英文标题：
《Implied Stopping Rules for American Basket Options from Markovian
  Projection》
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作者：
Christian Bayer, Juho H\\\"app\\\"ol\\\"a, Ra\\\'ul Tempone
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最新提交年份：
2017
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英文摘要：
  This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and the Black-Scholes models. In high dimensions, nonlinear partial differential equation methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. It is shown that the ability to approximate the original value function by a lower-dimensional approximation is a feature of the dynamics of the system and is unaffected by the path-dependent nature of the American basket option. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early-exercise boundary of the option by solving a Hamilton-Jacobi-Bellman partial differential equation in the projected, low-dimensional space. The resulting near-optimal early-exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow to assess the accuracy of the proposed pricing method. Indeed, our approximate early-exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to fifty. In these examples, the resulting option price relative errors are only of the order of few percent.
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中文摘要：
这项工作解决了多变量环境下的美式篮子期权定价问题，其中包括Bachelier和Black-Scholes模型。在高维情况下，由于维数灾难，用于解决问题的非线性偏微分方程方法变得成本高昂。相反，这项工作建议使用一个停止规则，该规则取决于给定一篮子资产的低维马尔可夫投影的动态。结果表明，通过低维近似逼近原始值函数的能力是系统动力学的一个特征，不受美式篮子期权的路径依赖性的影响。假设我们知道正向过程的密度，并使用拉普拉斯近似，我们首先有效地计算对应于篮子的低维马尔可夫投影的扩散系数。然后，我们通过在投影的低维空间中求解Hamilton-Jacobi-Bellman偏微分方程来逼近期权的最优提前行使边界。由此产生的接近最优的早期行使边界用于生成高维期权的行使策略，从而提供美式篮子期权价格的下限。还提供了相应的上限。这些界限允许评估拟议定价方法的准确性。事实上，我们的近似提前行使策略为美国篮子期权价格提供了一个简单的下限。根据Rogers提出的对偶论证，我们导出了仅解决低维最优控制问题的相应上界。从数值上看，我们使用尺寸高达50的篮子证明了该方法的可行性。在这些例子中，由此产生的期权价格相对误差只有几个百分点。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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