The Reinsurance-investment Problem And The American Options Pricing Problem Under The Lévy Process

In this dissertation,we consider the reinsurance-investment problem and the pricing problem of American options under the Lévy process.The first chapter of this article is the preparatory knowledge part,which roughly lists the relevant theories and methods needed in this article.In Chapter 2,we mainly studies the insurance company’s reinsuranceinvestment strategy choice.Reinsurance is an effective means for insurance companies to effectively avoid their own risks by purchasing insurance business,investment can help insurance companies increase their returns.We assume that the surplus process of an insurance company is driven by the jump diffusion process.Next,we add a jump term to the Heston model,so that the risk assets invested by insurance companies follow the jump-diffusion process and have random volatility.We take the sum of the surplus and the income as the wealth process of insurance companies.Consider if the insurance company is an aversion insurance company,we give an objective function with a penalty term and the HJB equation that it satisfies.Then,by solving the HJB equation,the insurance company’s robust optimal strategy and the corresponding objective function are obtained.In Chapter 3,we consider the pricing of multi-asset American options under the CGMYe process.The CGMYe process is composed of CGMY process and a diffusion process.This model has the nature of asymmetric spikes and heavy tails,it makes up for the shortcomings of the diffusion process that cannot accurately describe the dynamic changes of asset prices.We assume that the asset price obeys the multi-dimensional Lévy process,and give its characteristic function.The Fourier transform method is used for the price formula of the option to finally obtain the fractional partial differential equation that the price of the European option satisfies.Next,the free boundary problem satisfied by multi-asset American options is further given,and the American maximum option is taken as an example.With the help of the optimal-exercise boundary,the approximate analytical formula for the price of American maximal option options applicable to short-term and long-term can be obtained using the second approximation method.