The Problem of Time Arrow in Financial Time Series

According to the efficient market hypothesis, future movements of the market cannot be predicted. This introduces an intrinsic time asymmetry of the financial time series as there are no laws forbidding “predicting” past based on the current market fluctuations. This clear time asymmetry in the basic laws of finance raises a question which we shall be referring to as the problem of time arrow: are there any noticeable statistical differences between forward-in-time and reverse-in-time market data. Majority of the statistical methods used for financial time series are time-symmetric and hence, not usable for our purposes. The first method used in our study is the analysis of the length-distribution of periods with high variability. With this method, the markets are judged to have a high variability if the price movements difference from the local average exceeds a threshold. The time arrow enters the play via the local average which can be calculated over a retarded time window, centred time window, or advanced time window. The second method used here is borrowed from the turbulence studies: the odd order structure functions. These structure functions have been traditionally used to quantify the so-called small-scale anisotropy of temperature fields in a fully developed turbulence. Small-scale anisotropy means that if there is a global anisotropy, i.e. a global average temperature gradient, the anisotropy survives at the smallest Kolmogorov scales: the largest temperature jumps are taken in the direction of the global gradient. Our study shows that while typically, there is no noticeable time asymmetry in market data, under certain circumstances, there is indeed a statistically significant time asymmetry which might provide a key to predicting a forthcoming crisis.