Time Fractional Black-Scholes Complementarity Model For American Option Pricing

Black-Scholes(BS)option pricing model,as an important method of option pricing,has attracted much attention because of linking the option price with the stochastic volatility of the underlying asset price and the risk-free interest rate.Through the observation and research of many scholars on the stock market,it's found that the essential characteristics and the state of the capital market are stochastic fluctuations,which don't fully coincide with the assumptions of the BS model,and it makes quit different between BS model pricing and the actual market price.Many scholars began to consider revising the Brownian movement of the partial differential equation,in order to make the revised Brownian movement better reflect the properties of autocorrelation,long-term memory and incremental correlation.Therefore,how to build an option pricing model which is more suitable for real financial markets and relatively easy to solve has always been a concern.With the discovery of the fractal structure of differential equations in the financial field,more and more scholars begin to pay attention to the fractional partial differential equations in the financial field.This paper studies the complementarity model of American option pricing based on the theory of time fractional BS equation.Firstly,according to the meaning of risk-free portfolio in BS equation hypothesis and the properties of American option,the complementarity model of time fractional BS equation is given.Then,the complementarity problem is discretized by using L1 interpolation approximation of Caputo fractional derivative,the truncation error of the difference scheme is analyzed,and the discretized option pricing complementarity problem is transformed into an optimization problem.Finally,the numerical experiments are carried out by using MATLAB programming,and the numerical results and conclusions are given.The research contents of this paper mainly have two aspects:Firstly,we study the time fractional BS complementarity model and its solution for American option pricing.The BS equation is transformed into complementarity problem,which effectively improves the dependence of the solution of some American option pricing model on free boundary.Secondly,we use the data of one-year American put options,China Mobile and HSBC Holdings put options to make preliminary numerical experiments,and analyze the change trend of option price of the time fractional BS complementarity model in different fractional order and expiration date.The numerical results show that the option pricing results obtained by the time fractional BS complementarity model are better than those obtained by the traditional BS equation model,and some of them are better than those obtained by the binary tree method.The paper is organized as follows: The first chapter briefly introduces the development status of option pricing.The second chapter mainly introduces the basic concepts of option and time fractional BS option pricing model.The third chapter introduces the theory and solution of complementarity problem.The fourth chapter introduces the free boundary problem of American option,and gives the time fractional BS complementary model,discretization form and solution methods of American options pricing problem,and gives the numerical results calculated by using MATLAB programming.Last part is the summary and future research directions.