Abstract

This paper considers the application of the stochastic mesh method in solving the multidimensional optimal stopping problem for a diffusion process with nonlinear payoff functions. A special discretization scheme of the diffusion process is presented to solve the problem in the case of geometric average Asian option payoff functions. This discretization scheme makes it possible to eliminate singularities in transition probabilities. Next, two estimates are given of the solution of the problem by the stochastic mesh method for the case of the stochastic mesh transition probabilities defined as averaged densities. The consistency of the estimates is proven. It is shown that the variance of the estimates is inversely proportional to the number of points in each mesh layer. The result extends the application area of the stochastic mesh method and methods for treating Asian options. A numerical example of the result of applying the obtained estimates to the call and put options compared to the obtained option prices using a regular mesh is presented.

中文翻译：

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最优停止问题的随机网格方法

摘要

本文考虑了随机网格方法在求解具有非线性收益函数的扩散过程的多维最优停止问题中的应用。提出了一种特殊的离散过程离散化方案，以解决几何平均亚洲期权收益函数的问题。这种离散化方案可以消除转移概率中的奇异性。接下来，对于将随机网格转变概率定义为平均密度的情况，通过随机网格方法给出了对问题解的两个估计。估计的一致性已得到证明。结果表明，估计的方差与每个网格层中的点数成反比。结果扩展了随机网格方法和亚洲期权的处理方法的应用领域。给出了使用常规网格将获得的估计值应用于看涨期权和看跌期权的结果与获得的期权价格相比的数值示例。