Research On Barrier Option Pricing In The Sub-mixed Fractional Brownian Motion With Jump Environment

With the continuous improvement of the financial market,more and more exotic options begin to emerge.Compared with the traditional call and put options,they have more flexible exercise conditions and forms,and are favored by investors.Barrier options are representative of exotic options.Because of its additional vesting conditions,its price is cheaper than the corresponding vanilla options.Barrier options can be seen everywhere in the global exchanges and OTC markets.Many companies use various barrier options to hedge risks.In addition,the research on barrier option pricing can also promote the research of many structured financial products.Therefore,studying the pricing of barrier option has important theoretical significance and practicality.In order to explore the pricing formula of barrier option under the mixed fractional jump diffusion model.Through the sub-mixed fractional jump diffusion process formula and the self-financing trading strategy,the partial differential equation It(?) satisfied by the option value is obtained.Then combined with other conditions of downand-out call option,the original problem is transformed into a mixed problem of heat conduction equation.Further,the analytic solution of down-and-out call option pricing is obtained by using odd extension and possion formula.Then,through the linear relationship between various barrier options,the pricing formulas of the other three European down barrier options are deduced.Finally,taking down-and-out barrier option as an example,through numerical experiments,it is found that barrier price and Hurst index are negatively correlated with the value of European down-and-out call options,while jump intensity and volatility are positively correlated with them.In this paper,we study the pricing of barrier options in the mixed fractional jump diffusion environment,and give analytical solutions of barrier options,which can help investors grasp the risks related to barrier options more intuitively and effectively,and enrich the research system of barrier options.