基于非均匀离散Fourier变换的隐含Levy模型估计研究

市场上交易的期权价格蕴含了市场参与者对期权标的资产价格未来运动的预期信息，基于期权价格得到的标的资产价格运动模型被称为隐含模型，应用于衍生品定价与风险管理显著优于基于资产价格历史数据得到的模型，Levy模型近年来被广泛应用于描述金融资产的价格运动。Levy模型一般不存在解析形式的概率密度函数，但总是存在解析形式的特征函数。利用Levy模型下期权定价的Fourier变换理论，基于非均匀离散Fourier变换研究了隐含Levy模型的参数估计问题。首先，基于Fourier变换的欧式期权定价，给出了欧式看涨期权价格与特征函数之间的关系。接着，介绍了Levy过程基本性质及其特征函数。再接着，给出了非均匀Fourier变换；然后，给出了Fourier域的模型拟合与参数估计。最后，演示了本文方法的在估计隐含Levy模型中的应用，从模型参数估计与模型识别两个方面验证了方法的有效性。研究结果表明，本文方法能够解决Levy过程不存在着解析形式密度函数导致的难以估计隐含模型参数问题，能够处理市场上交易的期权执行价格分布不均匀，数据量少的问题，是一种有价值的参数法隐含概率密度估计方法。

Estimating Implied Levy Models based on non-uniform Discrete Fourier Transforms

The price of the option traded on the market contains the prospective information of market participants on future movements of the underlying asset price. The underlying asset price movement model based on the option price is called an implied model, which is obviously superior to the model based on the history data of asset price while applied to pricing derivatives and risk management. The Levy model has been widely used to describe the price movement of financial assets in recent years, and the general Levy model does not have an analytic form of the probability density function, but there is always an analytic form of the characteristic functions. Based on the Fourier transform method of option pricing under the Levy model, the problem estimating the parameters of the implicit Levy model is studied based on the non-uniform discrete Fourier transform (NDFT). First of all, the European options pricing is introduced based on the Fourier transform, and the relationship between the European call option price and the characteristic function is also given. Then, the basic properties of Levy process and its characteristic function are described. Then, the non-uniform Fourier transform is given. Then, the model fitting and parameter estimation of the Fourier domain are given. Finally, the application of this method to the estimation of the implied Levy model is demonstrated, and the effectiveness of the method is verified from two aspects of model parameter estimation and model identification. The results show that this method can solve the problem that the most of Levy process does not have the analytic form of density function, which cannot estimate the implied Levy model parameter problem, and can also deal with the problem of uneven distribution of option execution price in the market and less data quantity.