Financial LPPL Bubbles with Mean-Reverting Noise in the Frequency Domain

TL;DR

This paper investigates the presence of LPPL bubbles in asset prices by analyzing frequency domain properties and mean-reverting noise, proposing methods to reject noise and better identify LPPL signals.

Contribution

It introduces a frequency-domain approach to distinguish LPPL signals from mean-reverting noise in asset price data, enhancing bubble detection methods.

Findings

Noise often obscures LPPL signals in price data.

LPPL spectrum can be characterized by power law and log-periodicity.

Mean-reverting noise is significant at low frequencies.

Abstract

The log-periodic power law (LPPL) is a model of asset prices during endogenous bubbles. A major open issue is to verify the presence of LPPL in price sequences and to estimate the LPPL parameters. Estimation is complicated by the fact that daily LPPL returns are typically orders of magnitude smaller than measured price returns, suggesting that noise obscures the underlying LPPL dynamics. However, if noise is mean-reverting, it would quickly cancel out over subsequent measurements. In this paper, we attempt to reject mean-reverting noise from price sequences by exploiting frequency-domain properties of LPPL and of mean reversion. First, we calculate the spectrum of mean-reverting \ou noise and devise estimators for the noise's parameters. Then, we derive the LPPL spectrum by breaking it down into its two main characteristics of power law and of log-periodicity. We compare price spectra with noise spectra during historical bubbles. In general, noise was strong also at low frequencies and, even if LPPL underlied price dynamics, LPPL would be obscured by noise.