Nonlinear Memory and Risk Estimation in Financial Records [chapter]

Armin Bunde, Mikhail I. Bogachev
2010 Econophysics Approaches to Large-Scale Business Data and Financial Crisis  
It is well known that financial data sets are multifractal and governed by nonlinear correlations. Here we are interested in the daily returns of a financial asset and in the way the occurrence of large gains or losses is triggered by the nonlinear memory. To this end, we study the statistics of the return intervals between gains (or losses) above a certain threshold Q. In the case of i.i.d. random numbers the probability density function (pdf) of the return intervals decays exponentially and
more » ... exponentially and the return intervals are uncorrelated. Here we show that the nonlinear correlations lead to a power law decay of the pdf and linear long-term correlations between the return intervals that are described by a power-law decay of the corresponding autocorrelation function. From the pdf of the return intervals one obtains the risk function W Q .tI t/, which is the probability that within the next t units of time at least one event above Q occurs, if the last event occurred t time units ago. We propose an analytical estimate of W Q and show explicitly that the proposed method is superior to the conventional precursory pattern recognition technique widely used in signal analysis, which requires considerable fine-tuning and is difficult to implement. We also show that the estimation of the Value at Risk, which is a standard tool in finances, can be improved considerably compared with previous estimates. In the past years, the occurrence of rare (extreme) events has attracted much attention [1] [2] [3] . Usually, rare events with magnitudes considerably exceeding the average magnitude, have been considered as independent, since the typical time span A. Bunde ( )
doi:10.1007/978-4-431-53853-0_2 fatcat:bl6odbjvtrfqjg5q6tk2ooujdu