Numerical Methods For Option Pricing Problem Under Stochastic Volatility

In 1973, Fischer Black and Myron Scholes established the famous option pricing model, which is also called Black-Schoes model. In the Black-Schoes model, we suppose that volatility is a constant. However, volatility of the underlying asset is the only unobservable parameter. This assumption is not realistic. When scholars apply Black-Scholes option pricing formula, they often need time to adjust volatility values. In comparison, this is a reasonable consideration to assume volatility is random. Since 1987, many scholars have studied the issue of stochastic volatility. This paper mainly studies a class of option pricing equation with stochastic volatility. For the option pricing equation with stochastic volatility, in general, Monte Carlo(MC) simulation can be used to obtain its numerical solution. But the Monte Carlo requires thousands of samples, which will naturally lead to a great amount of computation. This paper adopts the Adaptive Stroud Stochastic Collocation Method (SCM) to solve this equation in order to overcome the defect of MC and improve the continuous solution and boundary problem greatly. We can summarize the main content about this paper in the following:Chapter 1 is preface. It mainly introduces the background of paper, the origin of SCM, current research situation in domestic and abroad, and the main content of this paper.Chapter 2 is basic knowledge. It mainly introduces the theory about the Black-Schoes equation and formula, stochastic differential equations(SDE), SDE applied to the Black-Schoes equation, common numerical methods for the Black-Schoes equation.Chapter 3 describes AC=BD model in mathematics mechanization. It mainly introduces basic knowledge of AC=BD model, some examples about AC=BD model, and AC=BD model applied to numerical solutions.Chapter 4 describes numerical methods about option pricing. It mainly introduces a class of numerical method:SCM, including standard SCM, Stroud SCM and adaptive SCM. And SCM and MC are both applied to a class of option pricing equation. Through the two (European option and American option) numerical results, we can see that the superiority of SCM in the accuracy, error, stability is obvious, and SCM can solve the continuous solution and boundary problem. SCM will be applied more widely in the option pricing equation.Chapter 5 is the summary and outlook.