随机波动率费曼路径积分股指期权定价

Pricing of stochastic volatility stock index option based on Feynman path integral

采用量子力学中的费曼路径积分方法,推导出了更符合市场一般化情形的随机波动率股指期权定价模型.在此基础上,以恒指期权为例进行实证研究预测30天的期权价格,同时将Heston模型作为对照组,并进行稳健性检验.研究结果表明,本文构建的股指期权定价模型通过求解费曼定价核的数值解,进而在线性算法上直接实现股指期权价格的预测,相比于Heston模型利用特征函数的方法,不论是在相同到期日不同执行价格下还是在相同执行价格不同到期日下,定价精度显著提高.费曼路径积分作为量子金融的主要方法,本文的研究将为其进一步应用于金融衍生品定价提供参考.

Under the background that stock index options urgently need launching in China, the research on stock option pricing model has important theoretical and practical significance. In the traditional B-S-M model it is assumed that the volatility remains unchanged, which differs tremendously from the market’s reality. When the market fluctuates drastically, it is difficult to realize the risk management function of stock index options.Although in the Heston model, as one of the traditional stochastic volatility option pricing models, the correlation risk between the volatility and underlying price is taken into consideration, its pricing accuracy is still to be improved. From the quantum finance perspective, in this paper we use the Feynman path integral method to explore a more practical stock index option pricing model.In this paper, we construct a Feynman path integral pricing model of stock index options with stochastic volatility by taking Hang Seng index option as the research object and Heston model as the control group. It is found that the Feynman path integral pricing model is significantly superior to the Heston model either at different strike prices on the same expiration date or at different expiration dates for the same strike price. The stock index option pricing model constructed in this paper can give the numerical solution of Feynman’s pricing kernel, and directly realizes the forecast of stock index option price. The pricing accuracy is significantly improved compared with the pricing accuracy given by the Heston model through using the characteristic function method.The remarkable advantages of Feynman path integral stock index option pricing model are as follows.Firstly, the path integral has advantages in solving multivariate problems: the Feynman pricing kernel represents all the information about pricing and can be easily expanded from one-dimensional to multidimensional case, so the change of closing price of stock index and volatility of underlying index can be taken into account simultaneously. Secondly, based on the relationship between the Feynman path generation principle and the law of large number, the mean values of pricing kernel obtained by MATLAB not only optimizes the calculation process, but also significantly improves the pricing accuracy. Feynman path integral is the main method in quantum finance, and the research in this paper will provide reference for its further application in the pricing of financial derivatives.