# Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets

## TL;DR

This paper applies concepts from physics to study how volatility spreads across different time scales in stock markets, revealing differences between London and Tokyo stock exchanges.

## Contribution

The novel contribution is applying stochastic thermodynamics and the Fluctuation Theorem to model and analyze volatility cascade processes in financial markets.

## Key findings

- Volatility cascades in the London Stock Exchange follow a causal pattern from larger to smaller time scales.
- Tokyo Stock Exchange data shows anti-causal behavior in volatility cascades at longer time scales.
- The Langevin-based model successfully reproduces empirical volatility distributions and satisfies the Integral Fluctuation Theorem.

## Abstract

This study investigates the properties of financial markets that arise from the multi-scale structure of volatility, particularly intermittency, by employing robust theoretical tools from nonequilibrium thermodynamics. Intermittency in velocity fields along spatial and temporal axes is a well-known phenomenon in developed turbulence, with extensive research dedicated to its structures and underlying mechanisms. In turbulence, such intermittency is explained through energy cascades, where energy injected at macroscopic scales is transferred to microscopic scales. Similarly, analogous cascade processes have been proposed to explain the intermittency observed in financial time series. In this work, we model volatility cascade processes in the stock market by applying the framework of stochastic thermodynamics to a Langevin system that describes the dynamics. We introduce thermodynamic concepts such as temperature, heat, work, and entropy into the analysis of financial markets. This framework allows for a detailed investigation of individual trajectories of volatility cascades across longer to shorter time scales. Further, we conduct an empirical study primarily using the normalized average of intraday logarithmic stock prices of the constituent stocks in the FTSE 100 Index listed on the London Stock Exchange (LSE), along with two additional data sets from the Tokyo Stock Exchange (TSE). Our Langevin-based model successfully reproduces the empirical distribution of volatility—defined as the absolute value of the wavelet coefficients across time scales—and the cascade trajectories satisfy the Integral Fluctuation Theorem associated with entropy production. A detailed analysis of the cascade trajectories reveals that, for the LSE data set, volatility cascades from larger to smaller time scales occur in a causal manner along the temporal axis, consistent with known stylized facts of financial time series. In contrast, for the two data sets from the TSE, while similar behavior is observed at smaller time scales, anti-causal behavior emerges at longer time scales.