Estimating Stable Fixed Points and Langevin Potentials for Financial Dynamics

ArXiv ID: 2309.12082 “View on arXiv”

Authors: Unknown

Abstract

The Geometric Brownian Motion (GBM) is a standard model in quantitative finance, but the potential function of its stochastic differential equation (SDE) cannot include stable nonzero prices. This article generalises the GBM to an SDE with polynomial drift of order q and shows via model selection that q=2 is most frequently the optimal model to describe the data. Moreover, Markov chain Monte Carlo ensembles of the accompanying potential functions show a clear and pronounced potential well, indicating the existence of a stable price.

Keywords: Geometric Brownian Motion (GBM), Stochastic Differential Equations (SDE), Potential Functions, Monte Carlo Ensembles, Model Selection, General (Quantitative Finance)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper features advanced mathematics including stochastic differential equations, likelihood functions, and polynomial drift terms, while also applying a method to real stock market data (TAQ and CRSP) with specific data processing steps.

  flowchart TD
    A["Research Goal: Estimating Stable Fixed Points<br>and Langevin Potentials"] --> B["Methodology: SDEs with<br>Polynomial Drift of Order q"]
    B --> C["Data: Financial Time Series<br>e.g., Stock Indices"]
    C --> D["Computational Process:<br>Model Selection via MLE/BIC"]
    D --> E{"Model Selection Outcome"}
    E -- Optimal Model --> F["Key Finding: q=2 SDE<br>Most Frequently Optimal"]
    F --> G["Key Finding: Markov Chain Monte Carlo<br>Reveals Potential Well/Stable Price"]
    E -- Other q Values --> D