Stochastic Volatility Jump Diffusion Model For Option Pricing

The option lies in an essential position in numerous financial products, its pricing is the problem of early attention. The breakthrough of option pricing resulted from the Black-Scholes model in the early1970s, it marked the arrival of the financial engineering era. The assumed conditions in the Black-Scholes pricing model didn’t accord with the actual situations gradually because the economic environment has been becoming more and more complex. In order to forecast more accurately the real price of options, many scholars constantly improved the option pricing models from the volatility, dividend, interest rate, trading fees, and so on. At the same time they explored the numerical methods for the corresponding models and obtained many valuable results. Among them, Eraker Johannes and Polson expanded the Bertrand’s model and proposed a kind of estimation strategy. They also studied the volatility structures from the standard&poor’s and Tusk index, and they pointed out the advantages of the volatility jump process in the models, but didn’t obtain a close-form option pricing formula. In this thesis, we firstly summarized the research achievements of predecessors, then studied the European put options considering the stochastic volatility jump-diffusion model under the condition that the underlying asset price followed the model proposed by Eraker Johannes and Polson, and obtained a close-form European put option formula.Assume that the underlying asset price meets: The PDE equation met by option value was obtained by Dynkin formula, the analytic solution was further got:where Pj(l,v,t;k,T):=Pj(el,v,t;ek,T),j=1,2. And it was validated by using the differential equation met by the risk neutral probability P1,P2. Finally, it was pointed out that the analytic solution can be calculated by the inverse transformation of the characteristic function.