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## CHARACTERISTIC FUNCTION OF TRUNCATED LEVY FLIGHTS

The paper of Potters and Bouchaud ( potters-bouchaud-1997

has good insights such as information regarding the fine-tuning of Levy stable fits to returns but it is wrong in asserting that after 15 minutes Markovian models are appropriate for finance.  Although returns are uncorrelated, their absolute value or any powers are correlated even for daily data for many years.  Markovian models are simply wrong for finance.

For cutoff $\chi>0$ the truncated Levy distribution characteristic function with Levy index $\alpha$ is

$\log \phi(z) = -c (\chi^2 + z^2)^{\alpha/2} \frac{\cos(\alpha \arctan(|z|/\mu)}{\cos(\pi \alpha/2)} - \chi^\alpha$

This, as they point out, finds a ‘fatness’ of tail distributions interpolating between Gaussian and $\alpha$-stable distributions whose tails are too fat.  Now the Hurst exponent for this model is $\alpha/2$.  The Mittag-Leffler function $E_\alpha(-\lambda*t^\alpha)$ solves the basic fractional ordinary differential equation $y^(\alpha) + \lambda*y =0$.  Suppose alpha is a vector of length K, and A is a KxK matrix.  Then it is easy to solve the system

$Y^(\alpha)(t) + AY(t) = 0$ (*)
using Laplace transform: if L denotes Laplace transform then the solution is
[Ly_1(s),…,Ly_K(s)]^T = L^{-1}(A + diag(s^{alpha_1},…,s^{alpha_K}))^{-1}[s^{alpha_1-1},…,s^{alpha_K-1}]

because the Laplace transform of y^(alpha) is just s^alpha Ly(s) – s^{alpha-1}.  Since our univariate fractional diffusion can be rewritten as

$D^{beta}_t p(x,t) + B p(x,t) = 0$ (**)
where $\beta$ is some real number $0<\beta<1$ B acts only on the x variable — the most appropriate operator B is a ‘truncated Levy’ characteristic function. The multivariate version of (**) after Fourier transform is (*).  Now there is content in the multivariate formulation only when the B operator contains some information about the covariance of the market.  The simplest thing to do is just take the covariance of returns C (which in our data is a 1900×1900 matrix and just consider the eigenvectors and a diagonal matrix with artificial assets corresponding to the eigenvectors; this is indeed fairly obvious way that Wall Street quants deal with multivariate returns.

A good univariate model for the pdf of returns could be the three-parameter model via characteristic function:

$E_{\beta}( -c log \phi_{\alpha}(\xi) t^{\gamma})$

This is the Grey brownian motion increment characteristic function with truncated Levy flights, the Fourier transform.  It is a modification of the fractional diffusion equation solution for
$D^\gamma_t p(x,t) + F^{-1}( \phi(\xi) Fp(\xi,t) ) = 0$