dorsal/arxiv
View SchemaA Quantum Approach to Stock Price Fluctuations
| Authors | Martin Schaden |
|---|---|
| Categories | |
| ArXiv ID | physics/0205053 |
| URL | https://arxiv.org/abs/physics/0205053 |
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.
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"abstract": "A simple quantum model explains the Levy-unstable distributions for\nindividual stock returns observed by ref.[1]. The probability density function\nof the returns is written as the squared modulus of an amplitude. For short\ntime intervals this amplitude is proportional to a Cauchy-distribution and\nsatisfies the Schroedinger equation with a non-hermitian Hamiltonian. The\nobserved power law tails of the return fluctuations imply that the \"decay\nrate\", $\\gamma(q)$ asymptotically is proportional to $|q|$, for large $|q|$.\nThe wave number, the Fourier-conjugate variable to the return, is interpreted\nas a quantitative measure of \"market sentiment\". On a time scale of less than a\nfew weeks, the distribution of returns in this quantum model is shape stable\nand scales. The model quantitatively reproduces the observed cumulative\ndistribution for the short-term normalized returns over 7 orders of magnitude\nwithout adjustable parameters. The return fluctuations over large time periods\nultimately become Gaussian if $\\gamma(q\\sim 0)\\propto q^2$. The ansatz\n$\\gamma(q)=b_T\\sqrt{m^2+q^2}$ is found to describe the positive part of the\nobserved historic probability of normalized returns for time periods between\nT=5 min and $T\\sim 4$ years over more than 4 orders of magnitude in terms of\none adjustable parameter $s_T=m b_T\\propto T$. The Sharpe ratio of a stock in\nthis model has a finite limit as the investment horizon $T\\to 0$. Implications\nfor short-term investments are discussed.",
"arxiv_id": "physics/0205053",
"authors": [
"Martin Schaden"
],
"categories": [
"physics.soc-ph",
"physics.data-an",
"q-fin.ST"
],
"title": "A Quantum Approach to Stock Price Fluctuations",
"url": "https://arxiv.org/abs/physics/0205053"
},
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