## Abstract

The Gaussian white noise modulated in amplitude is defined as the product of a Gaussian white noise and a slowly varying signal with strictly positive values, called volatility. It is a special case of the superstatistical systems with the amplitude as the single parameter associated to the environment variations. If the volatility is deterministic, then the demodulation, i.e., the separation of the two components from a measured time series, can be achieved by a moving average with the averaging window length optimized by the condition that the absolute values of the estimated white noise are uncorrelated. Using Monte Carlo experiments we show that the large scale deterministic volatility can be accurately numerically determined. The artificial deterministic volatilities have a variety of shapes comparable with those occurring in real financial time series. Applied to the daily returns of the S&P500 index, the demodulation algorithm indicates that the most part of the financial volatility is deterministic.

## Authors

## Keywords

Statistical and Nonlinear Physics; computational methods; superstatistics; volatility; artificial time series; Monte Carlo methods

## Paper coordinates

C. Vamoş, M. Crăciun, *Numerical demodulation of a Gaussian white noise modulated in amplitude by a deterministic volatility, *Eur. Phys. J. B (2013) 86: 166.

10.1140/epjb/e2013-31072-x

## References

see the expansion block below.

soon

## About this paper

##### Print ISSN

1434-6028

##### Online ISSN

1434-6036

## Google Scholar

soon

*Asset Price Dynamics, Volatility, and Prediction*(Princeton University Press, Princeton, 2007)

*Time Series Analysis*(Princeton University Press, Princeton, 1994)

*The Statistical Mechanics of Financial Markets*, 3rd edn. (Springer, Berlin, 2005)

*ARCH, Selected Readings*, edited by R.F. Engle (Oxford University Press, Oxford, 1995)

**1**, 223 (2001)

*Wiley Handbook in Financial Engineering and Econometrics: Volatility Models and Their Applications*, edited by L. Bauwens, C. Hafner, S. Laurent (Wiley, New York, 2011), p. 323

*Time Series Analysis: Methods and Applications*, edited by T.S. Rao, S.S. Rao, C.R. Rao (North-Holland Publ., Oxford, 2012), p. 351

**56**, 3623 (2012)

**32**, 577 (2004)

**21**, 1187 (2008)

**81**, 051125 (2010)

*A Practical Guide to Forecasting Financial Market Volatility,*Wiley Finance Series (Wiley, Chichester, 2005)

**78**, 036707 (2008)

*Automatic Trend Estimation*(Springer, Dordrecht, 2012)

**4**, 311 (2008)

**73**, 23 (2010)

**61**, 43 (2001)

**87**, 503 (2005)

**322**, 267 (2003)

**369**, 453 (2011)

**39**, 357 (2009)

**108**, 6390 (2011)

**37**, 1405 (2009)

**36**, 1879 (2008)

*Business Cycles. A Theoretical, Historical and Statistical Analysis of the Capitalist Process*(McGraw-Hill, New York, 1939)

**20**, 611 (2004)

**80**, 036108 (2009)

**80**, 065102 (2009)

**60**, 1390 (1999)

*Time Series: Theory and Methods*(Springer Verlag, New York, 1996)

**65**, 1509 (1970)

**151**, 140 (2009)

*Heavy tails Phenomena. Probabilistic and Statistical Modeling*(Springer, New York, 2007)

**376**, 46 (1995)

**60**, 5305 (1999)

soon