 A Quantum Approach to Stock Price FluctuationsMartin Schaden Abstract: 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. Date: 2002-05, Revised 2003-05 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:physics/0205053 Access Statistics for this paper More papers in Papers from arXiv.org |
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