Physica A: Statistical Mechanics and its Applications
Statistical properties of short term price trends in high frequency stock market data
Introduction
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], [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], [8], since the price returns form a truncated Lévy distribution [9], [10], [11]. As a result of this observation many non-Gaussian models were introduced [2], [3], [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], [15].
The issue of market memory was also considered by many authors (see references in Refs. [2], [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 is larger than . In this paper we investigated this effect for high frequency stock market data.
Section snippets
Empirical data
Let us consider short term price trends for high frequency stock market data. By short term uptrend/downtrend we mean such a sequence of prices that a price is larger/smaller than the preceding one (see below for a more precise definition).
First, having a time series , which is in our case a history of a stock price or a market index, we build a series of variables in the following way:
- •
if ,
- •
if ,
- •
if .
A positive value of the variable means that at
A phenomenological model of correlated market prices
In real markets variables and are correlated, although this correlations decay very fast. Let stand for a conditional probability , which is independent of . For processes where autocorrelations are present we can write a generalization of Eq. (1): for and .
Let us see that the result (2) is equivalent to (1) if for any there is . A key issue is to model in order to describe the characteristics of a
Measuring trends in volume and volatility times
In previous chapters we presented an analysis of price trends measured in real and transaction times. The WIG20 index is published every 15 s, and for this index the 15 s data are the most frequent possible. Investigating these data we naturally used the real time with a 15 s interval, in which the sequence “” means that the index did not decrease for one minute.
For the stocks of companies tick-by-tick data are accessible, thus a transaction time is a natural measure of a time
Conclusions
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 has similar behaviour to
Acknowledgments
This work as a project of the COST Action P10 “Physics of Risk” was partially supported by the Polish Ministry of Science and Higher Education, Grant No. 134/E-365/SPB/COST/KN/DWM 105/2005-2007.
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