基于参数学习的GARCH动态无穷活动率Lévy过程的欧式期权定价

在股票价格中引入漂移率、波动率和随机跳跃三种状态， 建立动态状态空间模型， 并通过局部风险中性定价关系（RNVR）推导无套利定价模型. 以非高斯条件ARMA-NGARCH为基准模型， 构建S&P500指数的离散动态Lévy过程， 并基于序贯贝叶斯的参数学习方法， 进行模型估计和期权定价研究. 结果表明： 动态Lévy过程能够联合刻画时变漂移率、条件波动率和无穷活动率等特征， 且贝叶斯方法的引入提高了期权隐含波动率的定价精度. 同时， 无穷活动率模型在期权定价方面具有显著优势. 在五类滤波中， 无损粒子滤波估计精度最高， 速降调和稳态过程（RDTS）的期权定价误差最小， 而非高斯模型在收益率预测方面没有表现出显著的差异.

European option pricing for GARCH dynamic infinite activity Lévy processes based on parameter learning

In this paper, we consider a three-dimension state space model for establishing a discrete-time dynamic Lévy process, including time-varying drift, conditional volatility and stochastic jump activity. Then we obtain the equivalent non-arbitrage pricing model through local risk-neutral valuation relationship (RNVR). Taking non-Gaussian ARMA-NGARCH model as our benchmark, we construct a discrete time dynamic Lévy process with GARCH effect for modeling S&P500 index. Furthermore we jointly estimate the parameters of the model and study the option pricing performance based on Bayesian learning approach. Research results show that our dynamic Lévy process can depict the time-varying drift rate, conditional volatility and infinite activity styles. Meanwhile, Bayesian approach improves the option valuation ability of our model. Infinite jump models are significant superior and increase the pricing accuracy of implied volatility. We also find that unscented particle filtering (UPF) has the best estimation performance, non-Gaussian models in the yield prediction are of no significant difference, but the rapidly decreasing tempered stable processes (RDTS) have minimum errors for option pricing.