随机利率随机波动率混合指数跳扩散模型下的期权定价  

Option Pricing Under the Mixed-Exponential Jump Diffusion Model with Stochastic Interest Rate and Stochastic Volatility

作　　者：吴胤昊 陈荣达 汪圣楠 俞静婧 WU Yinhao;CHEN Rongda;WANG Shengnan;YU Jingjing(School of Finance,Zhejiang University of Finance and Economics,Hangzhou 310018;Jiaxing University,Jiaxing 314001)

机构地区：[1]浙江财经大学金融学院,杭州310018 [2]嘉兴学院,嘉兴314001

出　　处：《系统科学与数学》2022年第8期2207-2234,共28页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金重点项目(71631005)资助课题.

摘　　要：文章在Heston模型的基础上同时将随机利率与混合指数跳引入模型,并且考虑到金融市场中已存在的负利率事实,因此不同于经典的CIR假设,文章放松了利率不能为负的限制,假定其满足Hull-White过程,提出了Heston-HW-MEJ混合模型.在模型的求解方面,文章推导了含对数价格特征函数的定价半闭解,并采用快速Fourier变换方法进行计算.数值分析的结果表明,文章提出的模型较好地捕捉了尖峰厚尾、盈亏不对称等市场特征与隐含波动率曲面形态,并且快速Fourier变换方法求解相比于Monte Carlo模拟快速、高效.最后通过香港恒生指数期权市场与上证50ETF期权市场数据对多个模型进行了实证比较,结果表明,文章的模型在低利率市场具有更好的定价效果,但在利率相对较高的市场略差于Heston-CIR-MEJ模型.Based on Heston's model,this paper introduces stochastic interest rate and mixed exponential jumps into the model.And taking into account the fact that there is already negative interest rate in the financial market,this paper tries to relax the restriction that interest rate cannot be negative,assuming that it satisfies the Hull-White process which is different from the classical CIR hypothesis.Finally,the hybrid Heston-HW-MEJ model is proposed.However,the complexity of the multi-stochastic element system makes the closed-form solution usually does not exist,so in terms of model solving,this paper adopts the fast Fourier transform method to obtain the semi-closed solution of pricing containing the characteristic function of logarithmic price,which is the main work of this paper to derive this characteristic function.The results of numerical analysis show that the model proposed in this paper captures the market characteristics well,such as heavy tail,leverage effect,profit and loss asymmetry,etc,and the fast Fourier transform method is fast and efficient compared with Monte Carlo simulation in solving.And in comparison with the Black Scholes model,the call option prices obtained from the Heston-HW-MEJ model are higher,indicating that the Black Scholes model underestimates the volatility of the underlying,and the Heston-HW-MEJ model implies greater market uncertainty.Finally,through the data of Hong Kong Hang Seng Index option market and Shanghai 50ETF option market,the empirical comparison of several models shows that the model in this paper has better pricing effect in low interest rate market,but it is slightly worse than the Heston-CIR-MEJ model in relatively high interest rate market.

关 键 词：期权定价 Heston-HW-MEJ 混合模型 特征函数 快速FOURIER变换 

分 类 号：O211.6[理学—概率论与数理统计] F832.5[理学—数学]