×
Müller, Hans-Georg; Sen, Rituparna; Stadtmüller, Ulrich

Functional data analysis for volatility. (English) Zbl 1441.62817

J. Econom. 165, No. 2, 233-245 (2011).
Summary: We introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data-based recovery of the process from repeated observations. A study of its asymptotic properties, as the frequency of observed trades increases, is complemented by simulations and an application to the analysis of intra-day volatility patterns of the S&P 500 index. The proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility, through the application of functional regression and prediction techniques.

Cited in 14 Documents

MSC:

62P20 Applications of statistics to economics
62G07 Density estimation
62P05 Applications of statistics to actuarial sciences and financial mathematics
62M05 Markov processes: estimation; hidden Markov models
62H25 Factor analysis and principal components; correspondence analysis
62R10 Functional data analysis

Keywords:

diffusion model; functional principal component; functional regression; high frequency trading; market returns; prediction; volatility process; trajectories of volatility

Software:

fda (R)
PDF BibTeX XML Cite
\textit{H.-G. Müller} et al., J. Econom. 165, No. 2, 233--245 (2011; Zbl 1441.62817)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Adler, R.J., ()
[2] Aït-Sahalia, Y., Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 527-560, (1996) · Zbl 0844.62094
[3] Aït-Sahalia, Y.; Mykland, P.A., The effects of random and discrete sampling when estimating continuous-time diffusions, Econometrica, 71, 483-549, (2003) · Zbl 1142.60381
[4] Aït-Sahalia, Y.; Mykland, P.A.; Zhang, L., How often to sample a continuous-time process in the presence of market microstructure noise, Review of financial studies, 18, 351-416, (2005)
[5] Aït-Sahalia, Y.; Mykland, P.A.; Zhang, L., Ultra high frequency volatility estimation with dependent microstructure noise, Journal of econometrics, 160, 190-203, (2011) · Zbl 1441.62577
[6] Andersen, T.G.; Bollerslev, T., Intraday periodicity and volatility persistence in financial markets, Journal of empirical finance, 4, 115-158, (1997)
[7] Andersen, T.G; Bollerslev, T., Answering the skeptics: yes, standard volatility models do provide accurate forecasts, International economic review, 39, 885-905, (1998)
[8] Bandi, F.; Phillips, P., Fully nonparametric estimation of scalar diffusion models, Econometrica, 71, 241-283, (2003) · Zbl 1136.62365
[9] Bandi, F., Renò, R., 2008, Nonparametric stochastic volatility. Working Paper.
[10] Barndorff-Nielsen, O.E.; Graversen, S.E.; Jacod, J.; Shephard, N., Limit theorems for realised bipower variation in econometrics, Econometric theory, 22, 677-719, (2006) · Zbl 1125.62114
[11] Barndorff-Nielsen, O.E.; Shephard, N., Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of royal statistical society series B, 64, 253-280, (2002) · Zbl 1059.62107
[12] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-654, (1973) · Zbl 1092.91524
[13] Castro, P.E.; Lawton, W.H.; Sylvestre, E.A., Principal modes of variation for processes with continuous sample curves, Technometrics, 28, 329-337, (1986) · Zbl 0615.62074
[14] Dacorogna, M.; Gencay, R.; Muller, U.; Olsen, R. B; Pictet, O.V., An introduction to high-frequency finance, (2001), Academic Press San Diego, CA
[15] Eubank, R.; Thomas, W., Detecting heteroscedasticity in nonparametric regression, Journal of royal statistical society series B, 55, 145-155, (1993) · Zbl 0780.62033
[16] Fan, J., A selective overview of nonparametric methods in financial econometrics with discussion, Statistical science, 20, 317-357, (2005) · Zbl 1130.62364
[17] Fan, J.; Jiang, J.; Zhang, C.; Zhou, Z., Time-dependent diffusion models for term structure dynamics, Statistica sinica, 13, 965-992, (2003) · Zbl 1065.62177
[18] Fan, J.; Gijbels, I., Local polynomial modelling and its applications, (1996), Chapman and Hall London, UK · Zbl 0873.62037
[19] Fan, J.; Wang, Y., Multiscale jump and volatility analysis for high-frequency financial data, Journal of American statistical association, 102, 1349-1362, (2007) · Zbl 1332.62403
[20] Fan, J.; Wang, Y., Spot volatility estimation for high-frequency data, Statistics and its interface, 1, 279-288, (2008) · Zbl 1230.91192
[21] Fan, J.; Yao, Q.W., Efficient estimation of conditional variance functions in stochastic regression, Biometrika, 85, 645-660, (1998) · Zbl 0918.62065
