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## careful construction of the market graph

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

1:58 PM (0 minutes ago)

 to David, bcc: aimee
Dear Professor Donoho,

I have refined the construction of the market graph structure using stochastic volatility correlations.  Let v be the noisy volatility defined by v_t = log( return^2 + eps) where eps=1e-6 is a small constant to avoid infinities.  The following R function is the exact process. Attached also is the univariate distribution of correlations in case a more appropriate thresholding exists.  The data vectors have more than 13000 days and it is not best to consider the market graph to be constant over time.  However, this is a reasonable hypothesis till we have results on the actual volatility modeling.  The graph limit and parameter estimation of Wolfe et al can be used on this constant graph till we have reason to modify it.
C<-cor( v, v)
n<-dim(X)[1]
# remove the diagonal
B<- X – diag(X)
# determine the soft thresholding by taking off the noise
noise<-3*sd(B)
A<- abs(B) – noise
A[A<0]<-0
A<-sign(B)*A
# set the remaining nonzero values to 1
A[abs(A)>0]<-1
A
}
This method keeps a dense set of edges of possible n*(n-1)/2 of a complete graph, 75% which suggests the extreme correlation of volatilities even with 1900 stocks.
> sum(A)/(1899*1897/2)
[1] 0.7591333
In other words the MARKET GRAPH is pretty dense.  There are two interesting consequences of this density.  First, it means that it is a good idea to seek models of volatility by a small set of parameters and that such searches are likely to succeed.  Second, it reinforces my earlier claims that it is meaningless to consider the volatility of an individual asset in isolation of the entire volatility ‘snapshot’ for the world.  Finally, this graph does represent a good space variable for partial differential equations approaches or with recent study of FRACTIONAL PDE which are more likely to fit the data.

Next we consider the graph eigenvalue basis representation of this vector time  series and we expect to establish that the spectrum of a typical time series has POWER LAW decay. This will then ensure that it is pointless to consider purely Markovian models in finance altogether for this dataset is pretty simple composed of daily closing prices.  But this is definitely not a simple problem without denoising for this is what the first graph Laplace eigenvector series for volatility looks like (see attached).

So denoising properly is key to being able to understand the underlying dynamics.
Attachments area
Preview attachment Screenshot from 2015-06-04 13:54:50.png

Screenshot from 2015-06-04 13:54:50.png