关键词: 非均匀离散Fourier变换, 隐含Levy模型, 期权, Fourier域

Abstract: 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.

Key words: non-uniform discrete Fourier transform, implied Levy model, options, Fourier domain