The First Passage Time Of Stochastic Volatility Models

Black and Scholes develop the celebrated Black-Scholes option pricing model under the assumption that the volatility of stock price is constant.But many researches indicate that the constant volatility assumption distinctly conflicts with the volatility smile/skew phenomenon observed in the real options markets.In order to improve the Black-Scholes model,researchers put forward a lot of improved model,the stochastic volatility model has been widely noted because of its ability to describe the time-varying fluctuation,and it is applied to the study of option pricing.This paper mainly studies the first passage times of several special types of stochastic volatility model and stochastic volatility CEV model.The first passage time(FPT)problem has a very important practical significance in the financial market.It has been widely noted in recent years.A great deal of researchers have made extensive studied in the problem of FPT of Brownian motion,Ornstein-Uhlenbeck process,reflection Ornstein-Uhlenbeck process and Constant Elasticity Variance process.However,most of the above studies are directed to the one-dimensional diffusion process,and there are few researches on the FPT problem of the two-dimensional diffusion process such as stochastic volatility model.To fill this vacancy,this paper focuses on the following two key elements:Firstly,we studied the FPT problem of the asset price process as the O-U process and the geometric Brownian motion of the stochastic volatility model.We use the It?formula to construct the martingale.By using the martingale method,the problem can be converted into solving a second order constant coefficients ordinary differential equation and Euler equation under condition ?c_Y+?-0.And solve the corresponding ordinary differential equation.Then,the Laplace transform explicit expression is obtained.Finally,draw the diagram with different correlation coefficients and analyze the variation trend of graph.Then,we discuss the FPT problem of stochastic volatility CEV model.Similarly,we use the martingale method to solve the joint Laplace transform of the stochastic volatility CEV model,and the problem is converted into an second order variable coefficient ordinary differential equation.By variable substitution,it is turned directly to the Whittaker's equation.So we can get the general solution of Whittaker's equation.Thus,the explicit expressions for the joint Laplace transform of the FPT of stochastic volatility CEV model can be derived.As a special case,we obtained the explicit expressions for the joint Laplace transform of the FPT when the asset price process of stochastic volatility CEV model is the square root process.Finally,draw the corresponding diagram and analyze the variation trend of graph.