خواص آماری روندهای قیمت کوتاه مدت در داده های بازار سهام با فرکانس بالا

We investigated distributions of short term price trends for high frequency stock market data. A number of trends as a function of their lengths were measured. We found that such a distribution does not fit to the results following from an uncorrelated stochastic process. We proposed a simple model with a memory that gives a qualitative agreement with the real data.

Statistical analysis of stock prices is a rich source of information about the nature of financial markets. It was Louis Bachelier who used a stochastic approach to model financial time series for the first time [1]. Since that time the statistical analysis of stock prices has become a widely investigated area of interdisciplinary researches [2], [3], [4] and [5]. In 1973, Fischer Black and Myron Scholes published their famous work [6] where they presented a model for pricing European options. They assumed that a price of an asset can be described by a geometric Brownian motion. However, the behaviour of real markets differs from the Brownian property [7] and [8], since the price returns form a truncated Lévy distribution [9], [10] and [11]. As a result of this observation many non-Gaussian models were introduced [2], [3] and [4]. Another divergence from the Gaussian behaviour is an autocorrelation in financial systems. Empirical studies show that the autocorrelation function of the stock market time series decays exponentially with a characteristic time of a few minutes, while the absolute values of the autocorrelation of prices decay more slowly, as a power law function, which leads to a volatility clustering [12], [13], [14] and [15]. The issue of market memory was also considered by many authors (see references in Refs. [2] and [3]). It was observed [16], that for certain time scales, a sequence of two positive price changes lead more frequently to a subsequent positive change than a sequence of mixed changes, i.e. the conditional probability P(+∖++)P(+∖++) is larger than P(+∖+−)P(+∖+−). In this paper we investigated this effect for high frequency stock market data.

We have investigated the short term price trends for high frequency stock market data. It turned out that the statistics for real markets is significantly different from the statistics of uncorrelated processes. Longer trends (of the order of several minutes) are much more frequent than they should be, if one used an uncorrelated model. The investigations have been repeated for trends measured in volume and volatility time. The distribution of trends in volume time N(τv)N(τv) has similar behaviour to the function N(l)N(l). We proposed a simple model that qualitatively captures the behaviour of the market. The model leads to a distribution of trend series N(l)N(l) that is similar to the distribution observed in market data. Our model produces also short range correlations. This behaviour is caused by the conditional probability of trend continuation that changes non-monotonically with a trend length. At the beginning of the trend, the probability of the trend continuation grows, then it hits the maximum and finally decreases. As a result, trends posses limited lengths.