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 |
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Keywords
- Born-Oppenheimer approximation
- Power-law distribution of returns
- Stochastic volatility
ASJC Scopus subject areas
- Mathematical Physics
- Statistical and Nonlinear Physics
Cite this
Microscopic origin of non-Gaussian distributions of financial returns. / Bíró, T.; Rosenfeld, R.
In: Physica A: Statistical Mechanics and its Applications, Vol. 387, No. 7, 01.03.2008, p. 1603-1612.Research output: Contribution to journal › Article
}
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
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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 -