European Option Pricing Under The Levy Process With Stochastic Volatility And Stochastic Interest Rate

As one of the important parts of financial derivative products, development of options will make the investment products of derivatives market more diversified, and China's financial markets more perfect. Since the 70 s of last century, the pricing of financial derivatives, especially the research around option pricing has been gradually attention and has flourished. Since 1973, the Black-Scholes model for studying option pricing made a great contribution, but as time goes on, a lot of research has proved that the traditional Black-Scholes model has been difficult to adapt to the ever-changing financial markets, which must be modified before they can better portray the true complexity of changing financial market characteristics.Firstly, this paper considers the stock returns of fat tail, the volatility smile effect, a finite number of large jumps and an infinite number of small stock price jump behavior, we establish a pure jump Levy process with stochastic volatility and stochastic interest rate European option pricing model based on the traditional Black-Scholes model, the new model can better characterize the market characteristics. The model, not only in mathematical theory but in economic theory, are both reasonable and superiority, which not only inherits the advantages of B-S model, while better portray the real market performance after the end of the peak and the share price jump behavior,in particular, it is portrayed very well on the rare large jumps and small diffusion characteristics which exist anywhere in the financial market. Secondly, using the Ito formula and measure conversion we apply the log-price under the risk-neutral measure, and by decomposition characteristic function form, characterized by assuming exponential affine function form, we solve differential equations corresponded, and get characteristic function of the log-price process. Then, using the inverse Fourier transform, the solution thus obtained in the form of the characteristic function representation. Finally, we obtained in the form of numerical solution methods using fast Fourier transform for the solution represented by the characteristic function.