# Another Way to Look at Volatility, from Nigel Davies

March 20, 2008 |

Another way to look at changes in volatility (increase/decrease in price swings per unit time) is to view it as having a typical overall move in price but with the time scale expanding and contracting. This might offer some additional insights, for example point and figure charts by ignoring time also ignore volatility. So if you test these they'll produce a quite different set of patterns which are essentially 'volatility blind.'

## Sushil Kedia replies:

A filtration process such as Point & Figure is essentially based on defining one's tolerance for noise in the price series. The box size achieves that. Point & Figure is the only method in Technical Analysis to not plot the time axis. It is perhaps the only method of looking at prices which has a nature similar to the time/distance equivalence in Einsteinian physics. There perhaps is a case for finding a congruence in defining a unit of time for a particular security based on a certain standard movement in its price.

## Bill Rafter explains:

Point & Figure purports to separate "signal" from "noise" and discard the latter. Every time you run data through a filter, you eliminate both some signal and some noise.  The short-term data that you assume to be all noise contain substantial signal.  Thus, it is illogical to assume that you will improve results by discarding information.  Further, and most-importantly, we applied Point & Figure filtering to all of our trading methodologies and the degradation of results was universal.

## Phil McDonnell expands: The Grand Master poses an interesting question and Mr. Kedia wisely suggested an analogy to Einstein's relativity theory. There is much to be learned from these ideas.

Suppose we have two assets which substitute for each other in investment portfolios. For example they might be stocks (s) and bonds (b). Given that there is only a constant supply of funds (m) available for these two investments we can posit that their combined value would be equal to:

m ~= ( s^2 + b^2 )^.5

The above is essentially the equation of a circle of radius m. One possible flaw is that the market for s may be much smaller than the market for b. Thus a given fixed disinvestment from b might move s by considerably more. Our model would therefore no longer be a circle. Without loss of generality we can assume the fixed quantity m to be 1 (100% of all the money). Then we have:

1 = ( s^2 / a^2 + b^2 / d^2 )^.5

where a and d are two constants of proportionality which relate to how quickly the two markets move. Thus each market now has its own ease of movement parameter in this new elliptical model.

One of the key properties of the theory of relativity is that as one approaches the speed of light both time and space are distorted. In particular the Lorentz transformation governs this process and is given by:

gamma = 1 / ( 1 - v^2 / c^2 ) ^ .5

where v is the velocity of the spaceship and c is the speed of light. This represents the transformation in the x direction which we shall assume is the direction of acceleration. Referring back to the elliptical model formula above we see that the one dimensional form looks remarkably similar to the denominator of the Lorentz transform (gamma).

Qualitatively such a model would be consistent with the Davies/Kedia conjectures. Time would slow down as the market moved faster. Magnitude in the price direction would dilate as well as a function of velocity.

Having a theoretical model of the market is all very nice but unless the market follows it then it is useless. To test this a study was done of the above stock to bond relationship using SPY and TLT ETFs. Fitting the parameters a and d to past data one finds that the constants were 200 and 100 respectively. Then the fitted model was compared to the actual past history of SPY and TLT and found to provide very good agreement. Perhaps we may look at these constants as the speed of financial information (light) in the stock and bond medium respectively.

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