On the short term stability of financial ARCH price processes

TL;DR

This paper examines the short-term stability of quadratic ARCH processes in financial modeling, revealing empirical deviations from theoretical assumptions such as variance and mean stability over decades.

Contribution

It provides an analysis of the theoretical conditions for stable ARCH processes and empirically assesses their validity using extensive financial data.

Findings

Empirical innovations' variance exceeds 1, indicating short-term instability.

Some time series show significant deviation from zero mean.

Theoretical conditions are often not met in real financial data.

Abstract

For many financial applications, it is important to have reliable and tractable models for the behavior of assets and indexes, for example in risk evaluation. A successful approach is based on ARCH processes, which strike the right balance between statistical properties and ease of computation. This study focuses on quadratic ARCH processes and the theoretical conditions to have a stable long-term behavior. In particular, the weights for the variance estimators should sum to 1, and the variance of the innovations should be 1. Using historical data, the realized empirical innovations can be computed, and their statistical properties assessed. Using samples of 3 to 5 decades, the variance of the empirical innovations are always significantly above 1, for a sample of stock indexes, commodity indexes and FX rates. This departure points to a short term instability, or to a fast adaptability due to changing conditions. Another theoretical condition on the innovations is to have a zero mean. This condition is also investigated empirically, with some time series showing significant departure from zero.