We propose a deep learning method for solving the American options model with a free boundary feature. To extract the free boundary known as the early exercise boundary from our proposed method, we introduce the Landau transformation. For efficient implementation of our proposed method, we further construct a dual solution framework consisting of a novel auxiliary function and free boundary equations. The auxiliary function is formulated to include the feed forward deep neural network (DNN) output and further mimic the far boundary behaviour, smooth pasting condition, and remaining boundary conditions due to the second-order space derivative and first-order time derivative. Because the early exercise boundary and its derivative are not a priori known, the boundary values mimicked by the auxiliary function are in approximate form. Concurrently, we then establish equations that approximate the early exercise boundary and its derivative directly from the DNN output based on some linear relationships at the left boundary. Furthermore, the option Greeks are obtained from the derivatives of this auxiliary function. We test our implementation with several examples and compare them with the existing numerical methods. All indicators show that our proposed deep learning method presents an efficient and alternative way of pricing options with early exercise features.
翻译:我们提出一个深层次的学习方法,以解决具有自由边界特征的美国选项模式。为了从我们提议的方法中提取被称为早期行使边界的自由边界,我们引入了Landau改造。为了高效实施我们提议的方法,我们进一步构建了一个由新型辅助功能和自由边界方程式组成的双重解决方案框架。辅助功能的制定包括进料深层神经网络输出,并进一步模仿远端边界行为、平稳的粘贴条件以及第二阶层空间衍生物和第一阶时间衍生物导致的剩余边界条件。由于早期行使边界及其衍生物并不为人所知,因此由辅助功能模拟的边界值是近似形式的。与此同时,我们根据左界的某些线性关系,建立接近早期行使边界及其直接来自DNN输出的方程式。此外,希腊人从这一辅助功能的衍生物中获得选择权。我们用几个实例测试我们的执行情况,并将它们与现有的数字方法进行比较。所有指标都表明,我们提议的深层次学习方法提出了一种高效的替代定价方法,并带有早期行使特性。