[论文解读] Machine Learning Tree and Exact Integration for Pricing American Options in High Dimension

Ludovic Goudenège, Andrea Molent+1 more|arXiv (Cornell University)|May 22, 2019

Stochastic processes and financial applications被引用 1

一句话总结

本文提出GPR-Tree与GPR-Exact Integration（GPR-EI）两种机器学习增强方法，用于在高维情形下定价美式篮子期权。通过将高斯过程回归与二项树或解析积分相结合，以在逆向动态规划中估计继续持有价值，该方法在大型篮子及粗糙伯格米等非马尔可夫过程下仍能实现高精度与高效率。

ABSTRACT

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black-Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. Moreover, these two methods solve the backward dynamic programming problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite this model is only bidimensional, the whole history of the process impacts on the price, and handling all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.

研究动机与目标

• 开发在多维布莱克-斯科尔斯动态下对大规模资产篮子的美式期权进行高效且精确定价的方法。
• 解决高维美式期权定价的计算挑战，特别是当标的资产数量较多时。
• 将这些方法扩展至非马尔可夫情形，例如粗糙伯格米模型，其中完整路径历史影响期权价格。
• 提供一个稳健的框架，将机器学习与动态规划相结合，以近似最优行权策略。

提出的方法

• 该方法采用美式期权的伯姆丹近似，逐步求解逆向动态规划问题。
• 在每个行权日，期权价值被计算为立即行权价值与继续持有价值的最大值。
• 继续持有价值通过高斯过程回归（GPR）估计，其训练基于标的资产路径的模拟结果。
• GPR-Tree通过单步二项树近似计算继续持有价值。
• GPR-EI通过多维过程的概率密度进行精确积分计算继续持有价值。
• 通过将完整路径依赖性纳入GPR输入特征，该方法被适配至粗糙伯格米模型。

实验结果

研究问题

• RQ1 如何有效结合机器学习与动态规划，以高效定价高维美式期权？
• RQ2 高斯过程回归能否在大规模资产篮子的高维情形下准确近似继续持有价值？
• RQ3 GPR-Tree与GPR-EI方法如何适配至粗糙伯格米模型等非马尔可夫过程？
• RQ4 与传统方法相比，这些方法在高维期权定价中的精度与计算性能如何？

主要发现

• GPR-Tree与GPR-EI方法在多维布莱克-斯科尔斯动态下对大规模资产篮子的美式期权定价中表现出高精度与高可靠性。
• 这些方法展现出计算效率，使其适用于传统方法变得不可行的高维问题。
• GPR-Tree与GPR-EI对粗糙伯格米模型的适配成功处理了路径依赖性波动，实现了非马尔可夫情形下的精确定价。
• 数值结果表明，当标的资产数量显著增加时，两种方法仍能保持稳定与精确。