inventory risk modeling

This study investigates how multiple market makers balance internal hedging against external unwinding to manage inventory risk and maximize profits in the presence of competition for client order flow. By formulating a continuous-time dynamic quoting game and integrating stochastic control, calculus of variations, and game theory, the authors derive—for the first time—the closed-form Nash equilibrium in this setting. The analysis reveals that competition driven by externalization compels otherwise internalizing market makers to increase external trading, thereby raising aggregate hedging costs and significantly widening the bid–ask spreads faced by clients. This mechanism uncovers an endogenous source of liquidity deterioration in hybrid market-making environments, offering new insights into market maker behavior and the microstructure of financial markets.

This paper addresses the optimal execution problem under time- and size-constrained trading scenarios. Methodologically, it proposes a reinforcement learning (RL)-based autonomous decision-making framework that pioneers an end-to-end training paradigm integrating the ABIDES multi-agent market simulator with a customized Markov Decision Process (MDP) formulation. The framework dynamically learns execution policies directly from real-time limit-order-book (LOB) states, eliminating reliance on historical market data. It incorporates domain-informed LOB feature engineering and adapts both Proximal Policy Optimization (PPO) and Deep Q-Network (DQN) algorithms. Empirical evaluation demonstrates significant and robust improvements over benchmark strategies—including TWAP and VWAP—across key metrics: transaction cost (market impact), order completion rate, and timing risk. These results validate the efficacy and practicality of simulation-driven RL for algorithmic trade execution.

This paper investigates the complete-information Nash equilibrium among brokers and two types of clients—informative and uninformed—in a limit-order-book market, focusing on dynamic strategic interactions under instantaneous and transient price impacts and exponential price resilience. We develop the first tripartite dynamic game framework encompassing brokers, informed traders, and uninformed traders, and derive closed-form equilibrium strategies by solving a coupled system of forward–backward stochastic differential equations (FBSDEs). Theoretically, we characterize, for the first time, the equilibrium trade-off among informational advantage, profit objectives, and inventory risk. Empirically, our model quantifies how information asymmetry systematically distorts broker market-making behavior and informed traders’ execution paths, thereby enhancing the predictability of market equilibria and the accuracy of policy simulations. (149 words)

This paper investigates liquidity competition between brokers and informed traders in a multi-market-maker environment, addressing the trade-off among adverse selection, inventory risk, and transaction costs. We formulate a sequential Stackelberg game model that captures market makers’ internalization–externalization strategies and commitment-based liquidity pricing. Methodologically, we integrate game theory, stochastic optimal control, and nonlinear equilibrium analysis. Our contributions are threefold: (i) we derive, for the first time, a closed-form analytical solution for the informed trader’s optimal cross-broker strategy; (ii) we construct and numerically solve a system of equilibrium classification equations for the multi-market-maker setting, rigorously establishing equilibrium existence; and (iii) we demonstrate that commitment-driven competition leads to a non-Pareto-efficient equilibrium—specifically, a liquidity price war inducing systemic inefficiency. The analysis provides novel theoretical insights into the microstructure implications of precommitment and strategic fragmentation in modern equity markets.

This paper addresses the optimal liquidation problem under progressively measurable price prediction signals, incorporating both instantaneous and Volterra-type transient price impact—including singular power-law kernels. We propose the first framework jointly modeling predictive signals and the Volterra propagator, and solve it via infinite-dimensional stochastic control, yielding explicit analytical solutions: a free-boundary backward stochastic differential equation (BSDE) and an operator-valued Riccati equation. Our theoretical contributions are threefold: (i) we establish, for the first time, existence and uniqueness of the optimal strategy under signal-driven liquidation with singular Volterra kernels; (ii) we derive a closed-form expression for the optimal trading rate; and (iii) the resulting algorithm is computationally efficient and applicable to a broad class of price impact kernels. These results provide a rigorous mathematical foundation and practical execution strategies for high-frequency trading and algorithmic order execution.

This study addresses the problem of optimal dynamic quoting at the individual order level in a limit order book, rather than merely controlling aggregate trading rates. By developing a stochastic dynamic programming framework that incorporates signal-dependent drift, price impact, inventory risk, and execution risk, the authors combine Hamilton-Jacobi-Bellman equations, point-process modeling of fill intensity, CARA utility, and a triangular finite-dimensional structural approach to derive, for the first time, explicit analytical solutions for four canonical execution objectives. The resulting fully explicit value functions and optimal quoting strategies not only reveal intrinsic connections among different objectives and enable asymptotic analysis over long horizons, but also demonstrate well-posedness and practical implementability through numerical experiments, highlighting the critical role of signal-dependent drift in shaping optimal execution strategies.

This study addresses the trade-off among execution probability, adverse selection, and opportunity cost in limit order trading by proposing a mesoscale optimal passive execution strategy. Embedding two empirically observed microstructural features—namely, the exponential decay of limit order fill probability with quote distance and the short-term linear price response to order flow imbalance—into a stochastic control framework, the work derives for the first time a passively induced market impact rate exhibiting exponential decay and solves for the corresponding optimal liquidation policy. The model is validated on both NASDAQ equity and foreign exchange data and extends naturally to settings involving heterogeneous decay rates, instantaneous impact, and target execution schedules, thereby establishing a theoretical foundation and practical mechanism for tactical passive execution.

This study addresses the optimal execution and fair pricing of financial derivatives—including total return swaps, collar options, and TWAP Asian contracts—in merger and acquisition transactions under price impact. Employing the principle of utility indifference alongside stochastic control and a linear market impact model, the paper derives optimal execution strategies and equitable fees for both cash-settled and physically delivered contracts. Its key contribution lies in demonstrating that linear cash-settled contracts are particularly vulnerable to market manipulation and statistical arbitrage, while also providing the first systematic quantification of the sensitivity of nonlinear and Asian-style contracts to such risks. These findings offer a theoretical foundation for the design and regulation of complex derivatives in markets with liquidity frictions.

This work derives, from first principles, a uniquely determined market maker quoting rule by imposing eight natural axioms and six environmental assumptions. It establishes that the mid-price is linear in inventory and that the bid-ask spread decomposes into components attributable to inventory risk and adverse selection. The study presents the first axiomatic framework guaranteeing the enforced uniqueness of the quoting rule, reveals that the limit order book encodes a recoverable inventory cost function, and uncovers a sharp phase transition between functional and frozen market states. Combining axiomatic derivation, functional equation analysis, and structural identification, the approach yields three identifiable parameters—each estimable from distinct observable moments—and establishes four meta-theoretic invariants that hold across all structural primitives.

This study addresses the challenge of executing large orders in continuous double-auction markets under time and liquidity constraints. The authors propose a risk-constrained model predictive control (MPC) framework that dynamically optimizes trading decisions via quadratic programming while tracking benchmark schedules such as TWAP or VWAP. The approach permits strategic deviations from the benchmark to minimize expected execution cost, explicitly incorporating benchmark residual cost into the objective function to enable modular, data-driven deployment in live trading environments. Empirical evaluation using six months of NASDAQ Level 3 data demonstrates that the method reduces execution schedule gaps by 40–50% compared to cross-price benchmarks and significantly mitigates slippage. Performance improves further when integrated with price forecasts.