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Equilibrium pricing under relative performance concerns. (English) Zbl 1367.91200

The authors study the effects of social interaction between economic agents on a market equilibrium, the efficiency of a securitization mechanism, and the global risk. They consider a finite set of agents having access to an incomplete market consisting of an exogenously priced liquidly traded financial asset. In an attempt to reduce the individual and overall market risks, a social planner introduces to the market a derivative written on the external risk source, allowing the agents to reduce their exposures by trading on it. The aim of the paper is to understand how a pricing mechanism and risk assessments are affected when the agents have relative performance concerns with respect to each other. In the first part of the paper, the authors investigate the existence of the Nash equilibrium in the problem and how to compute it for general risk measures induced by respective backward stochastic differential equations (BSDEs). The second part of the paper focuses on the case of agents using entropic risk measures, which can be treated more explicitly and allows for an in-depth study of the impact of the concern rates. In identifying the Nash equilibrium, a system of fully coupled multidimensional quadratic BSDEs appears, the analysis of which is, in general, quite involved. Among other results, it is established that the risk of a single agent increases if the other agents become more concerned with their relative performance but that it decreases as this agent becomes more concerned. Consequently, if the agents were to play this game repeatedly and their concern rate were to vary over time, they would both find it more advantageous to become more concerned (or jealous). In extreme situations, it is proved that the concern rates destroy the equilibrium while the risk measures themselves remain stable.

MSC:

91G70 Statistical methods; risk measures
91A10 Noncooperative games
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H07 Stochastic calculus of variations and the Malliavin calculus
60H20 Stochastic integral equations
65C30 Numerical solutions to stochastic differential and integral equations
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