First Passage Time Of Stochastic Volatility Models With Two Reflected Barriers

Local volatility models and stochastic volatility models are two kinds of effective method to overcome the constant volatility assumption in Black-Scholes model.But they have their own shortcomings.With the developing of model estimation,more and more practitioners and academics had begun to study the hybrid local and stochastic volatility models which have been widely used in the study of modeling interest rate term structure and option pricing.In this paper,we study the first passage time of the stochastic model which is combined with the constant elasticity volatility model with two reflected barriers.First passage time plays an important role in the field of option pricing.Many people have done a lot of research on the first passage time of local volatility models with reflected barriers.The first passage time of stochastic volatility models,however,has been rarely discussed.Firstly,this paper introduces the history of some famous diffusion processes in mathematical finance and the researches of first passage time of those processes.Then we introduce some basic knowledge of stochastic calculus,second order ordinary differential equations and reflected processes.Secondly,this paper transfers the problem of the first passage time of stochastic volatility CEV model to the problem of solving second order homogeneous differential equations by martingale methods.We use the confluent hypergeometric functions to express the solutions of the equations,then substitute into the previous conclusion,and get analytic formulae concerning a special kind of expectation of first passage time and volatility.We also give the results when one of the confluent hypergeometric functions does not exit as the elastic parameter takes special values by Frobenius methods.Finally,we discuss the situations that the elastic parameter is taken as 0,-1/2,-1,which the underlying assets correspondingly follow as reflected geometric Brownian motion,reflected square root process,reflected OU process respectively.And analyzes the influence of initial values of underlying asset and volatility on our results.