A glance at the spectrum of the Laplacian will convince you that for market graph networks, there’s no sharp drop from highest eigenvalues. Now this is the Laplace operator not the covariance of returns which are often treated as being approximable by some small number of eigenvectors of the covariance matrix of returns. We constructed our graph based not on returns defined as

r<-diff(log(prices))

but for

h<-log(r**2+0.00001)

which is the undenoised volatility in the model r=volatility*whitenoise, where we have for prices a N*(T+1) matrix. The heatmap of h is also attached, although it’s hard to interpret still. The Laplacian matrix containing the geometry is extremely helpful because then we can forget the graph and look at equations of type

d/dt Volatility = Laplacian*Volatility + F(Volatility) + fractionalNoise/whiteNoise

Now this is a pretty standard statistical model that will fit data not badly. Now the Laplacian spectrum in the graph shows that we can’t approximate this system easily by taking a subgraph without losing a lot of the determinants of volatility. So let’s check the total weight on edges just to make sure we’re right about the importance of these off-diagonal terms: the result is sum(diag(covariance))=0.073 while sum(covariance)-sum(diag(covariance)) is 3.46 which tells me that the off diagonals of the covariance gain a much stronger weight in volatility dynamics making the whole idea of volatility modeling on a single asset a silly exercise scientifically. Volatility is a single global beast and the graph topology is what is missing from being able to capture its law on a global market graph. Only in this scale can finanance even have any science. What science of automobiles comes from careful attention to the construction of bits and pieces of its engine? Even stochastic dynamical models can give us clues to parameters of phase transition. We could easily look at contageon phenomena and so on. The volatility storms cannot be managed — by anyone at all — unless this graph dynamics is quantified and real risks of the market under some control. Today people are so lost they are first assuming ridiculous Gaussian models of risk distribution as though we still lived in the age before computers and then they calculate tail probabilities for value-at-risk. This is so comical that if people’s lives were not destroyed by these bad practices we could use this to entertain children as the people who have lots of money and play their parlour games on chance to destroy lives of poor people mostly for fun while getting their sweethearts at the Fed keep their mansions and islands and their sex orgies.

2 Attachments

Preview attachment Screenshot from 2015-05-27 21:05:27.png

Screenshot from 2015-05-27 21:05:27.png

Preview attachment vol-heatmap.png

vol-heatmap.png

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