# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## Empirical Properties of Asset Returns: Stylized Facts

When thinking about empirical properties of asset returns, it’s always useful to have a structure in mind. One of the first published attempts to put structure to the empirical properties of asset returns was done by Paul Cootner. Since then we have learnt a lot more about the empirical properties of asset returns, thanks not only to the increasing sophistication of (and interest in) financial markets, but also in part to the increasingly sophisticated technology of statistical/econometric tools available compared to the 1960s.

Rama Cont in his 2000 paper, neatly summarizes a set of stylized facts (observed/assumed) common to time series properties of asset returns.

In one of our planned papers, we plan to use these stylized facts as a starting point to compare the time series properties of asset returns implied by popular stochastic volatility models. This post is an attempt to aide in starting to think about these stylized facts and how to implement them in Matlab.

Cont lists the following “statistical factscommon to a wide set of financial assets

1. Absence  of autocorrelations: Linear autocorrelations of asset returns are often insignficant except at very small time scales (high frequency data)

• This should be easy to check using MATLAB’s autocorr function

2. Heavy tails: The unconditional distribution of returns display a power-law tail

• This is new to us, but hopefully Gencay’s EVIM (Extreme Value Analysis in Matlab) toolbox should be some help here [we discuss this in a separate post]

3. Gain/loss asymmetry: Large downward movements are more frequent than equivalent upward ones

• This again shouldn’t be too difficult to check for, and we can do it in a variety of ways: just plot the histogram, look at the frequency of negative moves vs. positive moves, or the easiest – just measure statistical skewness

4. Aggregational Gaussianity: As one increases the time scales over which the returns are being measured, the probability distribution of returns tends to a “normal”/Gaussian distribution.

• Same as above

5. Intermittency: Presence of “irregular bursts” in time series at both fine/coarse time scales

• A naive (though not too problematic) way would be to see how often within a certain time scale returns “jump” by a certain standard deviation. A more econometric way to do would be to fit a time series that explicitly allows for jump estimation. Alternatively, we could also try using wavelets (though prima facie not sure how)
• Cont suggests using Holder’s exponent to characterize irregularity in the data – “in order for a model to represent adequately the intermittent character of price variations, the local regularity of the sample paths should try to reproduce those of empirically observed price trajectories”

6. Volatility Clustering: Different measures of volatility (returns squared/absolute returns)  display a positive correlation over several days – indicating the fact that high volatility events “come together”

• Same as 1

7. Conditional heavy tails: Even after correcting for clustering (using a GARCH type model, for example), the residual time series still exhibits heavy tails

• MATLAB has a GARCH toolbox so for this guess we just need to fit a GARCH model to the time series of asset returns and then studying the correlation in absolute residuals

8. Slow decay of autocorrelation in asset returns: As a function of time lag, effect as measured in6 using absolute returns decays slowly – interpreted as a sign of long range dependence

• Essentially should just involve regressing correlation over time scales

9. Leverage Effect: Most measures of volatility and asset returns are negatively correlated with asset returns

• One can do this in various ways. Two ways I can think are: a) look at the ‘spot skew’ from the options data; b) show that EGARCH gives a better fit than a simple GARCH

10. Volume/volatility correlation: Trading volume is correlated with all measures of volatility

• Irrelevant as far as we are concerned – we just don’t have volumes data

11. Asymmetry in time scales: Coarse-grained measures of volatility predict fine scales volatility better than the other way around

• Not sure if we are going to do this, but maybe regressions of these kinds can be done. Need a closer look at the paper for this