The Stock Price Distribution in Quantum Finance

A simple quantum model can explain the observed Levy-unstable distributions for individual stock returns. The tails of the short-term cumulative distribution for the logarithmic return, x, scale as x^{-3}, if the "decay rate" of a stock, \gamma(q), for large |q| is proportional to |q|, q being the Fourier-conjugate variable to x. On a time scale of a few days or less, the distribution of the quantum model is shape stable and scales with a single parameter, its variance. The observed cumulative distribution for the short-term normalized returns is quantitatively reproduced over 7 orders of magnitude without any free parameters. The distribution of returns ultimately converges to a Gaussian one for large time periods if \gamma(q\sim 0)\propto q^2. The empirical constraints suggest the ansatz \gamma(q)=b\sqrt{m^2+q^2}, which reproduces the positive part of the observed average cumulative distributions for time periods between 5 minutes and 4 years over more than 4 orders of magnitude with one parameter