Pricing Of Option Based On Fractional Differential Equation

Option pricing is the core issue of financial investment and an important part of modern financial theory research.Fractional option pricing model has been widely used in option pricing due to fractional derivative can describe the anomaly diffusion and transmission dynamics in complex systems.Most of the current analysis of option pricing is integer order partial differential equations,and volatility is considered as a constant value.In this paper,the fractional differential operator is introduced,and the infinite jump Levy process,random volatility and other factors are considered.The option pricing models are established respectively,and high-precision numerical algorithms are given.The main work of this article is as follows.(1)Time fractional CGMY model for the numerical pricing of European call options.The option pricing is subject to the infinite jump CGMY process,and the stock price fluctuation is regarded as a fractal transmission system,and derived a time fractional options European call pricing model(TFCGMY).Caputo fractional differential L1 approximation and modified GL approximation are used to discretize the time and space fractional order operators,and the central difference quotient is used to discretize the first order spatial derivatives.The TFCGMY model and the above numerical techniques are used for pricing analysis of European call options,and the stability and convergence of the format are analyzed separately.By comparing with the B-S model and the CGMY model,the analysis of the option pricing model proposed shows that the introduction of time fractional order can better capture the jump characteristics of the stock price of options.(2)The numerical solutions of tempered-TFCGMY model for European double barrier options.Using time fractional and spatial fractional order differentials to describe European double-barrier option pricing,a new option pricing tempered space-time fractional order model(T-TFCGMY)is obtained in order to improve the accuracy of capturing the thick-tailed features of option pricing and jumping.The L1 approximation of the Caputo fractional derivative and the tempered-weighted shift generalized difference with second-order approximation is used to discretize the time and space fractional order operators,and the first-order spatial derivatives are discretized by the central difference quotient to obtain the numerical dispersion of the T-TFCGMY model format.The T-TFCGMY model and the above numerical techniques are used for the pricing analysis of European double-barrier options,and the stability and convergence of the format are analyzed separately.The results of numerical examples show that the introduction of fractional derivatives of time and space can better capture the jump and thick tail of real and imaginary option stock prices.(3)Numerical approximation for European option pricing under stochastic volatility.Applying the Heston model driven by the Levy process of infinite activity to option pricing,a stochastic volatility pricing model for European call options(SVOP)was established.The five-point difference format and the backward difference quotient are used to discretize the spatial two-dimensional derivative and the time derivative,respectively,to obtain the discrete format of the stochastic volatility option pricing model,which is used in the pricing of European call options.The results of the numerical examples show that it's consistent with the phenomenon of jumps in option pricing observed in the underlying asset dynamics.The model can more deeply capture the jumping characteristics in asset returns and the volatility clustering effect in return differences and is effective to capture the volatility characteristics in option pricing.