Mean-Field Price Formation on Trees with a Network of Relative Performance Concerns

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The paper provides important theoretical insights into how relative performance concerns among institutional investors endogenously affect equilibrium prices and risk premia. The decomposition of equilibrium strategies into supply distribution and hedge distribution components (Remark 3.2) is directly relevant to understanding market-making and hedging behavior. The finding that θ=1 eliminates the risk premium entirely (price equals risk-neutral expectation) has practical implications. However, the paper is primarily theoretical and does not provide directly implementable trading algorithms. The binomial tree framework and exponential utility assumptions limit direct applicability to live trading systems.

Implementation requires: (1) constructing a recombining binomial tree with common and idiosyncratic noise processes; (2) backward induction over N time steps computing value functions Vp_n, functions fp_n-1, and transition probabilities at each node; (3) solving the mean-field fixed-point system involving the interaction matrix Θ and its inverse/pseudo-inverse; (4) handling singular cases via resolvent expansion when (I−Θ) is non-invertible; (5) computing equilibrium strategies for heterogeneous agents with distributed risk aversion. The multi-population case with network interactions adds significant algebraic complexity. Numerical stability near singular points requires careful treatment.

The paper provides explicit analytical formulas for equilibrium transition probabilities, optimal strategies, and mean-field dynamics via backward induction. Numerical examples include full parameter tables (Tables 1-4). However, no code repository is provided, and the backward induction algorithm on binomial trees with multiple populations requires careful implementation. The mathematical derivations are complete and self-contained.