Lookback Options Pricing And Statistical Simulation Analysis Under The Mixed Sub-fractional Jump Diffusion Model

Lookback options are exotic options,which have strong path dependence and are difficult and complex to price.How to reasonably price the option has been the focus of the financial industry and scholars,and is currently the forefront of theoretical research in the financial sector.At present,most of the research on the pricing of lookback options is based on the classical Black-Scholes option pricing model.However,the assumptions in this model are too stringent,such as the assumption that the price change of the underlying asset follows geometric Brownian motion and the market is frictionless,which does not characterize the fractal characteristics of financial asset prices and does not reflect the actual trading situation of the financial market.Therefore,this thesis uses the mixed sub-fractional Brownian motion to drive changes in the price of the underlying asset and combines the Poisson process to build a mixed sub-fractional jump diffusion model with three main research components.(1)The pricing model for perpetual American lookback options of stock paying continuous dividend is constructed under the mixed sub-fractional Brownian motion.Firstly,the partial differential equation satisfying the option and its boundary conditions are obtained by using the Delta hedging principle.Then,a closed form solution to the perpetual American lookback options as well as its optional exercise boundary are respectively obtained by using variable substitution method.Finally,numerical simulations are carried out to verify that the closed solution has the linear scaling property and it is discussed that the effects of Hurst index and volatility on the value of options.(2)The pricing model of European lookback options with transaction costs is established based on the mixed sub-fractional Brownian motion and the Poisson process.Firstly,the nonlinear partial differential equation satisfying the option and its boundary conditions are obtained by using the Delta hedging principle.Then,the partial differential equation is reduced by variable substitution method,and its numerical solution is obtained by constructing a Crank-Nicolson format.Finally,the validity of the numerical method is verified,and the effects of transaction costs,volatility and risk-free interest rate on the value of the option are discussed.(3)We select real stock data for statistical simulation analysis.Firstly,the required stock data are obtained from major stock trading platforms.Then,simple descriptive statistical analysis is performed on the data and the parameters in the pricing model are estimated.Finally,the simulated values of stock prices under different pricing models are obtained based on Monte Carlo simulations and compared with the real values to verify the validity of the mixed sub-fractional jump diffusion model.The study result shows that the mixed sub-fractional jump diffusion model can better characterize the fractal characteristics and discontinuous fluctuations of financial asset prices than other models and that it is reasonable and effective to price lookback options on the model.