基于模拟仿真方法的美式期权风险对冲参数计算问题研究

In this project we plan to study American option hedging problems, based on Monte Carlo simulation and dynamic programming methods from the perspective of Operations Research and Financial Engineering. Specifically, the research agenda mainly includes: (1) Establish a unified (two-dimensional) dynamic programming framework for the price and associated Greeks of an American option; (2) Develop efficient Monte Carlo simulation algorithms to estimate the price and Greeks simultaneously; (3) Conduct rigorous error and convergence analysis for the proposed estimates; (4) Develop lower/upper bounds simultaneously for option price and Greeks based on the emerging information relaxation techniques. The innovative contribution of this project is to simultaneously solve the problem of pricing and hedging an American option in a unified framework of dynamic programming. The research output of this project will substantially broaden the research domain of related literature. This project also has important implication to financial industry and regulation. American option is a huge class of financial derivatives which has been actively traded in major overseas securities and futures exchanges and over-the-counter markets. In Mainland China the authority just launched the stock option in Shanghai Stock Exchange and the regulation board plans to deeply develop our option market. Under this background, this prospective research project will improve the fair pricing and effective risk management abilities for financial industry and the regulators, and lay a solid foundation for the exchange trading of American options in Mainland China in the near future.

本项目拟从金融工程和运筹学的角度，运用模拟仿真以及动态规划的方法，研究美式期权风险对冲参数的计算问题，主要包括：（1）建立美式期权价格和风险对冲参数的二维动态规划统一框架；（2）提出基于模拟仿真的高效算法来同时估计价格和风险对冲参数的下界和上界；（3）给出价格及风险对冲参数的误差估计和收敛速度。本项目创新之处在于将在一个统一的动态规划框架下完整解决美式期权的价格和风险对冲的同时估计问题，所得成果将本质地推广现有相关文献的结果。本项目紧密联系实际，研究主题美式期权是一大类非常重要的金融衍生产品，在国外主要证券交易所以及期货交易所均有该产品活跃交易。在我国刚刚开始正式交易个股期权并即将大力发展场内和场外期权交易市场的大背景下，本前瞻性研究项目预期将提升监管当局和产业界对于该类产品的风险管理方式和定价水平，为我国未来开展美式期权交易打下基础。

本项目通过研究包括美式期权、欧式期权、亚式期权、障碍期权、股指期货在内的一系列重要的金融衍生产品的定价和风险对冲方法，取得了如下成果并发表于国内外主流期刊：1. 我们给出了美式期权风险对冲参数Vega的正确积分表示和计算方法；2.我们给出了在Levy过程下欧式期权风险对冲参数的模拟仿真估计；3. 我们给出了SABR模型下障碍期权的无套利定价公式以及连续监测型障碍期权的定价公式；4. 我们在一般性的马尔科夫过程框架下给出了亚式期权的基于单次数值变换的定价和风险对冲方法；5. 我们给出了美式期权的一种稳健性价格上界的计算方法；6. 我们发现了我国开展股指期货交易在一定程度上提高了股票现货市场的稳定性；7. 我们研究了我国金融市场中基金在泡沫资产配置上的模仿行为，发现我国开放式基金在泡沫资产配置上具有显著的同群效应。我们相信上述研究成果将在一定程度上丰富学术界关于金融衍生品定价和风险对冲的研究，为我国金融衍生品市场的健康发展提供了一些具有参考意义的学术成果。