×

Is Brownian motion necessary to model high-frequency data? (English) Zbl 1327.62118

Summary: This paper considers the problem of testing for the presence of a continuous part in a semimartingale sampled at high frequency. We provide two tests, one where the null hypothesis is that a continuous component is present, the other where the continuous component is absent, and the model is then driven by a pure jump process. When applied to high-frequency individual stock data, both tests point toward the need to include a continuous component in the model.

MSC:

62F12 Asymptotic properties of parametric estimators
62M05 Markov processes: estimation; hidden Markov models
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions . Dover, New York. · Zbl 0543.33001
[2] Aït-Sahalia, Y. and Jacod, J. (2008). Fisher’s information for discretely sampled Lévy processes. Econometrica 76 727-761. · Zbl 1144.62070
[3] Aït-Sahalia, Y. and Jacod, J. (2008). Testing whether jumps have finite or infinite activity. Technical report, Princeton Univ. and Univ. de Paris-6. · Zbl 1234.62117
[4] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency financial data. Ann. Statist. 37 2202-2244. · Zbl 1173.62060
[5] Aït-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184-222. · Zbl 1155.62057
[6] Ball, C. A. and Torous, W. N. (1983). A simplified jump process for common stock returns. Journal of Financial and Quantitative Analysis 18 53-65.
[7] Bates, D. S. (1991). The crash of ’87: Was it expected? The evidence from options markets. Journal of Finance 46 1009-1044.
[8] Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. Journal of Business 75 305-332.
[9] Carr, P. and Wu, L. (2003). The finite moment log stable process and option pricing. Journal of Finance 58 753-777.
[10] Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1 281-299. · Zbl 0836.62107
[11] Jacod, J. (2007). Statistics and high frequency data: SEMSTAT Seminar. Technical report, Univ. de Paris-6. · Zbl 1185.60031
[12] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517-559. · Zbl 1142.60022
[13] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes , 2nd ed. Springer, New York. · Zbl 1018.60002
[14] Madan, D. B. and Seneta, E. (1990). The Variance Gamma (V.G.) model for share market returns. Journal of Business 63 511-524.
[15] Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV 19-47.
[16] Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125-144. · Zbl 1131.91344
[17] Tauchen, G. T. and Todorov, V. (2009). Activity signature functions for high-frequency data analysis. Technical report, Duke Univ. · Zbl 1431.62483
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.