On a multi-timescale statistical feedback model for volatility fluctuations

Abstract

We study, both analytically and numerically, an ARCH-like, multiscale model of volatility, which assumes that the volatility is governed by the observed past price changes over different time scales. With a power-law distribution of time horizons, we obtain a model that captures most stylized facts of financial time series: Student-like distribution of returns with a power-law tail, long-memory of the volatility, slow convergence of the distribution of returns towards the Gaussian distribution, multifractality and anomalous volatility relaxation after shocks. At variance with recent multifractal models that are strictly time reversal invariant, the model also reproduces the time asymmetry of financial time series: past large scale volatility influence future small scale volatility. In order to quantitatively reproduce all empirical observations, the parameters must be chosen such that the model is close to an instability, meaning that (a) the feedback effect is important and substantially increases the volatility, and (b) that the model is intrinsically difficult to calibrate because of the very long range nature of the correlations. By imposing consistency of the model predictions with a large set of different empirical observations, a reasonable range of the parameters value can be determined. The model can easily be generalized to account for jumps, skewness and multiasset correlations.

An introduction to econophysics and quantitative finance

“…The latter are found to decay very slowly with τ , in agreement with previous discussions. Therefore, in a first approximation, the dominant feedback effect comes from the amplitude of daily returns only, with minor corrections coming from returns computed on large time spans, at variance with the assumption of the model put forward in [12]. We believe that this finding is unexpected and far from trivial.…”

“…• Second, there is no reason to single out the day as the only time scale to define the returns: in principle, returns over different time scales could also feedback on the volatility today [11,12,33,41]. For extended time scales longer than the day, this leads to another natural extension of the model as:…”

The Physics of Finance: Collective Dynamics in a Complex World

“…γ < 1 or approximately that g 1−γ < 1 and γ > 1 [18]. The model can be calibrated to real stock data yielding γ = 1.15 and g = 0.12, and it is seen that all of the main stylized facts of stock returns are reproduced.…”

“…Interestingly, the model also shows multi-fractal scaling akin to that seen in turbulent systems, although there is no explicit multi-fractality injected in the model. These results are discussed in [18]. In Figure(1) a simulation of that model is shown.…”

“…In this form it is clear that the volatility depends on the change in log-price at time t to the initial time t 0 , since Ω is essentially the log of the price. A natural extension is to generalize to include feedback over not just one but many time-scales, as discussed in [18]). The random return is constructed as the product of a time dependent volatility σ i and a random variable ω i of zero mean and unit variance:…”