Entropy-Regularized Certainty-Equivalent Bellman Policies for Risk-Sensitive Market Making

Battle Count: 54

Directly addresses the core problem of optimal market making with risk-sensitive objectives. Provides rigorous convergence guarantees for randomized quoting policies, which is highly relevant for practical algorithmic market-making systems. The Hamiltonian-Gibbs proxy offers a computationally tractable policy with provable performance bounds. The Avellaneda-Stoikov framework is the standard benchmark in quantitative market making. The entropy regularization connects to modern RL approaches while maintaining analytical tractability. The fresh-sampling implementation is directly applicable to real-time quoting systems.

The exact Bellman operator requires computing one-step certainty-equivalent scores for each deterministic quote via matrix exponentials on the (2Q+1)-dimensional inventory state space, making it computationally expensive. The Hamiltonian-Gibbs proxy reduces this to evaluating the soft Hamiltonian ODE via Euler discretization, which is more tractable. Implementation requires: (1) solving systems of ODEs (hard and soft Hamiltonian), (2) computing matrix exponentials for the Feynman-Kac representation, (3) Gauss-Legendre quadrature over the 2D quote rectangle, (4) implementing fresh-sampling marked point-process simulation with thinning. The theoretical framework involves advanced stochastic calculus, exponential semimartingale arguments, and backward Gronwall inequalities.

The paper provides detailed parameter tables (Table 1), specific numerical methods (RK4 solver, Gauss-Legendre quadrature with specified node counts), explicit formulas for all operators, and Monte Carlo implementation details (5000 paths, common random numbers, seed MC=12345). The soft-HJB Euler scheme and Hamiltonian-Gibbs proxy are fully specified. However, no code repository is explicitly linked, and some auxiliary proofs are referenced but not fully reproduced in the main text.