### Abstract

In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born-Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero-drift modified log-returns. We verify that the Dow-Jones Industrial Average index indeed follows this scaling. We then focus on two popular stochastic volatility models, the Heston and Hull-White models. In particular, we show that in the Hull-White model the resulting probability distribution of log-returns in this approximation corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are given in terms of the microscopic stochastic volatility model. Finally, we show that the log-returns for 30 years Dow Jones index data is well fitted by a Tsallis distribution, obtaining the relevant parameters.

Original language | English |
---|---|

Pages (from-to) | 1603-1612 |

Number of pages | 10 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 387 |

Issue number | 7 |

DOIs | |

Publication status | Published - Mar 1 2008 |

### Fingerprint

### Keywords

- Born-Oppenheimer approximation
- Power-law distribution of returns
- Stochastic volatility

### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*387*(7), 1603-1612. https://doi.org/10.1016/j.physa.2007.10.067

**Microscopic origin of non-Gaussian distributions of financial returns.** / Bíró, T.; Rosenfeld, R.

Research output: Contribution to journal › Article

*Physica A: Statistical Mechanics and its Applications*, vol. 387, no. 7, pp. 1603-1612. https://doi.org/10.1016/j.physa.2007.10.067

}

TY - JOUR

T1 - Microscopic origin of non-Gaussian distributions of financial returns

AU - Bíró, T.

AU - Rosenfeld, R.

PY - 2008/3/1

Y1 - 2008/3/1

N2 - In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born-Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero-drift modified log-returns. We verify that the Dow-Jones Industrial Average index indeed follows this scaling. We then focus on two popular stochastic volatility models, the Heston and Hull-White models. In particular, we show that in the Hull-White model the resulting probability distribution of log-returns in this approximation corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are given in terms of the microscopic stochastic volatility model. Finally, we show that the log-returns for 30 years Dow Jones index data is well fitted by a Tsallis distribution, obtaining the relevant parameters.

AB - In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born-Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero-drift modified log-returns. We verify that the Dow-Jones Industrial Average index indeed follows this scaling. We then focus on two popular stochastic volatility models, the Heston and Hull-White models. In particular, we show that in the Hull-White model the resulting probability distribution of log-returns in this approximation corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are given in terms of the microscopic stochastic volatility model. Finally, we show that the log-returns for 30 years Dow Jones index data is well fitted by a Tsallis distribution, obtaining the relevant parameters.

KW - Born-Oppenheimer approximation

KW - Power-law distribution of returns

KW - Stochastic volatility

UR - http://www.scopus.com/inward/record.url?scp=37349089091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37349089091&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2007.10.067

DO - 10.1016/j.physa.2007.10.067

M3 - Article

AN - SCOPUS:37349089091

VL - 387

SP - 1603

EP - 1612

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 7

ER -