Dec

7

Victor and Laurel have suggested that a fruitful area for market research may lie in replicating the methods of Brahe and Kepler. Brahe scrupulously gathered very precise data though years of observations. It was left to Kepler, his student, to develop the first model. Kepler first identified planetary orbits as elliptical.

Suppose we have two planetary bodies with periods P1 and P2 respectively. A quick review of Kepler's Laws reminds us that his third law is as follows:

P1^2 / P2^2 = R1^3 / R2^3

where R1 and R2 are the semi major axes of the two bodies. It is interesting to note that there are no linear terms in the above relationship. It can be read as the ratio of the squares of the periods are equal to the ratio of the cubes of the axes.

In the markets we know that the Efficient Market Hypothesis tells us that the market price change today should have no linear correlation with the price change tomorrow. Empirically this seems to be true most of the time for most markets. However a strict interpretation of EMH says nothing about the existence of non-linear relationships.

In particular when we evaluate the squares of changes we find they are significantly correlated. The same holds for the cubes at similar lags. It is left as an exercise for the reader to calculate the magnitude and direction of such correlations. So at first blush there may be an application for Kepler's third law in the markets.

In order to see if there is any similar Keplerian relationship in daily price series the data from the table on page 121 of Education of a Speculator hardback was studied. Using the midpoints of the classes in the table the model used only the squares and cubes of change to predict the next days performance. It turns out that the fit is statistically significant. Notably there is no linear term in the model. Checking whether a linear term would help, the data showed that it would not be helpful. Although the regression model was statistically significant it was based on out of date data and would have to be redone with current data.

Michael Cook remarks:

Kepler was not a student of Brahe; he came to Brahe's observatory because Brahe had good data, continuous night by night observations of the planets. Kepler was desperate to prove that the orbits of the planets were circles, because the circle is the perfect shape, consistent with the beauty of the divine Mind. He decided to work on Mars because it seemed to be closest. After much work he realized the ellipse was a better fit. His comment: "I set out to show that the universe was based on the eternal harmony of the spheres. Instead I showed that it rests on a carthill of dung [the ellipse]."

The other beautiful law of Kepler's is that the planets sweep out equal areas in equal times.
 
It is also significant that all of his laws can be deduced mathematically from the inverse square law of gravitation.

Adam Robinson replies:

As I'm sure Dr. Cook realizes, the point was that the law of gravitation can be deduced from Kepler's laws, as indeed Robert Hooke (whose insights into force and inverse square relationships were at least contemporaneous with Newton's) was able to do.  Newton's genius (in that regard, there were many instances of course) was in showing the equivalence of the acceleration of a falling object with the acceleration of an object in orbit.

Newton, in other words, gave the relationships a theoretic underpinning (until then Hooke's insights, along with Kepler's, were mere "curve fitting," in the literal sense of the phrase!), just as Einstein did, since numerous scientists at the time (Poincare for one) had come to similar conclusions (e.g., the Lorentz contraction), but lacked any overarching theory to explain why such phenomena had to occur.

Dr. Cook's quotation of Kepler also reveals the extent to which aesthetics can hinder the progress of theory as much as promote it.

Michael Cook responds:

Actually, I am not aware of any derivation of the inverse square law from Kepler's laws. I believe Hooke claimed to derive Kepler's laws from an inverse square law, which resulted in Newton's publishing his proof of the result. Hooke never published an actual proof — it's hard to do without calculus. Feynmann has a paper in which he does so, which I don't think he would have published it if it were already in the literature. 

It is incorrect to say the law of gravitation can be deduced from Kepler's laws — Kepler's laws are descriptive, and don't by themselves imply any causal mechanism.

Adam Robinson replies:

I refer Dr. Cook to the letters between Hooke and Newton; there was much controversy between the two about who had which insights, when. Hooke's insight was more of a conjecture, not a formal "derivation" as such. Not surprisingly, of course, since Hooke's inverse square law with springs contains a surprising analogue with gravitation.

Kim Zussman writes:

Jim SimonsA recent Bloomberg article on Jim Simons of RenTech mentions sunspots and markets, so along with Kepler's dung [see Dr. Cook's remarks above] this must explain the beauty of markets.

Recall that sunspots (which have been observed and recorded since well before Galileo) are magnetic storms on the sun, which appear dark in contrast to the photosphere because (though they are hot) they are relatively cooler.  And to the extent that there may be related effects on solar wind (solar ions flowing past the earth), radiation levels, and earth's ionosphere, and radio/satellite communications, here is a study.

Monthly average sunspot count (American, of course) 1944-2007 is available from the National Geophysical Data Center:

Regression of SP500 monthly index return vs. monthly avg sunspot count (1950-Oct 07) shows almost significant negative correlation (P=0.07):

Regression Analysis: SP CHG versus SPOT AV

The regression equation is
SP CHG = 0.0110 - 0.000052 SPOT AV

Predictor         Coef     SE Coef      T      P
Constant      0.010961    0.002534   4.33  0.000
SPOT AV      -0.0000518   0.000029  -1.79  0.074

S = 0.0405916   R-Sq = 0.5%   R-Sq(adj) = 0.3%

Analysis of Variance

Source           DF        SS        MS     F      P
Regression        1  0.005288  0.005288  3.21  0.074
Residual Error  691  1.138548  0.001648
Total           692  1.143836

Here is a plot of monthly avg sunspots vs date, which clearly shows the 11 year solar cycle. Note that we now near a minimum (good for stocks), and regardless of Fed actions relative to the housing market, explains the recent 5 year bull market (OK the last sunspot maximum was Sept 2001, so the prediction was off by about 1.5 yr).

Eric Falkenstein remarks:

One of the keys of finance is the implication that arbitrage implies that pricing is linear in 'risk', or whatever is priced. Otherwise, you could generate arbitrage by buying bulk and selling little bits, or vice versa. It is intriguing to think that there are nonlinear relations in markets, but these necessarily imply profits, so, to the degree they exist, they must not be too obvious (please email me the exceptions!).


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