Options Pricing Under Jump-diffusion Models With Transaction Costs In Fuzzy Environment

Due to the volatility of financial markets, many financial data can not be describedas crisp number, such as market risk-free rate, volatility rate. The interest rate is about5%and volatility is about3%. Both of which are ambiguous. Fuzzy Mathematics which candescribe these dates is introduced to financial theory. On the one hand, the asset priceoften occurs to jump. On the other hand, we need transaction costs, when make a dial. Inthis article, we will consider the option pricing with transaction costs in the fuzzyenvironment under the Merton jump-diffusion process.The paper is divided into four parts. In the first part, we will introduce the emergenceand development of option pricing theory including three aspects, the development ofoption pricing theory research under the jump-diffusion model, the development of optionpricing theory research with the transaction costs and the development of option pricingtheory research in fuzzy environment. In the second part, we will use the equivalentmartingale measure and probability to get the price of a call option under a jump-diffusionmodel with transaction costs.In the third part, the fuzzy theory is introduced into theoption pricing method. In this part, first, we derive the price range of option under theconfidence levela, that is, investors can choose a number as option price in the case ofsatisfaction for the confidence level; Second, we use optimization theory derive the fuzzymembership of the option price; Finally, to get uniform pricing, the fuzzy expectation ofthe fuzzy option price is described. In the last part, we will make some numerical analysisbased on the work done.