Researchers: Juuso Töyli, Laszlo Gillemot,
Janos Kertesz, Kimmo Kaski, and

Antti Kanto (Helsinki School of Economics and Business Administration)

Asset returns of stock markets have traditionally been modeled with a normal distribution. However, the statistical characteristics of empirical returns show longer tails than in normal distribution, the variances are auto-correlated though returns are not, large and small returns are clustered, and there are jumps and crashes. On the other hand, when the sampling interval is increased, the shape of the statistical distribution approaches normal distribution, i.e. monthly returns are regarded as normally distributed. These characteristics suggest that the return generating stochastic process is non-linear, time-dependent and complex. The left pane of figure 16 shows a time-dependent variance extracted from HEX (Helsinki Stock Exchange all shares daily index) data and the right pane a posterior distribution of power exponential distribution with shape parameter and location parameter estimated from S&P 500 index data of New York Stock Exchange. The value indicates normal distribution and larger values more peaked and long tailed distributions.

The research has so far concentrated on the understanding of the return generating process, by using the Standard & Poor 500 (New York) and HEX (Helsinki stock exchange) indexes as sources of data. Also a comprehensive toolbox has been developed to fit and simulate data according to the well-known time-independent models, which seem capable of capturing the long-term distribution but not yet the structure of the process. We are also studying the effect of different dependencies (linear and non-linear) on the shape of the statistical distribution and on the return generating process. Along with these, the possible biases resulting from different terms of measurements are looked at. In addition, we are studying the changes in the return generating process when the time interval between data point is changed. The main aims of the future research are as follows: First to understand the behaviour of returns and the return generating process. Second to understand the evolution of the generating process when time interval changes. Third to explore whether the shape of the distribution is time-dependent. Fourth to apply this understanding in developing adaptive methods for modeling of asset returns and stock indexes. These methods can be applied, e.g., in the financial risk management and possibly in developing asset pricing models.