Capital allocation for cash-subadditive risk measures: From BSDEs to BSVIEs

Abstract

In the context of risk measures, the capital allocation problem is widely studied in the literature where different approaches have been developed, also in connection with cooperative game theory and systemic risk. Although static capital allocation rules have been extensively studied in the recent years, only few works deal with dynamic capital allocations and its relation with BSDEs. Moreover, all those works only examine the case of an underneath risk measure satisfying cash-additivity and, moreover, a large part of them focuses on the specific case of the gradient allocation where Gateaux differentiability is assumed.

The main goal of this paper is, instead, to study general dynamic capital allocations associated to cash-subadditive risk measures, generalizing the approaches already existing in the literature and motivated by the presence of (ambiguity on) interest rates. Starting from an axiomatic approach, we then focus on the case where the underlying risk measures are induced by BSDEs whose drivers depend also on the y-variable. In this setting, we surprisingly find that the corresponding capital allocation rules solve special kinds of Backward Stochastic Volterra Integral Equations (BSVIEs).

Keywords

Risk measures / Capital allocation / BSDE / BSVIE / Cash-subadditivity / Subdifferential

Acknowledgements

The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA)-INdAM, Italy, and acknowledge the financial support of Gnampa Research Project 2024 (Grant No. PRR-20231026-073916-203). This research was funded in part by an Ermenegildo Zegna Founder’s Scholarship (Zullino).

The authors express their gratitude to two anonymous Referees whose insightful comments significantly enhanced the quality of the paper. The authors thank Ilaria Peri and the participants at the Quantitative Finance Workshop 2023 in Gaeta, Italy, for comments and discussions.

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