Currents Beneath Stability: A Stochastic Framework for Exchange Rate Instability Using Kramers Moyal Expansion

ArXiv ID: 2507.01989 “View on arXiv”

Authors: Yazdan Babazadeh Maghsoodlo, Amin Safaeesirat

Abstract

Understanding the stochastic behavior of currency exchange rates is critical for assessing financial stability and anticipating market transitions. In this study, we investigate the empirical dynamics of the USD exchange rate in three economies, including Iran, Turkey, and Sri Lanka, through the lens of the Kramers-Moyal expansion and Fokker-Planck formalism. Using log-return data, we confirm the Markovian nature of the exchange rate fluctuations, enabling us to model the system with a second-order Fokker-Planck equation. The inferred Langevin coefficients reveal a stabilizing linear drift and a nonlinear, return-dependent diffusion term, reflecting both regulatory effects and underlying volatility. A rolling-window estimation of these coefficients, paired with structural breakpoint detection, uncovers regime shifts that align with major political and economic events, offering insight into the hidden dynamics of currency instability. This framework provides a robust foundation for detecting latent transitions and modeling risk in complex financial systems.

Keywords: Fokker-Planck Equation, Exchange Rate Dynamics, Kramers-Moyal Expansion, Langevin Model, Regime Switching

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced stochastic calculus including Kramers-Moyal expansion and Fokker-Planck equations, demonstrating high mathematical density. It uses real-world currency data with rolling-window estimation and structural break detection, indicating substantial empirical implementation and backtest-ready methodology.

  flowchart TD
    A["Research Goal:\nModel Exchange Rate Instability"] --> B["Data Acquisition\nUSD Exchange Rate Data: Iran, Turkey, Sri Lanka"]
    B --> C["Methodology:\nKramers-Moyal Expansion & Fokker-Planck Formalism"]
    C --> D{"Markovian Analysis"}
    D -- Verified --> E["Computational Process:\nInfer Langevin Coefficients via Rolling-Window Estimation"]
    E --> F["Outcome 1:\nStabilizing Linear Drift & Nonlinear Diffusion"]
    E --> G["Outcome 2:\nRegime Shifts via Structural Breakpoint Detection"]
    F --> H["Final Insight:\nFramework for Latent Transition Detection & Risk Modeling"]
    G --> H