Manoj Ughade, Rahul Vishwakarma, and Manoj Kumar Shukla
ABSTRACT
We develop a mathematical fixed-point framework to study stress propagation in financial networks that combines interbank exposures and liquidity-induced market impact. By defining a stress-propagation operator on the space of institution-level stress vectors, we derive a sufficient, economically interpretable inequality that guarantees the operator is a contraction in the supremum norm. Through the Banach fixed-point theorem, we obtain rigorous existence, uniqueness, and geometric convergence results for the systemic-stress equilibrium. Explicit constants, detailed proofs, sensitivity analysis, and two real-world style applications a small banking contagion network and a portfolio-liquidity feedback model are presented. Numerical illustrations confirm convergence and provide insights into systemic stability under liquidity stress.
Keywords: Banach contraction principle; fixed point; financial networks; liquidity risk; market impact; systemic stability.