In this paper, we consider a two dimensional partial differential integral equation (PDIE) model for pricing American option. A nonlinear rationality parameter function for two asset problems is introduced to deal with the free boundary. The rationality parameter function is added in the PDIEs used for pricing American option problems under multi-state regime switching with jumps. The resulting two dimensional nonlinear system of PDIE is then numerically solved. Based on real poles rational approximation, a strongly stable highly efficient and reliable method is developed to solve such complicated systems of PIDEs. The method is build in a predictor corrector style which makes it linearly implicit, therefore, avoids solving nonlinear systems of equations at each time step in all regimes. The method is seen to maintain the stability and convergence for large jump sizes and high volatility in each regime. The impact of regime switching on option prices corresponding to different values interest rate, volatility, and rationality parameter is computed, illustrated by graphs and given in the tables. Convergence results in each regime are presented and time evolution graphs are given to show the effectiveness and reliability of the method.

中文翻译：

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多状态跳跃切换下美式期权定价的偏微分积分方程模型

在本文中，我们考虑了一个用于美式期权定价的二维偏微分积分方程 (PDIE) 模型。引入了两个资产问题的非线性合理性参数函数来处理自由边界。PDIEs中增加了合理性参数函数，用于在多状态跳跃切换下对美式期权问题进行定价。然后对所得到的 PDIE 二维非线性系统进行数值求解。基于实极点有理逼近，开发了一种强稳定高效可靠的方法来求解此类复杂的 PIDE 系统。该方法建立在预测校正器样式中，使其线性隐式，因此避免在所有方案的每个时间步求解非线性方程组。可以看出该方法在每个区域中保持大跳跃大小和高波动性的稳定性和收敛性。计算了制度转换对对应于不同值的利率、波动率和合理性参数的期权价格的影响，通过图表说明并在表格中给出。给出了每个方案的收敛结果，并给出了时间演化图，以显示该方法的有效性和可靠性。