  A simple quantum model explains the Levy-unstable distributions for
individual stock returns observed by ref.[1]. The probability density function
of the returns is written as the squared modulus of an amplitude. For short
time intervals this amplitude is proportional to a Cauchy-distribution and
satisfies the Schroedinger equation with a non-hermitian Hamiltonian. The
observed power law tails of the return fluctuations imply that the "decay
rate", $\gamma(q)$ asymptotically is proportional to $|q|$, for large $|q|$.
The wave number, the Fourier-conjugate variable to the return, is interpreted
as a quantitative measure of "market sentiment". On a time scale of less than a
few weeks, the distribution of returns in this quantum model is shape stable
and scales. The model quantitatively reproduces the observed cumulative
distribution for the short-term normalized returns over 7 orders of magnitude
without adjustable parameters. The return fluctuations over large time periods
ultimately become Gaussian if $\gamma(q\sim 0)\propto q^2$. The ansatz
$\gamma(q)=b_T\sqrt{m^2+q^2}$ is found to describe the positive part of the
observed historic probability of normalized returns for time periods between
T=5 min and $T\sim 4$ years over more than 4 orders of magnitude in terms of
one adjustable parameter $s_T=m b_T\propto T$. The Sharpe ratio of a stock in
this model has a finite limit as the investment horizon $T\to 0$. Implications
for short-term investments are discussed.

Physics and Society;Data Analysis, Statistics and Probability;Statistical Finance