Mapping Crisis-Driven Market Dynamics: A Transfer Entropy and Kramers-Moyal Approach to Financial Networks

Financial markets are dynamic, interconnected systems where local shocks can trigger widespread instability, challenging portfolio managers and policymakers. Traditional correlation analysis often miss the directionality and temporal dynamics of information flow. To address this, we present a unified framework integrating Transfer Entropy (TE) and the N-dimensional Kramers-Moyal (KM) expansion to map static and time-resolved coupling among four major indices: Nasdaq Composite (^IXIC), WTI crude oil (WTI), gold (GC=F), and the US Dollar Index (DX-Y.NYB). TE captures directional information flow. KM models non-linear stochastic dynamics, revealing interactions often overlooked by linear methods. Using daily data from August 11, 2014, to September 8, 2024, we compute returns, confirm non-stationary using a conduct sliding-window TE and KM analyses. We find that during the COVID-19 pandemic (March-June 2020) and the Russia-Ukraine crisis (Feb-Apr 2022), average TE increases by 35% and 28%, respectively, indicating heightened directional flow. Drift coefficients highlight gold-dollar interactions as a persistent safe-haven channel, while oil-equity linkages show regime shifts, weakening under stress and rebounding quickly. Our results expose the shortcomings of linear measures and underscore the value of combining information-theoretic and stochastic drift methods. This approach offers actionable insights for adaptive hedging and informs macro-prudential policy by revealing the evolving architecture of systemic risk.