Though the empirical analysis of asset price time series has been performed for more than half a century, only the development of computerized trading in 1980s allowed to record enough data for robust statistical analysis. Recent two decades of evolution of IT infrastructure opened new horizons for empirical finance by bringing huge amount of high-frequency data, numerical tools and computational power for analysis. Nowadays every large exchange record terabytes of high-frequency data at every trading session.
The study of these new financial datasets results in a number of empirical observations quantified with robust statistical methods. Interestingly, some of these statistical laws were found to be common across many types of sufficiently liquid instruments on many markets. These common statistical laws, discovered independently by many researchers, were called “stylized facts” of financial time series (Cont 2001; Bouchaud and Potters 2000; Lux 2009). Here we present a nonexhaustive list of most important of “stylized facts”:
1. Absence of linear autocorrelation in assets returns except short intra-day time scales, where effects of market microstructure plays substantial role. Absence of linear autocorrelation is perfectly described by the naive model of Bachelier (1900), though this is almost the only “stylized fact” that this model could reproduce. For this reason “stylized facts” could be viewed as a collection of facts that differ real price dynamics from the random walk. Despite its simplicity, absence of linear autocorrelation (which could be stated in other words as absence of linear predictability) of financial returns plays extremely important role in financial modeling. This observation was embedded in the so- called no arbitrage hypothesis and Efficient Market Hypothesis (EMH) (Fama 1970, 1991), that in a very broad sense claim impossibility of obtaining excess returns (more than a risk-free rate) without being exposed to risk.
2. Long memory in volatility. Despite the absence of linear autocorrelation in signed price returns, autocorrelation function of absolute (or squared) returns decays very slowly and are statistically significant even on scales of hundreds days for intra-day returns.
3. Heavy tails in probability distribution of asset returns. The Bachelier’s random walk model assume probability density function (pdf) of iid increments (returns) to be Normal. However empirical analysis of real financial time series at many time scales show that pdf of asset returns is very skewed, having narrow peak and tails decaying as a power law with exponent у in the range 2 < у < 4 for intraday and daily time scales. Such “heavy tails” of asset returns distribution accounts for the presence of extreme events (large positive or negative returns) in real time series in contrast to the idealized Gaussian random walk model.
4. Aggregational Gaussianity. The distribution function of the returns is not the same at different time scales and exponent у of the tail of pdf depends in fact on the scale over which returns are calculated and increase with increase of this time scale. With moving from intraday returns towards weekly or monthly returns the distribution slowly converges towards Gaussian distribution and for quarterly or annual returns one typically can not reject the null hypothesis of normal distribution. In a way this “stylized fact” is a direct consequence of the Central Limit Theorem and the fact that exponent у was never found smaller than 2, which ensures finite variance of returns.
5. Volatility clustering. Presence of long memory in volatility and heavy tails of returns merged in an interesting observation that of sufficient irregularity of returns time series. Its typical pattern has periods of high volatility, which are followed by periods of low volatility, and vice versa. In other words, volatility bursts tend to group into clusters.
6. Multifractal properties. Above properties (heavy tails, absence of autocorrelation, long-range memory in volatility) are observed at various time scales, implying scale invariance of financial time series. More specifically, financial time series are found to exhibit so-called multifractal properties (we discuss it in details in Sect. 4) (Muzy et al. 2000; Arneodo et al. 1998b; Calvet and Fisher 2002; Liu et al. 2008).
Leverage effect. One important observation of the financial time series is absence of time-reversal symmetry. In other words, statistical properties of time series in direct time and reversed time are different. Leverage effect describes particular aspect of time-reversal asymmetry in terms of correlation function between returns and volatility
which is negative for x >= 0 and decay to zero with x and almost zero for x < 0 (Bouchaud et al. 2001). In other words, past returns are negatively correlated with future volatility, but past values of volatility do not correlate with future signed returns, satisfying the absence of arbitrage hypothesis.
8. Gain-loss asymmetry. Leverage effect is tightly linked with another breaking of symmetry in behavior of asset prices. In stocks, indices and their derivatives one could observe large drawdowns in prices but not equally large drawups (this is typically not true for exchange rates that are highly symmetrical in price movements). Moreover, it typically takes longer time to reach a gain of a certain value than a loss of a same value (Siven and Lins 2009).
9. Volume-volatility correlation. Most of the statistical properties of volatility could be observed in series of trading volume as well. Moreover, trading volume is correlated with all measures of volatility.
10. Extreme events. Finally, more than 400 years of history of financial markets have shown that bubbles and crashes are not exceptions and observed extremely often in different markets. Statistical properties of such extreme events are nontrivial and differ from statistical properties of the financial time series in normal regimes (Sornette 2003).