Time reversal invariance in finance

TL;DR

This paper investigates time reversal invariance in financial time series, showing empirical foreign exchange data are not invariant and identifying which stochastic models can replicate these properties.

Contribution

It introduces new statistics to test time reversal invariance and evaluates various stochastic processes, highlighting structural differences that influence invariance.

Findings

Empirical FX prices are not time reversal invariant.

Only certain ARCH processes with multi-timescales reproduce empirical findings.

GARCH(1,1) processes show partial asymmetry, while stochastic volatility models are invariant.

Abstract

Time reversal invariance can be summarized as follows: no difference can be measured if a sequence of events is run forward or backward in time. Because price time series are dominated by a randomness that hides possible structures and orders, the existence of time reversal invariance requires care to be investigated. Different statistics are constructed with the property to be zero for time series which are time reversal invariant; they all show that high-frequency empirical foreign exchange prices are not invariant. The same statistics are applied to mathematical processes that should mimic empirical prices. Monte Carlo simulations show that only some ARCH processes with a multi-timescales structure can reproduce the empirical findings. A GARCH(1,1) process can only reproduce some asymmetry. On the other hand, all the stochastic volatility type processes are time reversal invariant. This clear difference related to the process structures gives some strong selection criterion for processes.