An Inverse Volatility Problem Of Financial Products Linked With Gold Price

With the increasing internationalization of economic activity,financial area has been im-mediately affected by the changes of international markets.People urgently need some neces-sary financial instruments to avoid risks,to ensure the value of assets.The financial products linked with gold price,as hedging or investment profit,are getting more and more attention.Based on the background,this paper devotes to consider the inverse problem of financial products linked with gold price.The profits of this product not only depends on some cumu-lative indexes of gold price but also some trigger indexes.Thus,this kind of problem can be regarded as a double barrier option question.Using the theories of partial differential equation,the problem becomes a inverse problem of determining the first coefficient from second order parabolic equation,the problem is ill-posed.The purpose of our studying is to eliminate the ill-posed of the inverse problem,to obtain the optimal solution.Being different from the common inverse volatility problems arising in the area of option pricing in which the underlying math-ematical model is a semi-infinite initial value problem,our model is a initial-boundary value parabolic equation defined on a bounded domain.Therefore,the inverse problem considered in the paper and the obtained results can be regarded as the beneficial supplement of the inverse option pricing problem.The main contents are arranged as follows:The first chapter is the introduction of paper,and it briefly introduces the development history of the inverse problem of financial derivatives pricing and PDE inverse problem,the research status of those problem has also been summarized.The second chapter,we present some needed preparatory knowledge in the process of model establishment,and briefly describes the establishment process of Black-Scholes option pricing model.The third chapter,linearization around constant volatility,the problem of the second chap-ter is translated into a heat function.Based on the optimal control framework,the existence of the minimizer is considered,and the necessary condition which must be satisfied by the minimizer is established.At last,the uniqueness and stability of the solution are also deduced.The fourth chapter,the model established in the second chapter is transformed into a op-timal control problem.We discuss the existence and necessary condition of the minimum for the control function.Since the optimal control problem is non-convex,one may not expect a unique solution in general.However,the local uniqueness of the solution is proved in this paper.In the end of the paper,an algorithm is proposed and some numerical experiments are given.Numerical results show that our algorithm is stable and effective.The fifth chapter is summary and disadvantages,we also list some expectation.