Option Pricing Model Whose Underlying Stock Pricing Process Is Mixed Process And Its Numerical Computation

"Financial Mathematics, Financial Engineering and Financial Managing" is one of the important items during the ninth Five-Year-Plan in China, and option pricing is being studied heartily in this item. Base on the detailed research of the characters of option and the affecting factors of option's price, the article gives us a new option pricing model and its numerical method using the mathematical tools. Firstly, the article studies the famous Black-Scholes Option Pricing Model in detail, including the inferring process of the B-S Differential Equation, and the method how to infer the B-S Model from the B-S Differential Equation. Secondly, the article gives us one of the important faults of this model, hi view of this fault, the article will infer a new type of option pricing model by means of changing one of the basic hypothesizes of B-S Model. In the third chapter, the article studies stock's changing tendency, researching the Markov jump process and Ito process. Afterwards, the article refers a new mixed process by mixing the two processes, saying that the stock's changing tendency is following the mixed process. At the last section of the third chapter, the article analyze the SHENZHEN and SHANGHAI stock markets' synthetic number and a piece of stock's (Shen-Nengyuan) number, proving the superiority of the mixed process. Base on this distribution model, the article infers the differential equation of derivative security whose basic stock's distribution is mixed process, the pricing model of European Call Option. Noting the Markov jump process is not continuous; this pricing model is also interrupted in view of time. At the end of this article, we solve the differential equation with the finite difference method and the finite element method, obtaining an executable method.