Pricing Option In A Mixed Bi-fractional Brownian Motion

Option is a kind of right to buy or sell a kind of underlying asset.In today's society,option has become a hot issue,mainly because it can effectively avoid some financial risks and better deal with economic downturn such as financial crisis.Therefore,many scholars and experts have great interest in option research.So how to make a reasonable pricing of options has become a process of trial and error.For example,the most classic Black Scholes pricing model(hereinafter referred to as B-S)is a good way to price options,but the B-S Option pricing model has some shortcomings,because its premise is that the financial market is an ideal state.However,there will always be various problems in the actual financial market,which will not be the perfect financial market in imagination.Therefore,in order to improve the fixed price model,this thesis mainly uses the improved option pricing model--mixed bifractional Brownian motion pricing model to analyze the option pricing,and considers that the major accident or disaster will have a certain impact on the underlying asset price,and then affect the option price.In order to better price the option,this thesis introduces jump diffusion into option pricing The main contents of this thesis are as follows:under the mixed fractional Brownian motion model and the two jump fractional Brownian motion modelFirst of all,we assume that our financial market is in an ideal state,that is,there is no friction and arbitrage in the market.Then we deduce the mixed bifractional Brownian motion pricing model.Then we use the risk hedging principle to derive a partial differential equation.Then we solve the partial differential equation.Finally,we get the European option pricing formula when we call or put.This thesis uses MATLAB software to analyze the impact of Hurst index,volatility and different risk-free interest rates on option pricing through numerical experiments.Secondly,we study the pricing of European options driven by mixed bifractional Brownian motion with jumps.Because the sudden accident will cause the jump change of the underlying asset price,and then affect the option value,the mixed bifractional Brownian motion equation with jump is derived through various formulas.Then according to the self financing trading strategy,the European option pricing model can be obtained,and the pricing model can be transformed into a solvable heat conduction equation by specific methods,so that the option can be calculated The theoretical price of the price.Then,Matlab is used to analyze the effect of different dividend,different jump intensity,different Hurst index and different exercise price on the option price in the real market.Finally,through empirical analysis,taking Shanghai Stock Exchange 50 ETF option as the research object,considering that there may be jumps in the event of major accidents,we choose the data from the COVID-19 period and use MATLAB to make the value of the mixed double fractional Brown motion pricing model and the mixed double fraction jump diffusion model,and compare the results with the traditional option pricing B-S formula.On the basis of the mean square error of the option price and the actual option price,this thesis measures the accuracy of the two options pricing models and obtains a better option pricing model.