Juuso Töyli

Characteristic times in stock market indices

In this study we analyze the Standard and Poors 500 index data of the New York Stock Exchange for more than 32 years. Using a simple random walk model we demonstrate that the proper variable to look at is the logarithmic return. In the statistical analysis we have done fittings to the Levy distribution using either the index data as such or pre-processing it with ARCH, GARCH or IGARCH methods, which tend to remove the time-dependent variance. For short times the truncated Levy distribution is found to fit the data quite well. Since this is not a stable distribution, the scaling behavior observed for short times should brake down for longer times. We demonstrate that the characteristic time where this cross-over starts is of the order of one day.

Time-independent models of asset returns revisited

In this study we investigate various well-known time-independent models of asset returns being simple normal distribution, Student t-distribution, Levy, truncated Levy, general stable distribution, mixed diffusion jump, and compound normal distribution. For this we use Standard and Poors 500 index data of the New York Stock Exchange, Helsinki Stock Exchange index data describing a small volatile market, and artificial data. The results indicate that all models, excluding the simple normal distribution, are, at least, quite reasonable descriptions of the data. Furthermore, the use of differences instead of logarithmic returns tends to make the data looking visually more Levy-type distributed than it is. This phenomenon is especially evident in the artificial data that has been generated by an inflated random walk process.

Models of asset returns: changes of pattern from high to low event frequency

In this paper we have analysed asset returns of the New York Stock Exchange and the Helsinki Stock Exchange using various time-independent models such as normal, general stable, truncated Levy, mixed diffusion-jump, compound normal, Student t distribution and power exponential distribution and the time-dependent GARCH(1, 1) model with Gaussian and Student t distributed innovations. In order to study changes of pattern at different event horizons, as well as changes of pattern over time for a given event horizon, we have analysed high-frequency or short-horizon intraday returns up from 20 s intervals to a full trading day, medium-frequency or medium-horizon daily returns and low-frequency or long-horizon returns with holding period up to 30 days. As for changes of pattern over time, we found that for medium-frequency returns there are relatively long periods of business-as-usual when the return-generating process is well-described by a normal distribution. We also found periods of ferment, when the volatility grows and complex time dependences tend to emerge, but the known time dependences cannot explain the variability of the distribution. Such changes of pattern are also observed for high-frequency or short-horizon returns, with the exception that the return-generating process never becomes normal. It also turned out that the time dependence of the distribution shape is far more prominent at high frequencies or short horizons than the time dependence of the variance. For long-horizon or low-frequency returns, the distribution is found to converge towards a normal distribution with the time dependences vanishing after a few days.

ON THE SHAPE OF ASSET RETURN DISTRIBUTION

ABSTRACT In this paper we investigate the shape of the asset return distribution using all shares index of Helsinki Stock Exchange and Standard & Poors 500 index of New York Stock Exchange. In both cases the power exponential distribution is used to model the shape of the return distribution and the inference is cross-checked with Student t-distribution. The possible dependencies in the data are studied by pre-whitening it with GARCH techniques and Cochrane-Orcutt correction. The parameters of the power exponential distribution are estimated with Bayesian approach and with maximum likelihood method. Kolmogorov-Smirnov test, for which the critical values are defined with simulation, is used to test the significance of power exponential fit. The results indicate that there are significant variations in the shape of the distribution over time, which cannot be explained by known time-dependencies. This finding suggests that the shape of distribution might be time-dependent or at least it is non-stationary. In contrast, differences in the shape of the distribution between weekdays are not observed but the tendency towards normality is observed, when the time interval is increased.

Break-down of Scaling and Convergence to Gaussian Distribution in Stock Market Data

We analyze the Standard and Poors 500 index data of the New York Stock Exchange for more than 32 years. It was suggested earlier that the high frequency data are well described by a truncated Levy distribution and scaling with respect to the sampling time differences was found. The truncated character of the distribution implies that scaling must break down and that the distribution ultimately converges to a Gaussian. We show by comparing Levy and Gaussian fits that the characteristic time of the break-down of scaling is of the order of few days. The analysis of the dependence of the kurtosis on the time differences shows that this is much shorter than the time needed for the convergence to the Gaussian being of the order of months.

Plan for profit and achieve profit: lessons learnt from a business management simulation

In this paper we introduce one new promising learning environment based on a business management simulation, discuss briefly how it was built, and how it has been used to cope with the business educational challenges. This learning environment has been used to train more than 1500 students and managers. In general, business education is research oriented, the knowledge base given to young professionals is abstract, and students lack the necessary skills to translate abstract knowledge to efficient practice. It is difficult to teach practical skills with traditional teaching methods. The learning environment was developed to answer to this challenge.