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## Empirical Ito densities show that BM-fBM type stochastic models are incorrect for univariate volatility on 1900 stocks daily data

mport pandas as pd
import numpy as np

def diff(x):
v = np.zeros(x.shape)
dx=np.diff(x)
v[1:]=dx
return(v)

def ito_density(x):
v = diff(x**2-x[0]**2 – 2*np.cumsum(x*diff(x)))
return(v)

ito=np.zeros((13434,1899))
for i in range(1899):
ito[:,i]=ito_density(v[:,i])

import matplotlib.pyplot as plt
def itoplot(n):
p=plt.plot(ito[:,n])
plt.show()

from matplotlib.backends.backend_pdf import PdfPages

with PdfPages(‘itodensities.pdf’) as pdf:
for k in range(1899):
plt.figure()
plt.plot(ito[:,k])
plt.title(‘Ito density %d’%k)
pdf.savefig()
plt.close()

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

5:11 PM (2 hours ago)

 to David, bcc: aimee
Sir,

In order to make immediate progress with new but not very incisive (as in discovering phase transition mechanisms in empirical volatility) we consider a much easier problem that is easy to complete:

The problem is to determine the Ito density of empirical volatility series for 1900 stocks.  The solution is to consider the Ito formula generalized:

f(X_t) – f(X_0) = \int_0^t f(X_s) dX_s + 1/2 \int_0^t f”(X_s) phi(s) ds

and use the function f(x)=x^2 to determine the empirical phi(s) which is a power of s for fractional Brownian motions and 1 for Brownian motion.  >>> def diff(x):

…     v = np.zeros(x.shape)
…     dx=np.diff(x)
…     v[1:]=dx
…     return(v)
>>> v1=v[:,3]
>>>
>>> def ito_density(x):
…     v = diff(x**2-x[0]**2 – 2*cumsum(x*diff(x)))
…     return(v)
>>> ito_density(v1)
Traceback (most recent call last):
File “<stdin>”, line 1, in <module>
File “<stdin>”, line 2, in ito_density
NameError: global name ‘cumsum’ is not defined
>>> def ito_density(x):
…     v = diff(x**2-x[0]**2 – 2*np.cumsum(x*diff(x)))
…     return(v)
>>> ito_density(v1)
array([ 0.    , -1.    , -0.1849, …,     nan,     nan,     nan])
So here you can see except for some nan the code produces results for Ito density.  So a simple exercise is to tabulate the Ito densities and analyze these.  This approach will immediately knock out general fractional Brownian motions and still leave space for stochastic diffusion models subordinated to the Brownian motion.  These models are interesting because they do deviate a great deal from theoretical stochastic volatility models which assume that the volatility processes follow some stochastic process that is BM or fBM and if we take a totally empirical view of defining stochastic processes by their Ito density then we have non-Markov long memory and so on but we do not gain deeper insight to whether phase transititon to turbulence is captured by a quant model.

So these are unsatisfactory but useful intermediate results.

>>> ito=np.zeros((13434,1899))
>>> for i in range(1899):
…     ito[:,i]=ito_density(v[:,i])
>>> import matplotlib.pyplot as plt
>>> p=plt.plot(ito[:,155])
>>> plt.show()

Well I will check now the code more carefully.  But this is a bit surprising.  Instead of obtaining a smooth Ito density we get a complex spectrum.

Attachments area
Preview attachment Screenshot 2015-07-03 17.07.24.png

Screenshot 2015-07-03 17.07.24.png

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

7:49 PM (0 minutes ago)

 to David, bcc: aimee
A clean result:  empirical Ito densities are zeros with periods of sharp jumps that are clustered.  In particular while fBM type stochastic models do not apply, these could be possibly explained by deterministic functions in the high-dimensional complex network.  This is a relief and the evidence is presented with data in ‘volatility.csv’ and a pdf file containing the graphs of the Ito densities of the 1900 volatility univariate series.

The data are here.

A quick examination will show that these Ito densities are spikes on a ZERO baseline and therefore are not well modeled by processes in the fBM class.

This could be good news, telling us that volatility series should be modeled not a stochastic process but as a deterministic process with fractional Poisson shocks or purely deterministic on a network with fractional Poisson noise.

This is a significant but narrow result.
2 Attachments

Preview attachment volatility.csv

volatility.csv
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Preview attachment itodensities.pdf

itodensities.pdf
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