The fine-structure of volatility feedback I: multi-scale self-reflexivity

TL;DR

This paper investigates the detailed structure of volatility feedback in financial markets using quadratic autoregressive models, revealing complex interactions, long-memory effects, and subtle time asymmetries in stock return volatility.

Contribution

It introduces a comprehensive quadratic autoregressive model for volatility, uncovering significant off-diagonal effects and complex spectral properties not captured by traditional models.

Findings

Off-diagonal quadratic coefficients are significant but smaller than diagonal ones.

The feedback kernel shows a power-law decay, indicating long-memory.

Empirical evidence of small violations of time reversal symmetry.

Abstract

We attempt to unveil the fine structure of volatility feedback effects in the context of general quadratic autoregressive (QARCH) models, which assume that today's volatility can be expressed as a general quadratic form of the past daily returns. The standard ARCH or GARCH framework is recovered when the quadratic kernel is diagonal. The calibration of these models on US stock returns reveals several unexpected features. The off-diagonal (non ARCH) coefficients of the quadratic kernel are found to be highly significant both In-Sample and Out-of-Sample, but all these coefficients turn out to be one order of magnitude smaller than the diagonal elements. This confirms that daily returns play a special role in the volatility feedback mechanism, as postulated by ARCH models. The feedback kernel exhibits a surprisingly complex structure, incompatible with models proposed so far in the literature. Its spectral properties suggest the existence of volatility-neutral patterns of past returns. The diagonal part of the quadratic kernel is found to decay as a power-law of the lag, in line with the long-memory of volatility. Finally, QARCH models suggest some violations of Time Reversal Symmetry in financial time series, which are indeed observed empirically, although of much smaller amplitude than predicted. We speculate that a faithful volatility model should include both ARCH feedback effects and a stochastic component.