The stylized empirical facts of financial time series is a collection of empirical observations and inferences of statistical properties common across a wide range of instruments, markets and time periods.
The main references for this section are: Black (1976), Christie (1982), Cont (2001), Erb (1994), Goetzmann (2005), Ledoit et al (2003), Longin and Solnik (1995), McNeil et al (2005) and Mandelbrot (1963).
The stylized facts are potentially very useful in determining how we should model financial risk, as they are able to give us guidelines to which properties our
models should exhibit. Cont (2001) argues that in order to let the data speak for itself as much as possible the facts should be formulated as qualitative
assumptions, which statistical models then can be fitted to, rather than assume that the data belongs to any pre specified parametric family.
Much has been written on stylized facts, and the exact formulation of each stylized fact varies from author to author, and some authors include “facts” that others don’t. With that being said, most of the stylized facts are reoccurring in most of the literature on the topic. The list we present are based on what seems to be the most reoccurring facts, with formulations inspired by McNeil et al (2005) and Cont (2001).
(1) Linear autocorrelations of asset returns are often insignificant. The exception to this is typically for small intraday timescales (~ 20 minutes) for which market microstructure effects enter (Cont 2001). This stylized fact is often cited as support for weak market efficiency, as significant autocorrelations would imply that previous returns could be used to predict future returns (and thus
“statistical arbitrage”
(2) Volatility appears to cluster and vary over time, and in a somewhat predictable manner. It is observed that series of absolute or squared returns show profound serial correlation. A commonly used metric to measure volatility clustering is the autocorrelation function of the squared returns;
𝐴𝐶2 = 𝑐𝑜𝑟𝑟(|𝑟𝑡+𝜏,∆𝑡|2, |𝑟𝑡,∆𝑡|2) (6)
9 (3) Asymmetric relationship between gains and losses. One can typically observe large drawdowns in stock prices but not equally large upward
movements. As the mean return tends to be positive or close to zero, this implies a skewed distribution.
(4) Slow decay of autocorrelation in absolute returns. The autocorrelation function of absolute returns decays slowly, often modelled by a power law with exponent 𝛽 ∈ [0.2, 0.4] (Cont 2001). This can be interpreted as a sign of long-range dependence.
(5) Return series are leptokurtic, i.e. heavy-tailed. The unconditional distribution of returns has excess kurtosis relative to that off the normal distribution (> 3). This effect is often still present even after correcting returns (e.g. via GARCH-type models), but reduced compared to that of the unconditional distribution.
(6) Leverage effects. Most measures of volatility of an asset are negatively correlated with the returns of that asset, e.g. 𝐿𝜏 = 𝑐𝑜𝑟𝑟(|𝑟𝑡+𝜏,∆𝑡|2, 𝑟𝑡,∆𝑡)
start from a negative value and decays to zero, suggesting that negative returns leads to increased volatility.
Black (1976) suggested that this could be attributed to the fact that bad news drives down the stock price, increasing the debt to equity ratio (i.e. the leverage) and thus causing the stock to be more volatile(risky).
In addition to these stylized facts (that reach across different asset classes and financial instruments) a lot of research has also been done on the individual asset classes. Of particular relevance to this thesis is the research stream surrounding the dynamic nature of equity correlations. Goetzmann et al. found that
correlations between equity returns vary substantially over time, and peak during periods of highly integrated financial markets (as one would expect). Longin and Solnik found evidence for rejection of the hypothesis of constant correlations among international stock markets, while Ledoit et al. and Erb et al. show time-varying (dynamic) correlations tend to be higher during periods of recession. The latter observation is particularly interesting (or worrisome) as it would imply that if we model financial risk in “normal” or “boom” periods, our correlations would
10 be understated and should a recession come, our risk measures would be
understated in the time we needed them the most.
Even if stylized facts can be a useful tool, the gain in generality across financial instruments, markets and time do come at the cost of precision of the statements that can be made about asset returns (this of course holds true in general to statistical models). Nevertheless, these stylized facts present properties that are regarded very constraining for a model to exhibit, even as an ad hoc stochastic process (Cont 2001). A question which should be noted in this regard is whether a stylized fact is relevant for the economic task at hand. If deemed not, it should not be a constraint to the model we are seeking either.
3.1 The normal distribution, i.i.d. assumption and the stylized facts
Figure 2 - The S&P 500 vs the normal distribution
The inappropriateness of modeling the marginal distribution of asset returns with the normal distribution was pointed out as early as 1963 by B. Mandelbrot. The properties of the normal distribution simply doesn’t reconcile with the stylized facts.
We can characterize the needs for a parametric model to be able to successfully reproduce the observed empirical features with it having at least four parameters;
a location parameter (e.g. mean), a scale parameter (e.g. standard deviation), a parameter describing the tail decay and eventually an asymmetry parameter allowing different behavior in each of the tails. The normal distribution only meets two of these requirements.
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3.2 The random walk hypothesis
The insignificance of autocorrelations in return gave support for the random walk hypothesis of prices, where returns are considered independent random variables.
However, the lack of linear dependence doesn’t imply independence: one also have to consider nonlinear functions of return. As we know from stylized fact 2, there is nonlinear dependence (which is exactly what is drawn from in order to create GARCH models for one). Log prices are therefore not properly modelled by random walks. Traditional tools of signal processing such as ARMA modeling and autocovariance analysis, can’t distinguish between asset returns and white noise. This points out the need for nonlinear dependence measures (e.g. GARCH modeling) to properly measure the dependence of asset returns (Cont 2001).
3.3 Assuming elliptical distributions in general
Much has been written on the validity of assuming that financial asset returns follow a normal distribution. A highly related and interesting discussion is that of the validity of assuming financial returns follow elliptical distributions in general.
For instance Owen and Rabinovitch (1983) take the position that non-normal elliptical distributions such as the student t can be useful as it allows for
describing tail decay through the degrees of freedom parameter, despite that the asymmetry parameter is still lacking (While this paper was written before high impact statistical methods such as ARCH rose to popularity in finance, we are of the opinion that the arguments are still valid). Chicheportiche and Bouchaud (2012) argue that elliptical distributions might be a fair assumption when assets are highly correlated, but also argue that it is very unrealistic when correlations are low. From our point of view, the only real consensus seems to be that these assumptions need to be assessed case-by-case. We’d also like to note that while some form of asymmetry parameter probably is desirable for most financial asset classes, it is absolutely necessary when modeling a joint distribution including non-linear assets such as options, almost regardless of the underlying asset. These assets are not considered in the real data application of this thesis.
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