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Early exercise premium method for pricing American options under the J-model

机译： J -模型下的美国期权定价的提前行使溢价法

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Abstract Background This study develops a new model called J -am for pricing American options and for determining the related early exercise boundary (EEB). This model is based on a closed-form solution J -formula for pricing European options, defined in the study by Jerbi (Quantitative Finance, 15:2041–2052, 2015). The J -am pricing formula is a solution of the Black & Scholes (BS) PDE with an additional function called f as a second member and with limit conditions adapted to the American option context. The aforesaid function f represents the cash flows resulting from an early exercise of the option. Methods This study develops the theoretical formulas of the early exercise premium value related to three American option pricing models called J -am, BS-am, and Heston-am models. These three models are based on the J -formula by Jerbi (Quantitative Finance, 15:2041–2052, 2015), BS model, and Heston (Rev Financ Stud, 6:327–343, 1993) model, respectively. This study performs a general algorithm leading to the EEB and to the American option price for the three models. Results After implementing the algorithms, we compare the three aforesaid models in terms of pricing and the EEB curve. In particular, we examine the equivalence between J -am and Heston-am as an extension of the equivalence studied by Jerbi (Quantitative Finance, 15:2041–2052, 2015). This equivalence is interesting since it can reduce a bi-dimensional model to an equivalent uni-dimensional model. Conclusions We deduce that our model J -am exactly fits the Heston-am one for certain parameters values to be optimized and that all the theoretical results conform with the empirical studies. The required CPU time to compute the solution is significantly less in the case of the J -am model compared with to the Heston-am model.
机译：摘要背景本研究开发了一种称为J -am的新模型，用于对美式期权定价并确定相关的早期行使边界（EEB）。该模型基于Jerbi的研究（Quantitative Finance，15：2041-2052，2015）中定义的用于定价欧洲期权的封闭式解决方案J公式。 J -am定价公式是Black＆Scholes（BS）PDE的一种解决方案，它具有一个称为f的附加功能作为第二个成员，并且其限制条件适用于美国期权环境。上述函数f代表早期行使期权产生的现金流量。方法本研究开发了与三个美国期权定价模型J -am，BS-am和Heston-am模型相关的早期行使保费价值的理论公式。这三个模型分别基于Jerbi的J公式（量化金融，15：2041-2052，2015），BS模型和Heston（Rev Financ Stud，6：327-343，1993）模型。这项研究执行了三种模型的通用算法，从而得出了EEB和美国期权价格。结果实施算法后，我们在定价和EEB曲线方面比较了上述三个模型。尤其是，我们研究了J-am和Heston-am之间的等价关系，作为Jerbi研究的等价关系的扩展（Quantitative Finance，15：2041-2052，2015）。这种等效性很有趣，因为它可以将二维模型简化为等效的一维模型。结论我们得出结论，我们的模型J -am对于要优化的某些参数值完全适合Heston-am模型，并且所有理论结果都与经验研究相符。在J -am模型的情况下，与Heston-am模型相比，计算解决方案所需的CPU时间大大减少。

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