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:: Non-Gaussian Behaviour of Markets ::

Most tradition approaches to market price fluctuation is based on geometric Brownian motion, thus, suggesting Gaussian returns. The return between time t and t+1 is defined to be log(P(t+1)/P(t)), where P(t) is the price at time t.

To test this hypothesis, we can take any data set of price quotations of an equilty or foreign exchange currencies, and look at the distributions of price changes over a certain period.  Below we have taken daily closing BP share prices from 1988 to 2001. (This data set is available on the quantnotes dataset page and is provided courtesy of Bloomberg).

There is a steady rise in the share prices from 1988 to 1997, thereafter, a highly volatile period from 1997 onwards.

By plotting the distribution of the price returns (below) we have fitted the corresponding Gaussian fit in blue using the mean and standard deviation estimators from the data set.
    

In red we have fitted a non-Gaussian distribution which improves on the Gaussian.   The shape of the distribution has a sharp peak and more weight to the far ends of the tails (fat-tails).  The Gaussian distribution underestimated the probability small price returns, and overestimates the mid-range values.  Extreme price jumps are hugely underestimated by the Gaussian.
      
Functions like the one in red are term leptokurtic, and appear to characterise the pricing dynamics far better then a Gaussian distribution on shorter timescales. At times, Levy distributions can be used to describe the statistics of these price returns.  Unfortunately, Levy-distributions have an infinite second moment (i.e. an infinite standard deviation) which is unphysical for price fluctuations.  For this reason, the Levy distribution distibution is sometimes truncated.
    
So what would happen if we take larger time lags for price changes, like every 10 days, as below.   The distribution peak is lowered, and there is a larger range of price returns.  The leptokurtic fit is no longer as good and the corresponded Gaussian becomes more reasonable.  
      
      Note: we have not adjusted the leptokurtic fit as we have done so for the Gaussian to account for the change in standard deviation and mean.  This is purley to emphasis the transition that is seen as the time lag increases.
            

In summary, we have seen a change in the distribution of price returns that evolves according to the relative timescales we are considering.  There is a gradual transition from a leptokurtic to a Gaussian distribution as expected according to the Central Limit Theorem (CLT).  Certainly, this is somewhat different to random walk approximation.  It is a problem that has bugged many in the financial community for years.  What statistics of price flutuations does one assume over various timescales?  The random walk is by far the most easiest to work with and agrees well over periods of years, especially for equities.  Other distributions are mathematically more complex to use for practical purposes.  Part of fat-tails can be attributed to external influences on the market dynamics (e.g. news or natural disasters).
The problem is not unique to financial time series - other physical systems exhibit this property.  Some interesting analogies exist between financial time series and other physical systems that exhibit leptokurtosis. Hydrodynamic turbulence is one of them.

Written by Alessio Farhadi

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