Research On The Pricing Of Exchange Options Under Two Types Of Mixed Subfractional Brownian Motion Model

The exchange option is a kind of exotic option.The option holder has the right to exchange one asset for another asset on the maturity date T.The exchange option is essentially a contract and is widely used in the fields of funds and international trade.Therefore,the study of exchange option has a very important practical role.Mixed sub-fractional Brownian motion model is a popular model for scholars to study exchange option pricing in recent years.It can well describe the long-term dependence and self similarity of the underlying asset price.However,there are relatively few studies on the pricing of exchange options under mixed sub-fractional Brownian motion with jumps and the pricing of exchange options under time-varying mixed sub-fractional Brownian motion with transaction costs.Therefore,based on the above two types of exchange options pricing under mixed sub-fractional Brownian motion,this thesis conducts the following research work:（1）In this thesis,considering that the price of financial assets such as stocks sometimes appears"jump"and other abnormal fluctuations,we use the Poisson processN_t with strengthλto describe this kind of abnormal jump.The pricing of exchange options in the mixed sub-fractional Brownian motion model with jumps is investigated,and the pricing formula of exchange options under mixed sub-fractional Brownian motion with jump is obtained.Python software is used for numerical simulation to fit the impact of jump intensityλ₁,λ₂and Hurst index H on option price.（2）The pricing problem of exchange options with transaction costs in the time-varying mixed sub-fractional Brownian motion model is studied.In order to reflect the constant value periodicity of asset prices,we consider introducing a time-varying process,that is,the t in the mixed sub-fractional Brownian motion is changed into a random processT_α（t）,thus a composite random process is formed and the transaction costs in the transaction process is considered.We obtain the pricing formula of exchange options with transaction costs in the time-varying mixed sub-fractional Brownian motion.And the impact of Hurst index H and volatilityσ₁,σ₂on option pricing is verified through the numerical simulation of Matlab software.