Optimal Control

Option market making with hedging-induced market impact ArXiv ID: 2511.02518 “View on arXiv” Authors: Paulin Aubert, Etienne Chevalier, Vathana Ly Vath Abstract This paper develops a model for option market making in which the hedging activity of the market maker generates price impact on the underlying asset. The option order flow is modeled by Cox processes, with intensities depending on the state of the underlying and on the market maker’s quoted prices. The resulting dynamics combine stochastic option demand with both permanent and transient impact on the underlying, leading to a coupled evolution of inventory and price. We first study market manipulation and arbitrage phenomena that may arise from the feedback between option trading and underlying impact. We then establish the well-posedness of the mixed control problem, which involves continuous quoting decisions and impulsive hedging actions. Finally, we implement a numerical method based on policy optimization to approximate optimal strategies and illustrate the interplay between option market liquidity, inventory risk, and underlying impact. ...

Gaining efficiency in deep policy gradient method for continuous-time optimal control problems ArXiv ID: 2502.14141 “View on arXiv” Authors: Unknown Abstract In this paper, we propose an efficient implementation of deep policy gradient method (PGM) for optimal control problems in continuous time. The proposed method has the ability to manage the allocation of computational resources, number of trajectories, and complexity of architecture of the neural network. This is, in particular, important for continuous-time problems that require a fine time discretization. Each step of this method focuses on a different time scale and learns a policy, modeled by a neural network, for a discretized optimal control problem. The first step has the coarsest time discretization. As we proceed to other steps, the time discretization becomes finer. The optimal trained policy in each step is also used to provide data for the next step. We accompany the multi-scale deep PGM with a theoretical result on allocation of computational resources to obtain a targeted efficiency and test our methods on the linear-quadratic stochastic optimal control problem. ...

Market Making with Fads, Informed, and Uninformed Traders ArXiv ID: 2501.03658 “View on arXiv” Authors: Unknown Abstract We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads – these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad, we model how informed traders and uninformed traders arrive in the market. The market maker knows of the two groups of traders but only observes the anonymous order arrivals. We study both, the complete information and the partial information versions of the control problem faced by the market maker. In such frameworks, we characterise the value of information, and we find the price of liquidity as a function of the proportion of informed traders in the market. Lastly, for the partial information setup, we explore how to go beyond the Kalman-Bucy filter to extract information about the fad from the market arrivals. ...

Finite Element Method for HJB in Option Pricing with Stock Borrowing Fees ArXiv ID: 2501.02327 “View on arXiv” Authors: Unknown Abstract In mathematical finance, many derivatives from markets with frictions can be formulated as optimal control problems in the HJB framework. Analytical optimal control can result in highly nonlinear PDEs, which might yield unstable numerical results. Accurate and convergent numerical schemes are essential to leverage the benefits of the hedging process. In this study, we apply a finite element approach with a non-uniform mesh for the task of option pricing with stock borrowing fees, leading to an HJB equation that bypasses analytical optimal control in favor of direct PDE discretization. The time integration employs the theta-scheme, with initial modifications following Rannacher`s procedure. A Newton-type algorithm is applied to address the penalty-like term at each time step. Numerical experiments are conducted, demonstrating consistency with a benchmark problem and showing a strong match. The CPU time needed to reach the desired results favors P2-FEM over FDM and linear P1-FEM, with P2-FEM displaying superior convergence. This paper presents an efficient alternative framework for the HJB problem and contributes to the literature by introducing a finite element method (FEM)-based solution for HJB applications in mathematical finance. ...

Ergodic optimal liquidations in DeFi ArXiv ID: 2411.19637 “View on arXiv” Authors: Unknown Abstract We address the liquidation problem arising from the credit risk management in decentralised finance (DeFi) by formulating it as an ergodic optimal control problem. In decentralised derivatives exchanges, liquidation is triggered whenever the parties fail to maintain sufficient collateral for their open positions. Consequently, effectively managing and liquidating disposal of positions accrued through liquidations is a critical concern for decentralised derivatives exchanges. By simplifying the model (linear temporary and permanent price impacts, simplified cash balance dynamics), we derive the closed-form solutions for the optimal liquidation strategies, which balance immediate executions with the temporary and permanent price impacts, and the optimal long-term average reward. Numerical simulations further highlight the effectiveness of the proposed optimal strategy and demonstrate that the simplified model closely approximates the original market environment. Finally, we provide the method for calibrating the parameters in the model from the available data. ...

Simultaneously Solving FBSDEs and their Associated Semilinear Elliptic PDEs with Small Neural Operators ArXiv ID: 2410.14788 “View on arXiv” Authors: Unknown Abstract Forward-backwards stochastic differential equations (FBSDEs) play an important role in optimal control, game theory, economics, mathematical finance, and in reinforcement learning. Unfortunately, the available FBSDE solvers operate on \textit{“individual”} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs, as these solvers must be re-run several times. \textit{“Neural operators”} (NOs) offer an alternative approach for \textit{“simultaneously solving”} large families of decoupled FBSDEs by directly approximating the solution operator mapping \textit{“inputs:”} terminal conditions and dynamics of the backwards process to \textit{“outputs:”} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. Upon making only a few simple theoretically-guided tweaks to the standard convolutional NO build, we confirm that ``small’’ NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms using a logarithmic depth, a constant width, and a polynomial rank in the reciprocal approximation error. This result is rooted in our second result, and main contribution to the NOs for PDE literature, showing that our convolutional NOs of similar depth and width but grow only \textit{“quadratically”} (at a dimension-free rate) when uniformly approximating the solution operator of the associated class of semilinear Elliptic PDEs to these families of FBSDEs. A key insight into how NOs work we uncover is that the convolutional layers of our NO can approximately implement the fixed point iteration used to prove the existence of a unique solution to these semilinear Elliptic PDEs. ...

Can market volumes reveal traders’ rationality and a new risk premium? ArXiv ID: 2406.05854 “View on arXiv” Authors: Unknown Abstract An empirical analysis, suggested by optimal Merton dynamics, reveals some unexpected features of asset volumes. These features are connected to traders’ belief and risk aversion. This paper proposes a trading strategy model in the optimal Merton framework that is representative of the collective behavior of heterogeneous rational traders. This model allows for the estimation of the average risk aversion of traders acting on a specific risky asset, while revealing the existence of a price of risk closely related to market price of risk and volume rate. The empirical analysis, conducted on real data, confirms the validity of the proposed model. ...