Stochastic volatility models can describe the evolution of financial assets, such as stocks, currencies, and commodities, better than the classic Black–Scholes model. Some strategic decision-making problems also involve path-dependent and American-style options. In this article, the authors propose a novel, efficient, accurate, and unified two-factor willow tree method to price exotic and American options under the stochastic volatility models, such as the Heston, 3/2, 4/2, Hull–White, Stein–Stein, and a-Hypergeometric models. They also present the convergence analysis of their proposed tree method. They then apply the tree method to price European and American options, and the expected present value and survival rate in a dividend-and-ruin problem. Numerical results demonstrate the efficiency, accuracy, and convergence of their method. TOPICS: Options, volatility measures, factor-based models, analysis of individual factors/risk premia Key Findings • The authors propose an efficient and unified two-dimensional willow tree structure for various stochastic volatility models. • The convergence rate of the two-dimensional willow tree method is O(Δt). • The authors apply the willow tree to evaluate the present firm value and survival rate of a dividend-and-ruin problem, which embeds the lookback, the reflecting and absorbing barrier, and the stopping time features.

中文翻译：

---

随机波动率模型下美国和异国期权定价的有效收敛柳树方法

随机波动率模型可以比经典的Black-Scholes模型更好地描述金融资产（例如股票，货币和商品）的演变。一些战略决策问题还涉及依赖路径和美国风格的选择。在本文中，作者提出了一种新颖，高效，准确和统一的两因素柳树方法，可以在随机波动率模型（例如Heston，3 / 2、4 / 2，Hull–White）下为奇异和美式期权定价，Stein–Stein和a超几何模型。他们还提出了他们提出的树法的收敛性分析。然后，他们将树法应用于欧洲和美国期权的定价，以及股息和破产问题中的预期现值和生存率。数值结果表明了该方法的有效性，准确性和收敛性。主题：选择权，波动率度量，基于因子的模型，单个因子/风险溢价的分析主要发现•作者针对各种随机波动率模型提出了一种有效且统一的二维柳树结构。•二维柳树法的收敛速度为O（Δt）。•作者运用柳树来评估股息和破产问题的当前公司价值和生存率，其中嵌入了回溯，反射和吸收障碍以及停止时间特征。