ CO }}_2\) emissions 2008 futures contract: Distribution, dynamics and forecasting. Annals of Finance, 7, 1–29.CrossRefGoogle Scholar
  20. Corsi, F. (2004). A simple long memory model of realized volatility. Workingpaper, University of Southern Switzerland.Google Scholar
  21. Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7(2), 174–196.CrossRefGoogle Scholar
  22. Dobrev, D., & Szerszen, P. (2010). The information content of high-frequency data for estimating equity return models and forecasting risk. Working Paper, Finance and Economics Discussion Series.Google Scholar
  23. Gelfand, A. E., & Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. Journal of the Royal Statistical Society Series B, 56(3), 501–514.Google Scholar
  24. Hansen, P. R., & Huang, Z. (2016). Exponential GARCH modeling with realized measures of volatility. Journal of Business & Economic Statistics, 34, 269–287.CrossRefGoogle Scholar
  25. Hansen, P. R., & Lunde, A. (2005). A realized variance for the whole day based on intermittent high-frequency data. Journal of Financial Econometrics, 3(4), 525–554.CrossRefGoogle Scholar
  26. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–795.CrossRefGoogle Scholar
  27. Kim, S., Shephard, N., & Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models. In N. Shephard (Ed.), Stochastic Volatility: Selected Readings. New York: Oxford University Press.Google Scholar
  28. Koopman, S. J., & Scharth, M. (2013). The analysis of stochastic volatility in the presence of daily realized measures. Journal of Financial Econometrics, 11(1), 76–115.CrossRefGoogle Scholar
  29. Masuda, H., & Morimoto, T. (2012). Optimal weight for realized variance based on intermittent high-frequency data. Japanese Economic Review, 63(4), 497–527.CrossRefGoogle Scholar
  30. McAleer, M., & Medeiros, M. (2008). Realized volatility: A review. Econometric Reviews, 27(1–3), 10–45.CrossRefGoogle Scholar
  31. Nugroho, D. B., & Morimoto, T. (2014). Realized non-linear stochastic volatility models with asymmetric effects and generalized Student’s t-distributions. Journal of The Japan Statistical Society, 44(1), 83–118.CrossRefGoogle Scholar
  32. Nugroho, D. B., & Morimoto, T. (2015). Estimation of realized stochastic volatility models using Hamiltonian Monte Carlo-based methods. Computational Statistics, 30(2), 491–516.CrossRefGoogle Scholar
  33. Nugroho, D. B., & Morimoto, T. (2016). Box–Cox realized asymmetric stochastic volatility models with generalized Student’s \(t\)-error distributions. Journal of Applied Statistics, 43(10), 1906–1927.CrossRefGoogle Scholar
  34. Nugroho, D. B., Mahatma, T., & Pratomo, Y. (2018). Modeling of stochastic volatility to validate IDR anchor currency. Gadjah Mada International Journal of Business, 20(2), 165–185.CrossRefGoogle Scholar
  35. Takahashi, M., Omori, Y., & Watanabe, T. (2009). Estimating stochastic volatility models using daily returns and realized volatility simultaneously. Computational Statistics & Data Analisys, 53(6), 2404–2426.CrossRefGoogle Scholar
  36. Takahashi, M., Omori, Y., & Watanabe, T. (2014). Volatility and quantile forecasts by realized stochastic volatility models with generalized hyperbolic distribution. Working Paper Series CIRJE-F-921, CIRJE, Faculty of Economics, University of Tokyo.Google Scholar
  37. Taylor, S. J. (1982). Financial returns modelled by the product of two stochastic processes–a study of the daily sugar prices 1961–75. In N. Shephard (Ed.), Stochastic Volatility: Selected Readings (pp. 60–82). New York: Oxford University Press.Google Scholar
  38. Watanabe, T., & Asai, M. (2001). Stochastic volatility models with heavy-tailed distributions: A Bayesian analysis. IMES discussion paper series no. 2001-E-17. Institute for Monetary and Economic Studies, Bank of Japan.Google Scholar
  39. Yu, J. (2002). Forecasting volatility in the New Zealand stock market. Applied Financial Economics, 12, 193–202.CrossRefGoogle Scholar
  40. Yu, J. (2005). On leverage in a stochastic volatility model. Journal of Econometrics, 127, 165–178.CrossRefGoogle Scholar
  41. Yu, J., Yang, Z., & Zhang, X. (2006). A class of nonlinear stochastic volatility models and its implications for pricing currency options. Computational Statistics & Data Analisys, 51(4), 2218–2231.CrossRefGoogle Scholar
  42. Zhang, L., Mykland, P., & Ait-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association, 100(472), 1394–1411.CrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Study Center for Multidisciplinary Applied Research and Technology (SeMARTy)Satya Wacana Christian UniversitySalatigaIndonesia
  2. 2.Department of Mathematical SciencesKwansei Gakuin UniversitySandaJapan

Personalised recommendations