In this paper, we develop the lower–upper-bound approximation in the space of Laplace transforms for pricing American options. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, e.g. a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the corresponding “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures in the Laplace space. Finally, we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option and further utilize the early exercise premium representation in the Laplace space to obtain the upper bound. Numerical examples are conducted to compare the method with a variety of existing methods in the literature as benchmark to demonstrate the accuracy and efficiency.

中文翻译：

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美式期权的拉普拉斯边界近似

在本文中，我们开发了在拉普拉斯变换空间中为美式期权定价的上界近似。当标的股票由一大类随机过程（例如时间均匀扩散过程和跳跃扩散过程）建模时，我们为有限期限美式期权的价格构建了严格的下限和上限。该方法的新颖之处在于首先对相应的“上限（障碍）期权”的价格对到期时间进行拉普拉斯变换，然后在拉普拉斯空间中进行优化程序。最后，我们对拉普拉斯变换进行数值反转以获得美式期权价格的下限，并进一步利用拉普拉斯空间中的早期行使溢价表示来获得上限。