In this paper, a continuous-time Markov chain (CTMC) approach is proposed to solve the problem of American option pricing under stochastic local volatility (SLV) models. The early exercise premium (EEP) representation of the value function, which contains the corresponding European option term and the EEP term, is in general not available in closed-form. We propose to use the CTMC to approximate the underlying asset, and derive explicit closed-form expressions for both the European option term and the EEP term, so that the integral equation characterizing the early exercise surface can be explicitly expressed through characteristics of the CTMC. The integral equations are then solved by the iteration method and the early exercise surface can be computed, and semi-explicit expressions for the values and Greeks of American options are derived. We denote the new method as the CTMC integral equation method, and establish both the theoretical convergence and the precise convergence order. Numerical examples are given for the classical Black-Scholes model and the general stochastic (local) volatility models, such as the stochastic alpha beta rho (SABR) model, the Heston model, the $4/2$ model and the $\alpha -$hypergeometric models. They illustrate the high accuracy and efficiency of the method.

在本文中，提出了一种连续时间马尔可夫链（CTMC）方法来解决随机局部波动率（SLV）模型下的美式期权定价问题。包含相应的欧式期权条款和 EEP 条款的价值函数的提前行权溢价 (EEP) 表示通常在封闭形式中不可用。我们建议使用 CTMC 来近似标的资产，并为欧式期权项和 EE​​P 项导出明确的封闭式表达式，以便通过 CTMC 的特征可以明确表达表征早期行权面的积分方程。然后通过迭代法求解积分方程，计算出早期行使面，并推导出美式期权的价值和希腊字母的半显式表达式。我们将新方法记为 CTMC 积分方程法，并建立了理论收敛性和精确收敛阶数。给出了经典 Black-Scholes 模型和一般随机（局部）波动率模型的数值例子，如随机 alpha beta rho (SABR) 模型、Heston 模型、$4/2$ 模型和 $\alpha -$超几何模型。它们说明了该方法的高精度和高效率。