Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets
Abstract
1. Introduction
2. Materials and Methods
2.1. Multiplicative Random Cascade Model of Volatility
2.2. Parameter Estimation
2.3. Integral Fluctuation Theorem
2.4. Empirical Study
2.4.1. Data
2.4.2. Data Processing
Data Set 1 (LSE Data Set: Normalized Average)
- From the one-minute price data of each stock’s intraday price fluctuations, the logarithmic return was calculated.
- The normalized average of the logarithmic return was computed as
- We cumulated to obtain the path of the process as follows:
Data Set 2 (TSE Data Set: Normalized Average)
Data Set 3 (TSE Data Set: Individual Stocks)
2.4.3. Wavelet Transform
2.4.4. Wavelet Transform Modulus Maxima Line
3. Results
3.1. Parameter Estimation Based on Integral Fluctuation Theorem
3.2. Details of Trajectories
- (A)
- Volatility: This represents the position of the Brownian particle, which corresponds to the system’s state variable.
- (B)
- Force applied to the system: This indicates the external force acting on the Brownian particle.
- (C)
- Inverse temperature of the heat bath: According to Equations (16) and (26), the temperature represents the trajectory of the variance of volatility.
- (D)
- Heat transfer: This describes the energy transferred from the heat bath to the system, that is, the energy received by the Brownian particle from the heat bath. A negative value indicates a positive amount of heat flowing from the particle back to the heat bath.
- (E)
- Change in the content of the information of the system: As expressed in Equation (32), this quantity can be interpreted as the change in the entropy of the system, a central concept in stochastic thermodynamics.
- (F)
- Change in total entropy: This indicates the change in the total entropy of the system and the heat bath.
4. Discussion
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
LSE | London Stock Exchange |
TSE | Tokyo Stock Exchange |
SDE | Stochastic differential equation |
Probability density function | |
WTMM | Wavelet transform modulus maxima |
WTMML | Wavelet transform modulus maxima line |
Appendix A. Relations Between Parameters
Appendix B. Wavelet Transformation Modulus Maxima Line
Appendix B.1. Wavelet Transformation Modulus Maxima (WTMM)
- The point is a local maximum or minimum of the wavelet transform coefficient :
- For a point u in the right or left neighborhood of ,and for a point u in the opposite neighborhood,
Appendix B.2. Wavelet Transformation Modulus Maxima Line (WTMML)
- If a point is included in the curve l, then , and the point is a WTMM.
- For any scale , there exists a point on the curve l.
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Maskawa, J.-i. Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets. Entropy 2025, 27, 435. https://doi.org/10.3390/e27040435
Maskawa J-i. Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets. Entropy. 2025; 27(4):435. https://doi.org/10.3390/e27040435
Chicago/Turabian StyleMaskawa, Jun-ichi. 2025. "Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets" Entropy 27, no. 4: 435. https://doi.org/10.3390/e27040435
APA StyleMaskawa, J.-i. (2025). Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets. Entropy, 27(4), 435. https://doi.org/10.3390/e27040435