Tuesday, 27 October 2020

Low volatility is past perfect

Three main subjects in this article: some explanations on historic stock market volatility and how it is derived, an overview of rallies and swoons both in the 21st and the 20th century and finally a discussion on the trend of volatility since the start of the 21st century.

Historic volatility 

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. 

The key example to studying historic volatility is the Dow Jones Industrial index. Data for a public series available on Yahoo Finance go back to Oct 1, 1928. Hence they contain all information on the 1929 stock market crash and the subsequent 1930's economic depression.

For this series, we have calculated all day-to-day variations at the close (Dvar) and made a statistical analysis on their distribution. Hereby you find the results:

Distribution analysis of daily variations

The UNIVARIATE Procedure

Variable: Dvar




Sum Weights




Sum Observations


Std Deviation








Uncorrected SS


Corrected SS


Coeff Variation


Std Error Mean


Some initial remarks: N is the number of day-to-day variations. Many of them evidently over a weekend or bridging a stock market holiday. The mean (0.0273%) is the average progression bringing the Dow Jones Industrial Average from 240 on Oct 1, 1928 to 28195 on Oct 19, 2020. The series now spans over 92 years. The 117.48 fold increase, requires a constant and regular daily advance of only 0.0206%. In this hypothetical case, volatility would remain zero. 

The standard deviation of 1.154% is the very long term daily historic volatility of the Dow Jones Industrial Average (DJIA). When multiplying the daily variance by the average number of trading days a year (252) we obtain a mean yearly variance (variances are additive if observations are non correlated). That also is equivalent to multiplying the historic daily volatility with sqrt(252) to obtain the 18.32% mean annualized volatility of the DJIA.

The negative skewness (-0.206) implies that large swoons are slightly more probable than large rallies. The excess kurtosis of 21.13 is a clear indication that the distribution is not Gaussian.

Basic Statistical Measures





Std Deviation












Interquartile Range



Basic Confidence Limits Assuming Normality



95% Confidence Limits





Std Deviation









Tests for Location: Mu0=0



p Value

Student's t



Pr > |t|





Pr >= |M|


Signed Rank



Pr >= |S|



Some comments on the below graph:

Small progressions (intervals centered 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 in the graph, but all were taken on board in calculations and tables.

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 far more frequent than suggested by the normal distribution.

This is something to keep in mind: the Black&Scholes formula for option valuations has been mathematically derived assuming a normal distribution function: a Gaussian random walk of variations. Consequently the B&S model seems built on rather shaky ground. Therefore a former article "Risk mispricing" still is a good read.

Distribution Graph 

Fig 1: Distribution of day-to-day percentage variations of the DJIA since Oct 1, 1928.
The interval [-12%, 12%] only leaves aside 6 observations.

Parameters for the fitted Normal Distribution of Dvar

Parameters for Normal Distribution







Std Dev




Goodness-of-Fit Tests for Normal Distribution



p Value




Pr > D


Cramer-von Mises



Pr > W-Sq





Pr > A-Sq



SAS System ('SASApp', X64_ES08R2) on 26OCT2020 at 3:05 PM

The final table includes a number of tests which all are rejecting the hypothesis of a normal (gaussian) distribution (with a less than 1% doubt for the Kolmogorov Smirnov test and less than 0.5% for both others). The Anderson-Darling test has its emphasis on data points in the tails of the distribution. Hence a distribution with too many 'outliers' will more readily be identified as non-gaussian. 

Extreme observations: rallies and swoons

In order not to blur or overload the picture, I have sectioned the observations into those of the 21st century, followed by those of the 20th century.

Fig 2: Top-20 major rallies (left) and swoons (right) from the 21st century with the date
when the occurred and the DJIA close.

Major swoons now are those of the outbreak of the Covid-19 pandemic in Europe and the US, eclipsing those of the 2008 financial crisis, which used to be so prominent before. The 7.13% decline upon reopening stock markets on 17 Sep 2001 after the 9/11 terrorist attacks are an 'isolated' event. Note the 'absence' of the popping of the tech bubble in 2002 to March 2003 among the swoons. The DJIA has been less impacted: all occurrences now are outside the 21st century top 20.

Major rallies occur in tandem with major swoons. Bear market 'relief' rallies indeed also are typical for periods with extreme volatility.

Fig 3: Top-20 major rallies (left) and swoons (right) from the 20th century with the date

when the occurred and the DJIA close.

Observations of the 20th century still are overloaded with bear market rallies and swoons going back to the Oct 1929 stock market crash and the 1930's depression. However the largest single day swoon in the 20th century will remain that of the Oct 19, 1987 stock market crash.

Graphing volatility

Volatility usually is calculated from the standard deviations on a limited number of past observations. There is no strict rule, but the longer you make your 'filter', the more reliable values are, however also the slower it reacts to changes of new observations taken on board. Therefore filters typically are from 20 to 25 observations, thus including data of about one month, give or take a few days.

For short series, shorter filters might be used, since the number of volatility observations is the total number minus the filter length. For our DJIA study, this of course is not an issue. A 24 days filter is used and volatility observations are subsequently annualized.

Fig. 4 : Long term annualized volatility (in percentage, red on the right axis) and
DJIA closing data in blue on the left axis.

Volatility discussion

The surge of US stock market volatility upon the outbreak of the Covid-19 pandemic has eclipsed that of the 2008 financial crisis. Yet if we go into more detail on last few years, the days of low stock market volatility ended early 2018 when US FED funds rates were being hiked and credit markets started tightening. As primary dealers no longer were able to market federal US ten year bonds, the FED stepped in and reversed policy. Quantitative tightening to compensate for the QE from 2008 till 2014 was suddenly off the table. Yet volatility did not slide to its appeased 2017 level. The trade war with China with escalating tariffs, the Brexit saga in Europe against the back scene of climate disasters and refugee flows all caused minor surges of volatility. Nevertheless, the FED balance sheet expansion and the funds rate gradually reduced to near zero cranked up stock markets. Very high valuations are what also fueled the sudden surge of volatility as the Covid pandemic hit the markets.

The rapid recovery of stock markets after the pandemic is the result of credit expansion and exploding sovereign deficits, not of a general recovery of corporate earnings. Both the second wave of corona infections and the risk of a disorderly 'changing of the guard' upon contested US presidential elections are major reasons for concern. Low stock market volatility now is past perfect and it doesn't look as if volatility will soon subside.

Earlier on

The dot-com bubble spooked the markets in 2000, with the 9/11 terrorist attacks causing another blow and surge in volatility. The unwinding of the tech bubble caused repetitive minor surges in volatility on the DJA (while Nasdaq suffered its major setback). Volatility was to appease during subsequent years as economic growth resumed (2004-2006).

Yet low interest rates also caused real estate prices to surge, with loose mortgage credit policies and repacking real estate debt in CDO's (securization) which started to be a growing concern in 2007. Volatility started rising well before the 2008 financial crisis hit the markets.

Again volatility was rising prior to the crash, as was the case more recently. The major difference being that the recent rise of volatility was caused a number of events not related to the cause of the major crash, whereas in 2007-2008 a succession of aberrations set the stage for the ultimate collapse.

After stock markets pulled themselves together in 2009, the sovereign debt crisis built up in Europe: remember the acronym 'PIGS' countries, for highly indebted EU member states (Portugal, Italy, Greece and Spain). The 'I' was later doubled to include Ireland after the 'Celtic tiger' lost its teeth with a major recession. The overextended construction boom and hot real-estated markets were aggravating the crises in Ireland and Spain. After 2012 days of low volatility returned.

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