dorsal/arxiv
View SchemaAn elementary model of price dynamics in a financial market: Distribution, Multiscaling & Entropy
| Authors | Stefan Reimann |
|---|---|
| Categories | |
| ArXiv ID | physics/0602097 |
| URL | https://arxiv.org/abs/physics/0602097 |
Abstract
Stylized facts of empirical assets log-returns $Z$ include the existence of (semi) heavy tailed distributions $f_Z(z)$ and a non-linear spectrum of Hurst exponents $\tau(\beta)$. Empirical data considered are daily prices of 10 large indices from 01/01/1990 to 12/31/2004. We propose a stylized model of price dynamics which is driven by expectations. The model is a multiplicative random process with a stochastic, state-dependent growth rate which establishes a negative feedback component in the price dynamics. This 0-order model implies that the distribution of log-returns is Laplacian $f_Z(z) \sim \exp(-\frac{|z|}{\alpha})$, whose single parameter $\alpha$ can be regarded as a measure for the long-time averaged liquidity in the respective market. A comparison with the (more general) Weibull distribution shows that empirical daily log returns are close to being Laplacian distributed. The spectra of Hurst exponents of both, empirical data $\tau_{emp}$ and simulated data due to our model $\tau_{theor}$, are compared. Due to the finding of non-linear Hurst spectra, the Renyi entropy (RE) $R_\beta(f_Z)$is considered. An explicit functional form of the RE for an exponential distribution is derived. Theoretical REs of simulated asset return trails are in good agreement with the RE estimated from empirical returns.
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"abstract": "Stylized facts of empirical assets log-returns $Z$ include the existence of\n(semi) heavy tailed distributions $f_Z(z)$ and a non-linear spectrum of Hurst\nexponents $\\tau(\\beta)$. Empirical data considered are daily prices of 10 large\nindices from 01/01/1990 to 12/31/2004. We propose a stylized model of price\ndynamics which is driven by expectations. The model is a multiplicative random\nprocess with a stochastic, state-dependent growth rate which establishes a\nnegative feedback component in the price dynamics. This 0-order model implies\nthat the distribution of log-returns is Laplacian $f_Z(z) \\sim\n\\exp(-\\frac{|z|}{\\alpha})$, whose single parameter $\\alpha$ can be regarded as\na measure for the long-time averaged liquidity in the respective market. A\ncomparison with the (more general) Weibull distribution shows that empirical\ndaily log returns are close to being Laplacian distributed. The spectra of\nHurst exponents of both, empirical data $\\tau_{emp}$ and simulated data due to\nour model $\\tau_{theor}$, are compared. Due to the finding of non-linear Hurst\nspectra, the Renyi entropy (RE) $R_\\beta(f_Z)$is considered. An explicit\nfunctional form of the RE for an exponential distribution is derived.\nTheoretical REs of simulated asset return trails are in good agreement with the\nRE estimated from empirical returns.",
"arxiv_id": "physics/0602097",
"authors": [
"Stefan Reimann"
],
"categories": [
"physics.soc-ph",
"q-fin.ST"
],
"title": "An elementary model of price dynamics in a financial market: Distribution, Multiscaling \u0026 Entropy",
"url": "https://arxiv.org/abs/physics/0602097"
},
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