Parameters Identification Problem In Option Pricing Based On Theory Of Degenerate Parabolic Equation

In this paper,the space-dependent inverse volatility problem in option pricing is studied.To reconstruct volatility over the entire domain,the original inverse problem is transformed into the following two problems by variable substitution and operator linearization,namely,iden-tifying the principal coefficient and source term problem of the bounded degenerate parabolic equation.The mathematical model involved in the study is a degenerate parabolic equation,un-like conventional problems,this equation may lack boundary information at some boundaries.As a result,the regularity of its solution is poor.In view of the ill-posed nature of the inverse problem itself and the various characteristics of the volatility,how to apply suitable regulariza-tion method to recover the unknown volatility is one of the important tasks of this article.The research results of this topic can help regulators better understand various phenomena and laws in the financial market,so as to formulate more scientific and reasonable regulatory policies and measures,and maintain the stability and healthy development of the financial market.This article mainly includes the following five parts:In the first chapter,the financial model in this study and some strategies of other scholars are introduced,the significance and innovation of this research are explained.Some lemmas,definitions and theorems that will be used later are given in the second chapter.In the third chapter,the problem of identifying the principal term coefficient of non-divergent degenerate parabolic equations with terminal data is discussed.Theoretically,the inverse volatil-ity problem on semi-unbounded regions can be transformed into an inverse principal term co-efficient problem of bounded degenerate parabolic equation by variable substitution.Based on the optimal control framework,inverse principal term coefficient problem is transformed into an optimization problem.The existence of the optimal solution of the cost functional is proved and necessary conditions satisfied by minimizers are established.In addition,bring in source con-dition,the minimizers of the optimal control problem converges to the solution of the original problem.Numerically,a gradient iterative algorithm is designed to solve the numerical solu-tion of the inverse problem and some numerical experiments are carried out.The experimental results show that the proposed algorithm has good robustness and fast convergence speed.In the fourth chapter,a linearized inverse volatility problem under the mean reversion of volatility is mainly investigated.Since the nonlinear inverse problem discussed in the third sec-tion is not convenient to conduct the numerical simulations,linearization techniques and variable substitution are introduced to transform the original inverse volatility problem into an inverse source problem of bounded degenerate parabolic equations.From the theoretical point of view,the Tikhonov regularization method with L₂penalty term and the Total Variation regulariza-tion with L₁penalty term are adopted to transform the inverse source problem into optimization problem,respectively.The existence,local uniqueness and stability of the control functional corresponding to the two regularization methods have been proved.From the numerical point of view,there are two contributions.First of all,a numerical method based on the finite integration method is introduced to obtain numerical solutions of forward problem,which has significant ad-vantages in numerical accuracy and distribution freedom.Secondly,a Landweber-type iterative method based on finite integration method is designed for the Tikhonov regularization method,while the Gauss-Jacobi iterative algorithm based on finite integration method is designed for the total variation regularization method.The core idea of the discretization algorithm is to obtain the Euler-Lagrange equation that satisfies the optimal solution by the necessary condi-tions.Finally,numerical experiments are carried out on two regularization algorithms based on finite integral method,which show that the total variation regularization can better reflect the overnight and jumping of volatility,while the Tikhonov regularization method is more suitable for continuous volatility function.In the fifth chapter,the content of the full paper is summarized and some future work is introduced.