Binary Options Pricing Under Three Lévy Processes

In the financial market,options are a very important financial derivative,so the pricing research on options is one of the links that traders are most concerned about and has important research significance.With the gradual development of the financial derivatives market,the types of options have gradually increased.In addition to the conventional European and American options,a more complex form of options has emerged,which called exotic options.As a typical exotic option,binary options have a lot of research results on its pricing in recent years.Based on previous studies,this thesis considers the Binary options with dividends pricing problem under the classical Lévy process.Firstly,under the risk-neutral measure,this thesis uses the Lévy process to replace the Brownian movement in the classic pricing model that is B-S model in order to derive the pricing equation and boundary conditions that the binary options satisfy through mathematical methods such as Fourier transformation.Secondly,this thesis selects the typical infinite jump process in the Lévy process that is the CGMY process to model and derives the fractional partial differential equations and boundary conditions satisfied by the binary options with dividends.Then,based on previous research results,this thesis tries to derive the closed or semi-closed analytical solution of the fractional partial differential equations satisfied by the binary options under the CGMY process.Through the aforementioned research process,this thesis generalizes the CGMY process,and selects two other typical infinite jump models in the Lévy process,namely the FMLS process and the KOBOL process.And then this thesis solves the fractional partial differential equations of binary options with dividends analytically under the two Lévy processes based on the pricing method of the CGMY process in order to derive the corresponding closed or semi-closed analytical solutions.Finally,this thesis performs numerical simulation to the closed or semi-closed analytical solution of the binary options pricing equation obtained above,and propose an effective numerical method to realize the analytical solution obtained above.This thesis changes the size of different parameters in the model to investigate the influence of different parameters on the model's pricing results.Compared with the previous studies,although the process of deriving the closed or semi-closed analytical solution of the pricing equation is cumbersome and complicated,this thesis proposes a pricing method for binary options with dividends,which provides reference on binary options pricing in future.