| The inverse problem of option pricing occupies an important place in the research field of modern financial theory. Both in theoretical discussion and practical applications, it has the great academic value and social and economic significance. In this paper, we consider an inverse problem of option pricing which has the stochastic volatility, and introduce the volatility driving factor, and then use stochastic optimal control theory to solve the stochastic volatility. We construct the corresponding numerical solution, and finally make problem more complex and practical.This paper's main work and results are mainly in chapter 3,4,5. Based on previous work in chapter 3, we introduce the volatility driving factor satisfied a certain stochastic differential equation and having its own volatility, which makes the problem be more close to the structure of financial market. We consider the underlying asset following the stochastic differential movement, which has the stochastic volatility on a function of the underlying asset, volatility driving factor and time. By the relative entropy, the inverse problem of option pricing under the stochastic is made into an optimal control problem. Finally, we derive the corresponding HJB equation using the stochastic optimal control theory, and then obtain the stochastic volatility's correct solution. In chapter 4, we give an implicit finite difference method to solve the HJB equation, and construct a Newton numerical method, and process the boundary conditions. In chapter 5, we prove that the finite difference method is monotone, stable and convergent. Then, we design its solution scheme, and finally show the method is effective through an example. |
