Fluctuation patterns in high-frequency financial asset returns

We introduce a new method for quantifying pattern-based complex short-time correlations of a time series. Our correlation measure is 1 for a perfectly correlated and 0 for a random walk time series. When we apply this method to high-frequency time series data of the German DAX future, we find clear correlations on short time scales. In order to subtract trivial autocorrelation parts from the pattern conformity, we introduce a simple model for reproducing the antipersistent regime and use alternatively level 1 quotes. When we remove the pattern conformity of this stochastic process from the original data, remaining pattern-based correlations can be observed. Copyright c © EPLA, 2008 Introduction. – Often the assumption is made that price dynamics of financial markets obey random walk statistics. However, real financial time series show deviations from this assumption [1–5], like fat-tailed price increment distributions [6–8]. Scaling behavior, shorttime anti-correlated price changes and volatility clustering [9,10] are also well known and can be reproduced, e.g., by a statistical model of the continuous double auction [11,12] or by various agent-based models [13–21]. Both price formation processes and cross correlations [22,23] between different stocks and indices have been studied with the intention to optimize asset allocation and portfolios. It is also known that stock markets display a reversion tendency after large price movements [24,25]. The rise of hedge fund industry in recent years and their interest in taking advantage of short-time correlations also boosted the analysis of the market microstructure, which is the study of the process of exchanging assets under explicit trading rules [26], and which is naturally studied and modeled intensively by the financial community [27–31] in order to minimize order execution costs. In this letter, we study autocorrelations of financial market data in the anti-persistent short-time regime. For this purpose we analyze the randomness of financial markets employing specific conditional probability distribution functions, which reflect the main market response on given price impacts. According to common wisdom, the (a)E-mail: preis@uni-mainz.de anti-persistence on short time scales is due to the bid ask bounce. In order to account for this effect, we introduce a simple stochastic model, in which the price is the sum of a random walk part and a second part describing the bid ask bounce. We show that beyond the correlations which are due to the bid ask bounce there are correlations in the fluctuation patterns, which we will call “complex correlations” in the following. In order to identify such complex correlations, we introduce a new method for quantifying pattern-based correlations of a time series on short time scales. Reproduction of the scaling behavior on short time scales. – Scientific market modeling can only be based on price time series, which are the outcome of the trading decisions of the market participants comprising the “many particle system” of a financial market. The following analysis is based on historic price time series of the German DAX future contract (FDAX) traded at the European Exchange (EUREX), which is one of the world’s largest derivatives exchanges. The time series, which is displayed in the inset of fig. 1, contains 2709952 trades recorded from 2 January 2007 to 16 March 2007. A future contract is a contract to buy or sell a proposed underlying asset —in this case the German DAX index— at a specific date in the future at a specified price. The time series analysis of futures has the advantage that the prices are created by trading decisions alone. Contrarily, stock index data are derived from a weighted summation