We consider a pairs trading stochastic control problem with transaction costs and constraints on the gross market exposure, and propose a new monotone Finite Difference scheme approximating the viscosity solution of the Hamilton–Jacobi–Bellman equation characterizing the optimal trading strategies. Given a fixed time horizon and a portfolio of two cointegrated assets, the agent trades the spread between the two assets and the trading strategy is defined as the possibly negative portfolio weight maximizing the expected exponential utility derived from terminal wealth. Furthermore, trades incur transaction costs comprised of explicit transactions fees and commissions and the implicit cost due to slippage. These costs are modeled as a linear or square root function of the trading rate and respectively added or subtracted from the observable asset price at the time when a buy or a sell order enters the market. Our main contribution is the derivation of a robust approximation for the nonlinear transaction cost term in the Hamilton–Jacobi–Bellman equation. Finally, we combine our monotone Finite Difference scheme with a Monte Carlo sampling method to analyze the effects of transaction fees and slippage on the trading policies’ performance.

我们考虑具有交易成本和总市场敞口约束的配对交易随机控制问题，并提出了一种新的单调有限差分方案，该方案近似于表征最优交易策略的 Hamilton-Jacobi-Bellman 方程的粘度解。给定一个固定的时间范围和两个协整资产的投资组合，代理交易这两种资产之间的价差，交易策略被定义为最大化来自终端财富的预期指数效用的可能为负的投资组合权重。此外，交易产生的交易成本包括显性交易费用和佣金以及由于滑点导致的隐性成本。这些成本被建模为交易率的线性或平方根函数，并分别从买入或卖出订单进入市场时的可观察资产价格中添加或减去。我们的主要贡献是推导出 Hamilton-Jacobi-Bellman 方程中非线性交易成本项的稳健近似。最后，我们将我们的单调有限差分方案与蒙特卡罗抽样方法相结合，分析交易费用和滑点对交易政策表现的影响。