教育部人文社科重点研究基地中央财经大学中国精算研究院学术活动

精算论坛第192期讲座

(2021年12月9日下午3:00-5:00)

讲座主题：Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models

摘要： In this talk we present different papers [1], [2], [3], [4] using Machine Learning techniques for option pricing. First [1], we develop an efficient approach based on a Machine Learning technique which allows one to quickly evaluate insurance products considering stochastic volatility and interest rate. Specifically, we apply Gaussian Process Regression to compute the price and the Greeks of a GMWB Variable Annuity. Numerical experiments show that the accuracy of the estimated values is high, while the computational cost is much lower than the one required by a direct calculation with standard approaches.
Second [2] [3], we consider three efficient methods which allow one to compute the price of American basket options in the multi-dimensional Black-Scholes model. The proposed methods, which are based on Gaussian Process Regression, are termed GPR Monte Carlo, GPR Tree and GPR Exact Integration. Specifically, they are backward dynamic programming algorithms which consider a finite number of uniformly dis tributed exercise dates. At each time step, the value of the option is computed as the maximum between the exercise value and the continuation value. This is done only for a finite set of points and then Gaussian Process Regression is exploited to approximate the whole value function. The technique employed to compute the continuation value identifies each of the three proposed methods: GPR-Monte Carlo employs Monte Carlo simulation, GPR-Tree a Tree step and GPR-Exact Integration a semi-analytical formula for integration. In order to improve the dimension range, we employ the European option price as a control variate, which allows us to treat very large baskets and moreover to reduce the variance of price estimators. Numerical tests show that the algorithms are fast and reliable, and they can be used to price American options on very large baskets of assets, overcoming the problem of the curse of dimensionality. Moreover, we also consider the rough Bergomi model, which provides stochastic volatility with memory, and we present how to adapt the GPR-Tree and GPR-Exact Integration methods for pricing American options in this non-Markovian framework.

Finally [4], we consider moving average options. Evaluating moving average options is a tough compu- tational challenge for the energy and commodity market as the payoff of the option depends on the prices of a certain underlying observed on a moving window so, when a long window is considered, the pricing problem becomes high dimensional. We present an efficient method for pricing Bermudan style moving aver- age options, based on Gaussian Process Regression and Gauss-Hermite quadrature, thus named GPR-GHQ. Specifically, the proposed algorithm proceeds backward in time and, at each time-step, the continuation value is computed only in a few points by using Gauss-Hermite quadrature, and then it is learned through Gaussian Process Regression. We test the proposed approach in the Black-Scholes model, where the GPR- GHQ method is made even more efficient by exploiting the positive homogeneity of the continuation value, which allows one to reduce the problem size. Positive homogeneity is also exploited to develop a binomial Markov chain, which is able to deal efficiently with medium-long windows. Secondly, we test GPR-GHQ in the Clewlow-Strickland model, the reference framework for modeling prices of energy commodities. Finally, we consider a challenging problem which involves double non-Markovian feature, that is the rough-Bergomi model. In this case, the pricing problem is even harder since the whole history of the volatility process impacts the future distribution of the process. The manuscript includes a numerical investigation, which displays that GPR-GHQ is very accurate and it is able to handle options with a very long window, thus overcoming the problem of high dimensionality.

References

[1] L.Goudenege A.Molent A.Zanette 2021 Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate. Decisions Economics and Finance, Volume 44, Issue 1, 57–72

[2] L.Goudenege A.Molent A.Zanette 2020 Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models. Quantitative Finance, Vol. 20, No. 4, 573-591.

[3] L.Goudenege A.Molent A.Zanette 2021 Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension. Applications of Levy Procesess Edited by O. Kudryavtsev and A. Zanette, Nova Science Publishers, Inc New York Chapter 1 1-32.

[4] L.Goudenege A.Molent A.Zanette 2021 Moving average options: Machine Learning and Gauss-Hermite quadrature for a double non-Markovian problem. arXiv:2108.11141

报告人：Antonino Zanette

意大利乌迪内大学（University of Udine）经济与统计学院教授，法国国家信息与自动化研究院（INRIA）创办的金融衍生品定价与风险管理平台PREMIA的学术主管。他的研究领域是数量金融，主要从事二叉树方法等数值方法的设计与优化。他提出的混合树方法（Hybrid tree method）解决了很多复杂结构的金融衍生品和保险产品的定价问题，在Insurance：Mathematics and Economics、 Quantitative Finance等期刊发表了关于该方法及其应用的一系列成果。

报告人：Ludovic Goudenege

法国国家科学研究院（CNRS）研究员，巴黎中央理工大学（École Centrale Paris)教授。他博士毕业于法国卡尚高等师范（École Normale Supérieur de Cachan），随后在法国巴黎六大（Université Pierre Marrie-Currie）师从法国著名金融数学大师Nicole El Karoui进行博士后研究工作。现任职于法国国家科学研究院，同时也是巴黎中央理工大学的教授，从事的研究领域主要包括随机过程、遍历理论、金融衍生品及保险产品定价、随机偏微分方程、数值模拟方法等。目前已在《Stochastic Processes and its Applications》、《SIAM Journal of Mathematics Analysis》《Risk Magazine》《Journal of Scientific Computation》等学术期刊发表多篇论文。除此之外，Ludovic Goudenege 教授也致力于将学术研究应用于保险金融业界，他兼职于法国安盛保险集团AXA，为其提供风险管理咨询，同时也是法国国家信息与自动化研究院金融软件平台PREMIA的研发团队成员。

讲座时间：2021年12月9日 下午3:00-5:00

报告地点：Zoom会议（会议ID：964 1553 7094，密码：957084。国内用户无须登陆账号，直接进入会议即可。）

邀 请 人：韦晓

欢迎各位老师和同学积极参加！