Statistical Inference And Derivatives Pricing In Stochastic Volatility Model

It is widely recognized that the popular Black-Scholes model, which relates option prices to current stock prices and quantifies risk through a constant volatility parameter, is no longer sufficient to capture modern market phenomena. The nature extension of the Black-Scholes model is to modify the specification of volatility to make it a function of the stock price or a stochastic process. In this thesis, the problem of statistical inference and option pricing in financial markets with three kind of stochastic volatility is studied. The thesis contains two parts. In the first part of this thesis we study the problem of parametric and nonparametric methods for estimating coefficients in level-dependent volatility, Complete model with stochastic volatility, and Cox-Ingersoll-Ross model, based on discrete observations. In the second part of this thesis we study the problem of option pricing in the three kind of stochastic volatility models. The main results are following.1. The problem of nonparametric estimation of the diffusion coefficients in a diffusion process is studied. A wavelet estimator of the diffusion coefficients is constructed. And the rate of convergence of the wavelet estimator is given. The results is used to estimate the parameters in level-dependent volatility model, Complete model with stochastic volatility, and Cox-Ingersoll-Ross model.2. The mean and variance ergodic property is proved according to the construct characteristic of the CIR process. And the moment estimator of the equilibrium mean m and the equilibrium variance v of the CIR process are given. Having estimated the parameters m and v , the relation of the scale parameter a and the volatility β is obtained. And the problem is now to estimate the unknown parameter a in drift and diffusion. The condition moment estimate and the convergence maximum likelihood estimate are discussed. The simulation indicates that the condition moment estimation is more accurate than the convergence maximum likelihood estimation.3. The law of stock price of the level-dependent volatility and complete stochastic volatility model with dividend-paying and placing is derived. And the problem of option pricing on dividend-paying and placing stocks is discussed. It is concluded by the discussion of American option with a convex paying function, that the optimal exercising time of American call options can only at time immediately before payment of the dividend or expiration time. We also give the stochastic differential equation of the value function of the option .4. The asymptotic pricing of European option in the DIR model is derived. And the asymptotic pricing of American option on dividend-paying and placing stocks is discussed using the stochastic differential equation of the value function and the asymptotic pricing formula of the European option .