Financial Resilience Evaluation: From Conditional Expectations to Dynamic Convex Risk Measures

2026-06-29

AI Summary

This study addresses the limitation of traditional resilience metrics, which rely solely on conditional expectations and thus fail to distinguish financial positions with identical expected recoveries but differing risk profiles. To overcome this, the paper introduces dynamic convex risk measures into resilience assessment for the first time, establishing a nonlinear evaluation framework driven by backward stochastic differential equations (BSDEs). By modeling asset dynamics via Itô processes and leveraging BSDEs with Lipschitz or quadratic generators—combined with duality representations and measure-change techniques—the authors construct a well-defined resilience operator and derive its explicit dual formulation. The work establishes the attainability of optimal scenarios and validates the method’s efficacy and the necessity of key assumptions through illustrative examples, significantly enhancing the sensitivity and discriminatory power of resilience measures with respect to tail risk.

Abstract

Financial resilience concerns the rate at which a position recovers, or further deteriorates, in response to adverse conditions. As a first step, Laeven, Ferrari, Rosazza Gianin, and Zullino (arXiv:2505.07502) introduced the resilience rate, defined as the expected instantaneous rate of (favorable) change of a price or risk-assessment process. Since this quantity captures only the conditional mean of future increments, it cannot distinguish between positions having the same expected recovery but different conditional risk profiles. We obtain a richer characterization by evaluating such increments through a genuine, possibly nonlinear, dynamic risk measure. More precisely, for an Itô process $π$ and a normalized, cash-additive dynamic risk measure $ρ$, we define the resilience evaluation by \[\mathcal D_s^ρπ_t := L^1\text{-}\lim_{\varepsilon\to0^+} \frac{1}{\varepsilon}ρ_s(π_{t+\varepsilon}-π_t), \qquad 0\leq s\leq t<T,\] whenever the limit exists. When $ρ$ is a convex dynamic risk measure induced by a BSDE with a Lipschitz or quadratic driver, we prove that this limit is well-posed and admits an explicit dual representation. It is given by the worst-case conditional expectation, over a zero-penalty class of measure changes, of an effective drift combining the drift of $π$ with the risk adjustment assigned by $ρ$ to its volatility. We further establish attainment of the optimal scenario and illustrate the scope of the construction, as well as the role of the assumptions, through examples and counterexamples.

Problem

• financial resilience
• dynamic convex risk measures
• conditional risk profiles
• risk evaluation
• Itô processes

Innovation

• dynamic convex risk measures
• financial resilience
• BSDE
• dual representation
• Itô process