On quantiles of the temporal aggregation of a stable moving average process and their applications

Abstract

A stochastic volatility model is proposed for the daily log returns of a financial asset based on conditional log quantile differences, assuming the availability of high frequency intraday log returns. Calculation of the conditional log quantile differences is performed with the assumption that the intraday log returns follow a stable moving average process. The use of conditional log quantile difference in the proposed model, rather than conditional variance in standard models, offers an increase in flexibility, with the potential for a different dependency structure at different parts of the conditional distribution. The proposed model makes use of high frequency intraday log returns which are generally neglected in standard models. Formulae for the calculation of the conditional log quantile differences are provided and a method for their estimation is described The proposed model was applied to the ASX200 index from 2009 and 2010. References

• R. J. Adler, R. E. Feldman, and C. Gallagher. Analysing stable time series. In R. J. Adler, R. E. Feldman, and M. S. Taqqu, editors, A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkhauser, 1998.
• A. W. Barker. Estimation of stable distribution parameters from a dependent sample. Technical report, 2014. http://arxiv.org/abs/1405.0374
• A. W. Barker. On the log quantile difference of the temporal aggregation of a stable moving average process. Technical report, 2014. http://arxiv.org/abs/1404.6875
• T. Bollerslev. Generalized autoregressive conditional heteroskedasticity. J. Econometrics, 31:307–327, 1986. doi:10.1016/0304-4076(86)90063-1
• R. Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quant. Financ., 1:223–236, 2001. doi:10.1080/713665670
• R. A. Davis. Gauss–Newton and M-estimation for ARMA processes with infinite variance. Stoch. Proc. Appl., 63:75–95, 1996. doi:10.1016/0304-4149(96)00063-4
• R. A. Davis and S. Resnick. Limit theory for the sample covariance and correlation functions of moving averages. Ann. Stat., 14:533–558, 1986. http://www.jstor.org/stable/2241234
• Y. Dominicy, S. Hormann, H. Ogata, and D. Veredas. On sample marginal quantiles for stationary processes. Stat. Prob. Lett., 83:28–36, 2013. doi:10.1016/j.spl.2012.07.016
• J. McCulloch. Simple consistent estimators of stable distribution parameters. Commun. Stat. Simulat., 15:1109–1136, 1986. doi:10.1080/03610918608812563
• J. P. Nolan. Parameterizations and modes of stable distributions. Stat. Prob. Lett., 38:187–195, 1998. doi:10.1016/S0167-7152(98)00010-8
• S. J. Taylor. Modelling stochastic volatility: A review and comparative study. Math. Financ., 4:183–204, 1994. doi:10.1111/j.1467-9965.1994.tb00057.x
• R. S. Tsay. Analysis of Financial Time Series. Wiley, 3rd edition, 2010.
• K. Zhu and S. Ling. The global weighted LAD estimators for finite/infinite variance ARMA(p,q) models. Economet. Theor., 28:1065–1086, 2012. doi:10.1017/S0266466612000059