Research On The Pricing Of Lookback Options Under Several Mixed Bi-fractional Brownian Motion Models

Lookback options are a strong path-dependent option.The holders of lookback options have the right to buy at the lowest price or sell at the highest price during the lookback period,which provides investors with a way to choose the best market trading opportunity.In any case,it can bring the greatest benefits.The price of lookback options is very expensive,it is of great practical significance to research the pricing of lookback options.Mixed bi-fractional Brownian motion,as a newly proposed Gaussian process,not only has the self-similarity and long memory of fractional Brownian motion but also does not have an arbitrage opportunity.It is a semi-martingale under certain conditions,which can be solved by stochastic analysis theory.Therefore,the mixed bi-fractional Brownian motion is more suitable for describing the price changes in financial assets.We establish a mixed bi-fractional Brownian motion model and generalize it to a mixed bi-fractional jump-diffusion model.The results of this paper have promoted the research of lookback options and have a reference for the research of other path-dependent options.Lookback options can be divided into fixed strike lookback options and floating strike lookback options.However,fixed strike lookback options are not common in the market.Floating strike lookback options are usually called standard lookback options,so we only study floating strike lookback options in this thesis.In the actual financial trading market,the distribution of asset returns tends to show a ‘peak and thick tail’ shape and asset prices will appear intermittent and infrequent ‘jumps’,which is different from the traditional research on geometric Brownian motion and the actual situation.Therefore,in the case of considering continuous payment of dividends,we establish the mixed bi-fractional jump-diffusion model to study the pricing of European floating strike lookback options.The main results are as follows:(1)The pricing of the European floating strike lookback option is studied when the price of the underlying asset(stock)with dividends under the mixed bi-fractional Brownian motion model with normal constant parameters.We obtain the partial differential equations for the option price by riskless hedging principle,then we transform it into the classical Cauchy problem of heat conduction equation through variable substitution,finally,we obtain the analytical solution of the price of European floating strike lookback option.Besides,Matlab software is used to study the effect of different HK indexes and initial stock prices on option value.(2)The pricing of the European floating strike lookback option is studied when the price of the underlying asset(stock)with dividends under the mixed bi-fractional Brownian motion model with parameters of time-deterministic functions.Risk neutral measure can be transformed from actual measure by the method of equivalent martingale measure,finally,the analytical solution of the price of the European floating strike lookback option is obtained by the properties of conditional expectations.Also,Matlab software is used to study the effect of time deterministic parameters and constant parameters on option value.(3)The situation of ‘jumps’ in the actual financial market is studied and a mixed bi-fractional jump-diffusion process is introduced.A mixed bi-fractional jump-diffusion model is established for the European lookback option pricing based on some properties of the jump-diffusion process.The analytical solution of the price of European floating strike lookback options is obtained by the method of equivalent martingale measure.Also,Matlab software is used to study the effect of different jump times on option value.