HJB Equation

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. ...

Optimal consumption under loss-averse multiplicative habit-formation preferences ArXiv ID: 2406.20063 “View on arXiv” Authors: Unknown Abstract This paper studies a loss-averse version of the multiplicative habit formation preference and the corresponding optimal investment and consumption strategies over an infinite horizon. The agent’s consumption preference is depicted by a general S-shaped utility function of her consumption-to-habit ratio. By considering the concave envelope of the S-shaped utility and the associated dual value function, we provide a thorough analysis of the HJB equation for the concavified problem via studying a related nonlinear free boundary problem. Based on established properties of the solution to this free boundary problem, we obtain the optimal consumption and investment policies in feedback form. Some new and technical verification arguments are developed to cope with generality of the utility function. The equivalence between the original problem and the concavified problem readily follows from the structure of the feedback controls. We also discuss some quantitative properties of the optimal policies, complemented by illustrative numerical examples and their financial implications. ...

On optimal tracking portfolio in incomplete markets: The reinforcement learning approach ArXiv ID: 2311.14318 “View on arXiv” Authors: Unknown Abstract This paper studies an infinite horizon optimal tracking portfolio problem using capital injection in incomplete market models. The benchmark process is modelled by a geometric Brownian motion with zero drift driven by some unhedgeable risk. The relaxed tracking formulation is adopted where the fund account compensated by the injected capital needs to outperform the benchmark process at any time, and the goal is to minimize the cost of the discounted total capital injection. When model parameters are known, we formulate the equivalent auxiliary control problem with reflected state dynamics, for which the classical solution of the HJB equation with Neumann boundary condition is obtained explicitly. When model parameters are unknown, we introduce the exploratory formulation for the auxiliary control problem with entropy regularization and develop the continuous-time q-learning algorithm in models of reflected diffusion processes. In some illustrative numerical example, we show the satisfactory performance of the q-learning algorithm. ...

Optimal execution and speculation with trade signals ArXiv ID: 2306.00621 “View on arXiv” Authors: Unknown Abstract We propose a price impact model where changes in prices are purely driven by the order flow in the market. The stochastic price impact of market orders and the arrival rates of limit and market orders are functions of the market liquidity process which reflects the balance of the demand and supply of liquidity. Limit and market orders mutually excite each other so that liquidity is mean reverting. We use the theory of Meyer-$σ$-fields to introduce a short-term signal process from which a trader learns about imminent changes in order flow. Her trades impact the market through the same mechanism as other orders. With a novel version of Marcus-type SDEs we efficiently describe the intricate timing of market dynamics at moments when her orders concur with that of others. In this setting, we examine an optimal execution problem and derive the Hamilton–Jacobi–Bellman (HJB) equation for the value function of the trader. The HJB equation is solved numerically and we illustrate how the trader uses the signals to enhance the performance of execution problems and to execute speculative strategies. ...