European Option Pricing And Related Problems Under The G—Environment

With the rapid development of the financial markets,option pricing problem has become more and more important.Black and Scholes found the explicit solutions for the European call option by using dynamic replication.The simplicity and definiteness of the form of Black-Scholes options pricing formulas made them extremely valuable tools widely used in the study of mathematical finance.However,there are some differences between assuptions and reality.For instance,the financial asset prices is log-normally distributed with constant volatility,but the volatility need not to be constant in many cases.It may be uncertain.Thus,we find that the assumptions of the Black-Scholes options pricing formulas to be hold just provide an inaccurate description of the real world.In 2006,Peng introduced the concepts of G-expectations,which is actually a kind of nonlinear expectations.A very essential setting of G-geometric Brownian motion is that its quadratic variation<B>t satisfies σ2t≤<B>t ≤σ2t.This inequality characterizes the part of statistic uncertainty of G-Brownian motion.Hence it is very necessary to establish a certain theory under the formulation of G-expectations.The main results of this paper are as follows:(i)We prove the boundedness of the norm of stochastic integrals.For any n ≥ 1,p>1,there exist Ap,n,Bp,n such that （？）(ii)There exist A and B such that the following ratio inequality(?)holds true.(iii)We prove an extended version of Girsanov’s theorem and soften its condition to(?)(iv)By introducing G-geometric Brownian motion,we first describe the fluctuation of underlying asset prices,and then give the dynamic pricing formulas for European call option and European call option with a constant power bonus rates.