This paper investigates the concept of time reversal invariance in financial markets, specifically examining whether price time series exhibit the same statistical properties when run forward or backward in time. The research addresses the challenge that financial price series are dominated by randomness that may obscure underlying structures, requiring careful statistical analysis to detect potential deviations from time reversal invariance.

The authors develop specialized statistical measures designed to be zero for time series that are time reversal invariant and apply these to high-frequency foreign exchange price data. Their empirical analysis reveals clear deviations from time reversal invariance in real market data. To understand these findings, they conduct Monte Carlo simulations of various mathematical processes commonly used in finance, including ARCH, GARCH, and stochastic volatility models. The results show that only ARCH processes with a multi-timescale structure can reproduce the empirical asymmetries observed in foreign exchange data, while GARCH(1,1) processes only partially capture these effects, and all stochastic volatility processes remain time reversal invariant. This clear differentiation provides valuable selection criteria for financial time series models based on their ability to reproduce the time reversal properties observed in real market data.