[22] Fan, J.; Yao, Q.W., Nonlinear time series: nonparametric and parametric methods, (2003), Springer · Zbl 1014.62103
[23] Faraway, J.J., Regression analysis for a functional response, Technometrics, 39, 254-262, (1997) · Zbl 0891.62027
[24] Florens-Zmirou, D., On estimating the diffusion coefficient from discrete observations, Journal of applied probability, 30, 790-804, (1993) · Zbl 0796.62070
[25] Foster, D.; Nelson, D., Continuous record asymptotics for rolling sample variance estimators, Econometrica, 64, 139-174, (1996) · Zbl 0860.62101
[26] Genon-Catalot, V.; Laredo, C.; Picard, D., Non-parametric estimation of the diffusion coefficient by wavelet methods, Scandinavian journal of statistics, 19, 317-335, (1992) · Zbl 0776.62033
[27] Jacod, J.; Shiryaev, A., Limit theorems for stochastic processes, (2003), Springer New York · Zbl 1018.60002
[28] Kogure, A., Nonparametric prediction for the time-dependent volatility of the security price, Financial engineering of the Japanese market, 3, 1-22, (1996) · Zbl 1153.91527
[29] Kristensen, D., Nonparametric filtering of the realized spot volatility: a kernel-based approach, Econometric theory, 26, 60-93, (2010) · Zbl 1183.91189
[30] Malliavin, P.; Mancino, M., A Fourier transform method for non-parametric estimation of volatility, Annals of statistics, 37, 1983-2010, (2009) · Zbl 1168.62030
[31] Mancino, M.; Sanfelici, S., Robustness of Fourier estimator of integrated volatility in the presence of microstructure noise, Computational statistics and data analysis, 52, 2966-2989, (2008) · Zbl 1452.62780
[32] Malfait, N.; Ramsay, J.O., The historical functional linear model, Canadian journal of statistics, 31, 115-128, (2003) · Zbl 1039.62035
[33] Müller, H.G.; Chiou, J.M.; Leng, X., Inferring gene expression dynamics via functional regression analysis, BMC bioinformatics, 9, 60, (2008)
[34] Müller, H.G.; Stadtmüller, U.; Yao, F., Functional variance processes, Journal of American statistical association, 101, 1007-1018, (2006) · Zbl 1120.62327
[35] Ogawa, S., Sanfelici, S., 2008, An improved two-step regularization scheme for spot volatility estimation. Working paper.
[36] Ramsay, J.O.; Dalzell, C.J., Some tools for functional data analysis, Journal of royal statistical society series B, 53, 539-572, (1991) · Zbl 0800.62314
[37] Ramsay, J.O.; Ramsey, J.B., Functional data analysis of the dynamics of the monthly index of non durable goods production, Journal of econometrics, 107, 327-344, (2001) · Zbl 1051.62118
[38] Ramsay, J.O.; Silverman, B.W., Functional data analysis, (2005), Springer New York, NY · Zbl 0882.62002
[39] Renò, R., Nonparametric estimation of the diffusion coefficient of stochastic volatility models, Econometric theory, 24, 1174-1206, (2008) · Zbl 1284.62232
[40] Rice, J.A.; Silverman, B.W., Estimating the Mean and covariance structure nonparametrically when the data are curves, Journal of royal statistical society series B, 53, 233-243, (1991) · Zbl 0800.62214
[41] Speight, A.E.H.; McMillan, D.C.; Gwilym, O.A.P., Intra-day volatility components in FTSE-100 stock index futures, Journal of futures markets, 20, 425-444, (2000)
[42] Stanton, R., A nonparametric model of term structure dynamics and the market price of interest rate risk, Journal of finance, 52, 1973-2002, (1997)
[43] Wang, L.; Brown, L.D.; Cai, T.T; Levine, M., Effect of Mean on variance function estimation in nonparametric regression, Annals of statistics, 36, 646-664, (2008) · Zbl 1133.62033
[44] Yao, F.; Müller, H.G.; Clifford, A.J.; Dueker, S.R.; Follett, J.; Lin, Y.; Buchholz, B.; Vogel, J.S., Shrinkage estimation for functional principal component scores, with application to the population kinetics of plasma folate, Biometrics, 59, 676-685, (2003) · Zbl 1210.62076
[45] Yao, F.; Müller, H.G.; Wang, J.L., Functional data analysis for sparse longitudinal data, Journal of American statistical association, 100, 577-590, (2005) · Zbl 1117.62451
[46] Yao, F.; Müller, H.G.; Wang, J.L., Functional linear regression analysis for longitudinal data, Annals of statistics, 33, 2873-2903, (2005) · Zbl 1084.62096
[47] Yao, Q.W.; Tong, H., Nonparametric estimation of ratios of noise to signal in stochastic regression, Statistica sinica, 10, 751-770, (2000) · Zbl 0952.62040
[48] Zhang, L.; Mykland, P.; Aït-Sahalia, Y., A tale of two time scales: determining integrated volatility with noisy high-frequency data, Journal of American statistical association, 100, 1394-1411, (2005) · Zbl 1117.62461
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.