## The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc. | |

Dissertation for the degree of Doctor of Science in Technology to be presented
with due permission of the Department of Electrical Engineering, for public
examination and debate in Auditorium S4 at Helsinki University of Technology
(Espoo, Finland) on the
20^{th} of December, 2002, at 12 noon.

Overview in PDF format (ISBN 951-22-6254-1) [1738 KB]

Dissertation is also available in print (ISBN 951-22-6253-3)

In this thesis we discuss the asset returns. Our work was initially motivated by Mantegna's and Stanley's results (1995) that put forward the stable distribution as a model of asset returns and demonstrated the scaling property that seemed to be present in the data. Nevertheless, that work raised several questions both theoretically interesting and practically challenging such as: what is the effect of measurement quantity on the inference concerning asset returns, which are the proper quantities to look at, does the scaling exists, and if so what are its limits, are there characteristic times on asset returns, how the possible time-dependent variance affects the inference?

When exploring these issues, we became concerned about the possible variability and time-dependency of the shape of the asset return distribution in addition to the time-dependent variance. Thus, we speculated that the possible variability of the shape could have been one reason behind the contradictory results concerning the best fitting model of asset returns. Furthermore, the anomalies related to the mean returns and standard deviations led us to raise the question whether the shape of the asset return distribution shows similar kinds of anomalies. Finally, since we noticed that much debate has been had about the time-independent and time-dependent models but there has been relative few studies where these various models have been compared using the same datasets, especially high frequency data, this has been done is thesis and quite surprising results were obtained.

In order to address these questions we studied Standard & Poor's 500 daily index data of the New York Stock Exchange from more than 32 years. In addition, we used a high frequency data recorded on about 20 seconds time-interval over three years time period. For comparison reasons we also studied a small market, namely the Helsinki Stock Exchange all shares return index (HEX) over seven year period. Moreover, we used an artificial data to demonstrate some effects of measurement quantities.

Our results show that the proper variable to look at is the logarithmic return. Initially, for short time horizon or holding periods, the truncated Lévy distribution was found to fit the data quite well. Since this is not a stable distribution, the scaling behaviour observed for short times should break down for longer times. Thus, we demonstrated that the characteristic time of the break-down of scaling is of the order of few days. Furthermore, the analysis of convergence of the kurtosis showed that it takes place within few months.

When we investigated the various time-independent models of asset returns being simple normal distribution, Student t-distribution, Lévy, truncated Lévy, general stable distribution, mixed diffusion jump, power exponential distribution, and compound normal distribution, the results indicated that all models, excluding the simple normal distribution, are, at least, quite reasonable descriptions of the data. Surprisingly, however all other time-independent models except the normal distribution usually outperform the time-dependent GARCH(1,1) model for time horizons shorter than about four hours although the fine grained data evidently includes time dependencies. However, the GARCH-model is on average the best model for daily returns, and especially for periods of time when the return generating process cannot be assumed normal.

In the case of the variability of the shape of asset return distribution, our results showed that the shape of the distribution does not vary from one weekday to another. However, substantial deviations over time were observed while there are also temporal periods when it is reasonable to assume the return generating process as normal. The known time-dependencies were found inadequate in explaining these deviations. Furthermore, the results indicate that the return distribution approaches normal when the time interval used to calculate returns is increased.

Finally, our findings led us to raise three questions for future research to address. First, we speculated that there seem to be periods of "business as usual" when the return generating process is well described by the normal distribution. However, for some reason - for example, external shock, bubble formation - every now and then also periods of ferment emerge. These periods are characterised by higher volatility and increased time-dependencies. Second, the poor performance of GARCH(1,1) model on high frequencies lead us to question whether the assumption of GARCH that returns are normally distributed with time-dependent parameters is reasonable and whether it should be substituted with some other model where also the shape is allowed to vary over time. Such a model could, at least in theory, capture the business as usual periods and periods of ferment. Third, although we were surprised by the poor performance of GARCH(1,1) on high frequencies, we were reluctant to generalise this finding before a more detailed analysis. However, if this behaviour is typical for financial data, it could also be a source for further insight to the return generating process.

This thesis consists of an overview and of the following 6 publications:

- Kullman, L. - Töyli, J. - Kertesz, J. - Kanto, A. - Kaski, K. (1999) Characteristic times in stock market indices. Physica A: Statistical Mechanics and its Applications. Vol. 269:1, 98-110.
- Kullman, L. - Töyli, J. - Kertesz, J. - Kanto, A. - Kaski, K. (2000) Break-Down of Scaling and Convergence to Gaussian. International Journal of Theoretical and Applied Finance. Vol. 3:3, 371-373.
- Gillemot, L. - Töyli, J. - Kertesz, J. - Kaski, K. (2000) Time independent models of asset returns revisited. Physica A: Statistical Mechanics and its Applications. Vol. 282:1-2, 304-324.
- Töyli, J. - Kaski, K. - Kanto, A. (2002) Variability of Hex Return Distribution. The Finnish Journal of Business Economics. Vol. 51:1, 64-89. © 2002 LTA. By permission.
- Töyli, J. - Kaski, K. - Kanto, A. (2002) On the shape of asset return distribution. Communications in Statistics - Simulation and Computation. Vol. 31:4, 489-521.
- Töyli, J. - Sysi-Aho, M. - Kaski, K. (2002) Models of Asset Returns: Changes of Pattern from Tick by Tick to 30 Days Holding Period. Submitted for publication in Quantitative Finance.

**Keywords:**
asset returns, time dependency, modeling, distributions, scaling,
stock exchange, data analysis

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© 2002 Helsinki University of Technology

Last update 2011-05-26