Research On European Option Pricing And Hedging Strategies Under Sub-fractional Brownian Motion

With the prosperous development of financial markets,all kinds of financial derivatives came into being,options and exchanged options,in the form of combination financial derivatives,have been also more and more popular.The classical Black-Scholes(shorthand for B-S)model drived by geometric Brownian motion assumes that underlying assets logarithmic yield is normal distribution,and do not pay dividends.However,many studies indicate that the model can not accurately describe the asset prices' characteristics,such as long dependency,short time invariance properties.Therefore,it is necessary to extend the classic Brownian motion Black-Scholes model to sub-fractional Brownian motion Black-Scholes model,under the conditions of relaxing some assumptions,so that the asset price is more reflective of financial reality.The work of this paper has three main parts.In the first part,we consider option pricing with dividends based on time-changed sub-fractional Brownian motion model.First of all,studying the option pricing problem in discrete time,assuming that the price of the underlying assets follows a time-changed sub-fractional Brownian motion,get subdiffusion sub-fractional Brownian motion B-S model.Secondly,on the basis of the model,the option pricing formula for European in discrete time environment is obtained.In the second part,we consider European exchange option pricing and hedging problem with dividends under the sub-fractional Brownian motion.The assets of exchange options are proposed to follow stochastic differential equations derived by sub-fractional Brownian motion,using two quadratic hedging strategies obtains explicit solutions of exchange option,which are expressed in terms of the hedged exchange options pricing function and the price of the underlying assets.Compared with the general solution based on Monte-Carlo simulation,this solution will be well suited for computer implementation and reduce compute time.In the third part,the numerical simulation analysis is given.Firstly,by setting the initial value and its related parameters,simulate inverse stable subordinator path and standard diffusion path driven by sub-fractional Brownian motion using Euler method and Monte Carlo simulation,respectively.Then according to the characteristics of the subordinate process,using linear interpolation method to superpose the two paths,get sample path to the asset price in subdiffusion market model.Finally,the Shanghai 50 ETF transaction data is selected to simulate the path of the sub-fractional B-S model.Compared with the real value,the validity and feasibility of the model are verified through the above simulations.