We present a new approach to bounding financial derivative prices in regime-switching market models from both above and below. We derive sufficient conditions under which a particular class of functions act as bounds for the prices of financial derivatives in regime-switching market models. Using these sufficient conditions, we then formulate, in a general setting, optimization problems whose solutions can be identified with tight upper and lower bounds. The problems are made numerically tractable by imposing polynomial structures and employing results from the theory of sum-of-squares polynomials to arrive at a semidefinite programming problem that is implementable by existing software. The bounds obtained take the form of smooth polynomial functions and are valid for a continuous range of initial times and states. Moreover, they are obtained without recourse to sample path simulation or discretization of the temporal or spatial variables. We demonstrate the effectiveness of the proposed method on European-, barrier- and American-style options in several regime-switching settings with and without jumps.

中文翻译：

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制度转换下金融衍生品价格函数的多项式上下界

我们提出了一种新的方法来从上方和下方限制制度转换市场模型中的金融衍生品价格。我们推导出了充分条件，在该条件下，特定类别的函数作为制度转换市场模型中金融衍生品价格的界限。使用这些充分条件，我们然后在一般情况下制定优化问题，其解决方案可以通过严格的上限和下限确定。通过强加多项式结构并使用平方和多项式理论的结果来解决现有软件可实现的半定规划问题，这些问题在数值上变得易于处理。获得的边界采用平滑多项式函数的形式，并且对初始时间和状态的连续范围有效。而且，它们无需借助样本路径模拟或时间或空间变量的离散化即可获得。我们证明了所提出的方法在有和没有跳跃的几种政权转换设置中对欧式、障碍式和美式选项的有效性。