Tuesday, 16 November 2010

Volatility persists

Historic volatility is calculated using a time series of index (or stock price) close values. It is defined as the standard deviation on the series of percentage day-to-day variations. When considering all day-to-day variations, we obtain one single historic volatility, which - while illustrative - raises a mainly academic interest.

Figure 1:      Historic volatility: The columns represent the percentage fraction of variations of the Eurostoxx-50 blue chip index in half percentage point intervals. The red curve is the normal distribution fitted. The central value ‘0’ contains the variations between -0.25% and +0.25%. (click to enlarge)
Small progressions (intervals centred around +0.5% and 1.0%) are more frequent than small declines. This is no longer true for larger progressions, balancing the frequency of larger declines. Moreover, index swoons of over 4% even outnumber the rallies of that size category. As such the distribution is slightly skewed to the left. Note: the few observations outside the interval shown have been left aside.
The red curve (normal Gaussian function) gives but a poor fit to the observed frequency column plot of day-to-day variations. For the geeks: the distribution is leptokurtic. What does this mean? Tiny variations are more frequent than the normal (Gaussian) distribution suggests. Small to average variations are less frequent and – most importantly – you find fatter tails: excessive variations are more frequent than suggested by the normal distribution.

This is something to keep in mind: the Black&Scholes formula for option valuations has a normal distribution function at its very heart. Therefore, its premise - a Gaussian random walk of variations - allowing the mathematical derivations made in the B&S model, seems built on rather shaky ground.

Distribution analysis of: Day to Day variation

The UNIVARIATE Procedure
Variable: Day2Day

Basic Statistical Measures
Std Deviation

Interquartile Range

Basic Confidence Limits Assuming Normality
95% Confidence Limits
Std Deviation
Figure 2: Distribution-analysis of the day-to-day variations of the Eurostoxx-50
Historic volatility is 1.332% and the mean day-to-day variation is very low: 0.0277%.  Pure statistical analysis doesn't even certify the mean to be positive, since the lower 95% confidence limit is negative!
With a normal (Gaussian) distribution, the measure of historic volatility (1.332%) suggests that variations in excess of 4% should not occur more than once every 375 working days. Sharp corrections and manic rallies are more frequent, hence the shortcoming of the normal distribution.
Volatility time series for the Eurostoxx-50
Usually volatility is calculated as the standard deviation of a progressing series of day-to-day variations. (Are you still there?) Longer series make the volatility measure more reliable, but they also tend to be less sensitive for a volatility change. Throughout the data discussed volatility is calculated over 25 working days.

Figure 3: Historic series of Eurostoxx-50-fixings (left axis) and the volatility "Vol-E50" (right-axis) - Click to enlarge.

It is obvious that volatility peaks during a stock market crisis, while also corrections show a smaller peak. Market crises typically show a succession of catastrophic swoons, intermitted by single day manic relief rallies, leading to the epic volatility levels observed in October 2008.
During quiet bull market volatility fades away to a 50 basis points complacency level. Remarkable is that every major stock market crisis: both the 2000 techno-bust and the 2008 bankers implosion have been preceded by a series of successive higher minima of the volatility graph (Vol E50). Especially the techno-bust has been preceded by a prolonged period of increased volatility, with successive volatility peaks and higher troughs in between. “Irrational exhuberance” used to be Alan Greespan’s favourite phrasing, those days.
In 2007, successive higher volatility peaks and higher troughs are again building up to the October 2008 crisis with a volatility level not seen before in the nearly quarter century of historic Eurostoxx-50 data.

Volatility seems to persist ever since, with a 3% peak during the June 2010 downturn, a volatility level comparable to the LTCM crisis in 1998 or even the 9/11 twin tower terrorist attack. While below 1% by early November 2010, we still haven’t yet seen the 50 basis points complacency level of a nice and steady bull market. New fierce downturns seem more imminent than a sustained recovery.

Volatility timeseries for the SP500 
Conclusions aren’t any different when looking what happened across the Atlantic: index level and volatility of the broad SP500 market index have been plotted since the beginning of 1990: over almost 21 years. Volatility peaks are observed on the same days, showing that the world stock markets of developed nations have been quite intertwinned throughout the past decades.

We observe the same rising level of the troughs between successive volatility peaks in the period preceding a major market crisis.

Fig 4:   The broad SP500 market index (blue, left axis) and the volatility level derived (red on the right axis). The level of the troughs starts rising in the period of “irrational exhuberance” by the end of the 20th century, preceding the collapse of the techno-bubble over 2000-2002. Volatility rises significantly over 2007 and into 2008 preceding the implosion of the financial sector. (click to enlarge)

Observations on volatility:

1)  Volatility values calculated on day-to-day variations are by definition daily volatility measures: σd. They relate to longer period volatility values (monthly or yearly) by multiplication by √T. Thereby T is expressed in working days of the stock exchange. For a monthly volatility T is typically chosen = 21 For a yearly volatility it is chosen 252. This conversion implicitly relies on the ‘random walk’ hypothesis, whereby daily variations are non-correlated and normally distributed. This same hypothesis is used by the Black-Scholes method for option valuation. The conversion formula is therefore 'flawed' in the same way. I came across suggestions to use a non-integer exponent of 0.55 instead of 0.5 (square root). This is however purely based on observation and curve fitting. The volatility value you would plug into an option value calculator is the annual volatility. The option value calculated as such is Black&Scholes model based.
2)   Generally real option prices are higher than those B&S model based values. When taking option prices for granted, a volatility value can then reversely be calculated. This is then called an "implied volatility". It's no more than a fancy name for a 'curve fitting parameter' imho.
3)   Implied volatility values are often higher for an option that is far ITM or far OTM, whereas it is lowest for an at the money option. Another fancy name for this phenomenon is "the volatility smile". It must be caused by the leptokurtic characteristics of the distribution of day-to-day variations, though this statement is hard to prove mathematically.
4)   Historic volatility and Statistics on the Eurostoxx-50 were calculated on the full range of data available: starting on 31-Dec-1986, the series runs to 11-Nov-2010 (date of database extraction) which adds up to 6090 day-to-day variations. SP-500 data run between 2-Jan-1990 and 15-Nov-2010 over 5263 observation days.

The preliminary of this article (though less detailed) has been published in Dutch on this blog.
This index chosen is the BEL20 large cap index. Data run from Jan 1991 till Nov 2009.

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