Several Source Term Inversion Problems In Option Pricing

This paper is mainly based on the theory of option pricing,and the inverse problems in the pricing of several types of financial derivatives is studied.Different from the usual parameter identification problems,these are non-standard inverse problems.Using the(35)-hedging and Green function methods,it can be transformed into the inverse problem of parabolic equations with terminal observations,but the degree of nonlinearity and ill-posedness is serious,which is not conducive to theoretical analysis.In this paper,the partial differential equation linearization technique is used,which means the unknown parameter is assumed to be composed of known functions and unknown small disturbances,so as to transform the original problem into a source term identification problem.Based on the optimal control theory,we prove the existence,uniqueness and stability of the optimal control solution.These results have very important theoretical and practical significance,and can be widely applied to the inverse problem of pricing of various financial derivatives.The full paper is mainly composed of the following five chapters:In chapter one,we mainly describe the research background of the inverse problem and the current research results,and introduce the research content of this paper.In chapter two,the inverse problem of using market observation data to reconstruct the drift rate is mainly studied.After proper function changes,the drift rate function can be characterized as the first-order term coefficient in the parabolic equation,and then the method in the differential equation is used to convert it into the source term coefficient in the parabolic equation.Firstly we use the Lebesgue control convergence theorem and Schauder theory to get the existence of the solution to the inverse problem,and finally obtain the uniqueness and stability of the solution.In chapter three,we discuss the convergence of the optimal solution of an inverse problem with a developmental source term.Since the observation data is given discretely,we use the piecewise difference technique to give an approximate continuous observation function,and then use the optimization theory to establish the necessary conditions for the optimal solution based on continuous observations,and finally prove the convergence of the optimal solution.In chapter four,we consider the inverse problem of reconstructing local volatility from the market observation data of Schwartz model.Firstly,the linearization method is introduced to transform this problem into a problem of identifying the source term of a parabolic equation.Secondly,a penalty functional is constructed under the optimal control framework,and then the existence of the minimal element is proved.Finall y,the uniqueness and stability of the optimal solution are obtained through a series of derivations.In chapter five,we summarize the work of this paper and narrate the work arrangement for the next step.