Avellaneda–Stoikov and Cartea–Jaimungal as One Framework: A Forced Uniqueness Theorem for Inventory Market Making

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Highly relevant to quantitative trading, particularly for market-making desks. The paper provides a principled foundation for the choice between AS and CJ frameworks, forces the relation ϕ=γσ²/2 (eliminating over-parametrization), provides a calibration cross-check (γ=2ϕ/σ²), corrects stochastic-volatility inventory penalties to use quadratic-variation time, identifies CVaR time-inconsistency with dynamic optimization, and gives operational prescriptions for desk implementation. The forced uniqueness result directly impacts how practitioners calibrate and implement market-making strategies.

The theoretical framework requires deep understanding of dynamic risk measures, axiomatic preference theory, stochastic control (HJB equations), and the Kupper–Schachermayer representation theorem. The practical implementation of the forced coefficient relation (ϕ=γσ²/2) and the calibration inversion (γ=2ϕ/σ²) is straightforward. However, implementing the clock-coherent stochastic-volatility penalty (integrating against quadratic variation rather than wall-clock time) requires careful numerical handling. The multi-asset extension with forced covariance matrix Φ=γΣ/2 adds further complexity.

The paper is purely theoretical with complete mathematical proofs. The Monte Carlo simulation in Appendix D uses analytically specified parameters from Avellaneda and Stoikov (2008, Section 3.2) with N=10,000 trajectories and 1,000 time steps. All parameters are explicitly stated. However, no code repository is provided, and the simulation details (random seed, exact implementation of the CJ-formula heuristic) are not fully specified for exact reproduction.