Fully-dynamic risk measures: horizon risk, time-consistency, and relations with BSDEs and BSVIEs

TL;DR

This paper introduces the concept of horizon risk in dynamic risk measures, proposes fully-dynamic measures to address it, and explores their properties and representations via BSDEs and BSVIEs, including new theoretical results.

Contribution

It defines horizon risk, develops fully-dynamic risk measures, and establishes new theoretical results for BSVIEs and their relation to risk assessment.

Findings

Horizon risk affects dynamic risk measures significantly.

Fully-dynamic risk measures can be generated by BSDEs and BSVIEs.

New results include a converse comparison theorem and dual representation for BSVIEs.

Abstract

In a dynamic framework, we identify a new concept associated with the risk of assessing the financial exposure by a measure that is not adequate to the actual time horizon of the position. This will be called horizon risk. We clarify that dynamic risk measures are subject to horizon risk, so we propose to use the fully-dynamic version. To quantify horizon risk, we introduce h-longevity as an indicator. We investigate these notions together with other properties of risk measures as normalization, restriction property, and different formulations of time-consistency. We also consider these concepts for fully-dynamic risk measures generated by backward stochastic differential equations (BSDEs), backward stochastic Volterra integral equations (BSVIEs), and families of these. Within this study, we provide new results for BSVIEs such as a converse comparison theorem and the dual representation of the associated risk measures.