# Exponential Volatility Decay, from Kim Zussman

March 14, 2010 |

Since we're back to school lately, went back and brushed up to check whether stock market volatility spike-decay follows an exponential-decay law, of the general form:

dV/dt = -L*V(t), where

V=volatility
dV/dt is change in volatility with change in time
-L is the "decay constant"  for V

This can be rearranged as dV/V(t) = -L*dt

And integrated:

ln(V) = -L + C  (c= integration constant)

Which is a linear equation that can be used to evaluate the decay constant (slope). Using DJIA daily closes 1929-2009, every (non-overlapping) 10-day period I calculated stdev for the prior 10D. Then I identified volatility peaks >=3%, finding these:

date     max D10

10/20/08    0.059
09/25/01    0.030
10/29/97    0.030
10/28/87    0.089
09/11/39    0.031
10/30/29    0.074

For each spike, checked 10D stdev for 36 subsequent periods, and plotted ln(10D stdev) for each date. For the log-transformed six historical volatility spikes, used linear regressions to fit lines to the data: Independent variable = date. Dependent = ln(10D stdev). The first regression is for the recent spike ca 2008:

Regression Analysis: ln2008 versus date2008

The regression equation is ln2008 = 143 - 0.00369 date2008

Predictor            Coef       SE Coef      T      P
Constant            143.29      16.35   8.77  0.000
date2008   -0.0036902  0.0004087  -9.03  0.000

S = 0.369075   R-Sq = 70.6%   R-Sq(adj) = 69.7%

Note the slope (-0.0037) is highly significant, and negative; following
exponential decay (see plot above).  Here are the results for the other
years:

Regression Analysis: ln2001 versus date2001

The regression equation is
ln2001 = - 30.0 + 0.000685 date2001

Predictor           Coef    SE Coef      T      P
Constant           -29.95      16.16  -1.85  0.072
date2001   0.0006853  0.0004319   1.59  0.122

S = 0.390306   R-Sq = 6.9%   R-Sq(adj) = 4.2%

///////////

Regression Analysis: ln1997 versus date1997

The regression equation is
ln1997 = - 14.6 + 0.000282 date1997

Predictor             Coef    SE Coef      T      P
Constant           -14.65      15.38  -0.95  0.348
date1997   0.0002819  0.0004274   0.66  0.514

S = 0.387066   R-Sq = 1.3%   R-Sq(adj) = 0.0%

/////////////

Regression Analysis: ln1987 versus date1987

The regression equation is
ln1987 = 80.1 - 0.00262 date1987

Predictor            Coef    SE Coef      T      P
Constant            80.11      14.09   5.69  0.000
date1987   -0.0026168  0.0004357  -6.01  0.000

S = 0.392181   R-Sq = 51.5%   R-Sq(adj) = 50.1%

//////

Regression Analysis: ln1939 versus date1939

The regression equation is
ln1939 = - 2.10 - 0.000184 date1939

Predictor        Coef    SE Coef      T      P
Constant            -2.101      9.765  -0.22  0.831
date1939   -0.0001840  0.0006618  -0.28  0.783

S = 0.602352   R-Sq = 0.2%   R-Sq(adj) = 0.0%

//////////

Regression Analysis: ln1929 versus date1929

The regression equation is
ln1929 = - 3.24 - 0.000074 date1929

Predictor        Coef    SE Coef      T      P
Constant            -3.237      6.889  -0.47  0.641
date1929   -0.0000739  0.0006175  -0.12  0.905

S = 0.562338   R-Sq = 0.0%   R-Sq(adj) = 0.0%

Of the six historically large volatility spikes, only 2008 and 1987 followed exponential decay. This doesn't seem to be a result of the size of the spike, as 1929 was bigger than 2008 and smaller than 1987. To the extent that volatility proxies fear, Is the current volatility decay in some way similar to 1987, and different from the others? Unlike 1987, 1929 spike was the beginning of a long period of economic turbulence. In 1997 the market was already volatile, and went on to become more so. 2001 featured 911, followed by further stock declines through early 2003. In 2008, the banking system teetered on the edge of what now looks to be a fake precipice - with the only real consequences being higher debt/gdp, less home ownership, and higher taxes.

I am clueless on the science of volatility (ie approaching it with any quantitative proficiency). as a market participant though, i have found that there are three kinds of volatility.

one is VIX volatility. this seems to be negatively correlated to liquidity of risk assets, which sounds obvious to the point of tautology as i'm typing it out, but ive been surprised how insensitive it is to other metrics i've though should matter, e.g., estimate revision momentum (for example: once a sector's estimate revisions are X standard deviations above the historical average, particularly as correlations have risen to recent highs in an upward trending market, shouldn't this matter? apparently animal spirits among option market makers is a lot more important.) and so on.

two is pnl volatility, which for me is very positively correlated to volatility of volatility (in either direction, but especially volatility that trends down for surprising lengths of time), much more so than volatility alone. as a firm we very carefully watch the PnL volatility of the 30 separate books of pairs, particularly during a rising market, as a sort of jerry rigged predictor of market volatility. when it gets to intolerable levels during a rising market that sometimes signals that a trend is about to reverse, although it's not consistent enough to be reliable as a frequent go-to indicator, I believe it did work well for the firm in several very critical situations (in 2007, 2008 and 2009) when other indicators were not working.

another is "long term volatility" which i think is best captured by the seasonally adjusted price of gold. e.g., "how big will the nuclear explosion be when the world's imbalances eventually, inevitably?! readjust". the VIX seems to totally ignore that.

`SELECT * FROM wp_comments WHERE comment_post_ID = '4521' AND comment_approved = '1' ORDER BY comment_date`