Journal of Investment Strategies

On a multi-timescale statistical feedback model for volatility fluctuations

Lisa Borland and Jean-Philippe Bouchaud


We study, both analytically and numerically, a multiscale quadratic autoregressive conditional heteroskedasticity (QARCH) model of volatility, which assumes that the volatility is governed by the observed past price changes over different timescales.With a power-law distribution of time horizons, we obtain a model that captures most stylized facts of financial time series (ie, Student t -like distribution of returns with a power-law tail, long memory of the volatility, slow convergence of the distribution of returns toward the Gaussian distribution, multifractality and anomalous volatility relaxation after shocks). In contrast 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 influences 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 the feedback effect is important and substantially increases the volatility, and that the model is intrinsically difficult to calibrate because of the 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' values can be determined. The model can easily be generalized to account for jumps, skewness and multiasset correlations.


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