Memory Effects, Multiple Time Scales and Local Stability in Langevin Models of the S&P500 Market Correlation

ArXiv ID: 2307.12744 “View on arXiv”

Authors: Unknown

Abstract

The analysis of market correlations is crucial for optimal portfolio selection of correlated assets, but their memory effects have often been neglected. In this work, we analyse the mean market correlation of the S&P500 which corresponds to the main market mode in principle component analysis. We fit a generalised Langevin equation (GLE) to the data whose memory kernel implies that there is a significant memory effect in the market correlation ranging back at least three trading weeks. The memory kernel improves the forecasting accuracy of the GLE compared to models without memory and hence, such a memory effect has to be taken into account for optimal portfolio selection to minimise risk or for predicting future correlations. Moreover, a Bayesian resilience estimation provides further evidence for non-Markovianity in the data and suggests the existence of a hidden slow time scale that operates on much slower times than the observed daily market data. Assuming that such a slow time scale exists, our work supports previous research on the existence of locally stable market states.

Keywords: Generalised Langevin Equation, Memory Effects, Market Correlation, Non-Markovianity, Portfolio Selection, Equities

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced stochastic processes (Generalized Langevin Equations with memory kernels) and Bayesian inference for parameter estimation, indicating high mathematical complexity. It demonstrates strong empirical rigor through detailed data processing (S&P 500 returns over 20 years), model fitting, and comparative forecasting accuracy, moving beyond pure theory toward practical implementation.

  flowchart TD
    A["Research Goal: Analyze S&P500 Mean Correlation & Memory Effects"] --> B{"Methodology"}
    B --> C["Data: S&P500 Daily Prices"]
    B --> D["Generalized Langevin Equation GLE Fitting"]
    B --> E["Bayesian Resilience Estimation"]
    
    C --> D
    C --> E
    
    D --> F["Computational Process: Determine Memory Kernel Properties"]
    E --> F
    
    F --> G["Key Findings & Outcomes"]
    
    G --> H["Significant Memory Effect<br/>lasts >= 3 trading weeks"]
    G --> I["GLE with Memory ><br/>Standard Models in Forecasting"]
    G --> J["Evidence for Non-Markovianity &<br/>Hidden Slow Time Scale"]
    G --> K["Supports Existence of<br/>Locally Stable Market States"]