Research On Option Pricing With Incomplete Hedging

Option pricing is one of the hot issues in the field of financial mathematics,which has attracted the attention and research of many experts and scholars.One of the most classical models of option pricing is the Black-Scholes option pricing model,which established by Black and Scholes.This model can be said to be the basic model of option pricing.Many modern option pricing models are popularized on Black-Scholes option pricing model.As we all know,the Black-Scholes option pricing model is based on an ideal market,that can be completely hedged.Because the real financial markets has friction,so they are incomplete and not idealized.Therefore,it is almost impossible to realize the hedging without arbitrage in the complete sense.Therefore,it has theoretical and practical significance to study the option pricing problem in the sense of incomplete hedging.In this paper,based on the portfolio theory and the partial differential equation method,we give the option pricing model in the incomplete hedging markets,and prove the existence of the solution and the uniqueness of the solution.Firstly,in the financial market with incomplete hedging,the option pricing equation is established,that is,the option pricing equation with damping.Secondly,a kind of option pricing equation with certain damping is discussed,and the nonlinear heat conduction equation is used to solve the problem.The analytical solution of the equation is given.Then,the option pricing equation with random damping is discussed.By using martingale representation theorem,the existence and uniqueness of the martingale solution of this equation are proved.Finally,a simple example is used to illustrate the similarities and differences between the option pricing differential equation with damping and the traditional Black-Scholes option pricing differential equation.The study of this paper can be regarded as a natural generalization of the classical Black-Scholes option pricing problem.