Researchers: Juuso Töyli, Janos Kertesz, Marko Sysi-Aho, Tommi Nykopp, Kimmo Kaski, and Antti Kanto (Helsinki School of Economics and Business Administration)
Asset returns of stock markets have traditionally been modelled 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 13 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 Standard & Poor 500 index data of New York Stock Exchange. The value indicates normal distribution and larger values more peaked and long tailed distributions.
Figure 13: On the left: Time-dependent variance extracted from HEX (Helsinki). On the right: A posterior distribution of power exponential distribution with shape parameter and location parameter estimated from S&P 500 data (New York). |
The research has concentrated on the understanding of the return generating process, by using the S&P 500 and HEX indexes as sources of data. 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 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 consequtive data points is changed. The main aims are first to understand the behaviour of returns and the return generating process,second to understand the evolution of the generating process when the time interval changes, third to explore whether the shape of the distribution is time-dependent, and 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.
An additional perspective is to consider these stock market indexes as very noisy time-series, in which from the information theory point of view the noise does not include information or it is misinformation. In order to get better insight to the underlying return generating process random noise should be reduced or the misinfornation should be reduced and thus increase 'pure' or meaningful information of the stock market. This reduction of noise can be done effectively by information theory based 'Normalized Maximum Likelihood - based Wavelet Denoising' method developed by J. Rissanen. This approach is applied to S&P 500 data with different sampling frequencies.