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MEIXNER LEVY PROCESS FITS TO RANDOMLY SELECTED STOCK DISTRIBUTIONS

Wim Schoutens and Teugels introduced the Meixner process in 1998, and in 2001 showed some good fit to some index distributions.  Schoutens wrote a book detailing option pricing via Meixner processes.  The probability has four parameters

$f(x; a, b, m, d) = \frac{(2 \cos(b/2))^{2d}}{2\pi\Gamma(2d)} \exp(b(x-m)/a) |\Gamma(d+i(x-m)/a)|^2 MEAN:$latex m+a d \tan(b/2)$VARIANCE:$latex q^2 d/2 (\cos^{-1}(b/2))

KURTOSIS: 3 + \frac{3-2 \cos^2(b/2)}{d}

In this exercise we fit the parameters directly on empirical distributions with minor data preprocessing removing outliers and excess weight at zero returns.

Python code:

def meixnerpdf( x, a, b, m, d):
G1 = np.abs(SS.gamma(d + (m+1j*x))/a)**2
G2 = SS.gamma(2*d)
A = (2*np.cos(b/2))**(2*d)/(2*a*np.pi*G2)
B = np.exp(b*(x-m)/a)
return A*B*G1

The following is a fit of parameters of the distribution for MSFT.  We don’t worry about whether it is better than normal since that’s well established by the work of Schoutens but focus on whether it’s the correct model.  In the figure below we see that the fit is not bad but it’s not quite the right distibution.  In the background is the empirical daily return distribution for several years of MSFT in blue and the Meixner distribution fit in green.

This fit uses a mild L1 regularization and uses the Nelder-Mead algorithm to fir parameters.  The parameters are:
(a,b,m,d) = 0.01*(3.24,-6.58, 1.25,0.00).  For MSFT, the tail behaviour may be appropriate but the mass near the zero center (which is not marked in the graph here is shifted to the right).  We can check a few other examples to verify this observation.  MSFT may be special since for QGEN the fit seems to be better for the mass near zero.

Now the Meixner process is a Levy process, and we know that the return processes in finance are not Markovian, etc. but most importantly, none of these Levy processes can explain turbulence in the financial markets.  So we plan to consider the problem of nonlinearities with Meixner process.  In other words, we consider a parabolic equation followed by the Meixner distribution of the form (d/dt+A)u=0 and consider the modification (d/dt+A)u=F(u).