Pricing European Option Under Time-changed Mixed Fractional Brownian Motion,Stochastic Interest Rate,and Jump-diffusion Models

In order to capture long-range correlations and periods of constant values of the financial market,Guo(2014) established a time-changed mixed fractional Brownian motion model and studied the pricing formula of European option.Considering the randomness of interest rate,Guo(2017) established Merton short rate model under time-changed Brownian motion and obtained the pricing formula of European option.Based on Guo(2014) and Guo(2017),this paper studies the pricing of European options under time-changed mixed fractional Brownian motion,stochastic interest rate,and jump-diffusion models.Firstly,this paper studies the pricing formula of European options under the time-changed mixed fractional Brownian motion model with jumps,and applies the model to the pricing of real estate options with default risk.Then,under Merton short rate model by time-changed Brownian motion,the Brownian motion is extended to mixed fractional Brownian motion,and the pricing under Merton short rate model is extended to geometrically average Asian option pricing with floating execution price under Vasicek short rate model.Finally,considering periods of constant values,long-range correlations,stochastic interest rate and jump,this paper studies the pricing of European option the pricing of European options under time-changed mixed fractional Brownian motion,stochastic interest rate,and jump-diffusion models,and derives the pricing analytical solution of the model,which extends the Guo(2014)and Guo(2017)models.In this paper,the partial differential equation method is used to solve the pricing formula.By hedging the portfolio,the corresponding partial differential equation is obtained,and the pricing formula is obtained by variable substitution.Because the partial differential equation corresponding to Asian options is three-dimensional,it is necessary to reduce the dimension by multiple variable substitution,and get the solution of the corresponding partial differential equation by Fourier transform and its inverse transformation.Because of regulations of the government,house prices remain unchanged for a period of time,but at the same time,there will be large fluctuations due to the changes in financial markets.Therefore,this paper applies jump-diffusion model by time-changed mixed fractional Brownian motion to solve the pricing formula of house price option.Finally,the pricing formulas corresponding to each class of models generalized in this paper are analyzed by numerical examples,and it is found that the model is very close to the option price obtained by B-S model.The shortcomings of this paper are that the model is not fitted with the real data,and the change of jump intensity is not considered.