Applications Of Lévy Processes In Fnance And Insurance

Compared with the Black-Scholes model, the risk model driven by Lévy processeswith jumps seems to provide a better ft to the market data of fnance and insurance.In the risk theory, Lévy processes with jumps can capture two features. One featureis leptokurtosis and unsymmetrical, and the other is discontinuities in the trajectory.Thus, Lévy processes can not only overcome the default of continuities of normaldistribution, but also describe the efect of the jumps well. Therefore, Lévy processeswith jumps has become more and more important in modeling fnance and insurance.In this dissertation, we mainly study the following two problems by Lévy processestheory.One is to study the problem of the neutral pricing in the credit derivatives. Creditderivatives are increasingly important in fnancial markets. They provide methodsto hedge credit risks that arise during everyday trading, as well as more chances topromote investment return. In this paper we propose a new model with dependentrisks in the Cox framework, while the jumps of the default intensity are describedby Lévy processes. Specially, we consider that the default intensity processes aresubordinator processes and certain Vasicek processes driven by subordinator processes,respectively. Correspondingly, we give the closed forms of joint survival probability andthe fair spread of some credit derivatives. The other is to study the optimal reinsuranceproblem under the risk model with the thinning dependence structure. Many authorshave studied the optimal reinsurance problem under the classic risk model. As thescale of the insurance expands, the business varieties of insurance companies diversify.We introduce the risk model with the thinning dependence structure and study theoptimal proportional reinsurance strategy and the excess of loss reinsurance strategyby Lévy processes theory.The dissertation is organized as follows:In the frst part, we mainly study the problem of the neutral pricing in the creditderivatives. We consider a correlated reduced form credit risk model. While the de-fault intensity processes are assumed to be a subordinator processes and certain Vasicek processes which is the solution of some stochastic equation driven by subordinator pro-cesses, respectively. We obtain the explicit expression for the joint Laplace transformof the default intensity processes and the cumulative default intensity processes, andthen we get the closed form for the joint distribution of default times. We also give theclose form of the nth default probability for a portfolio, and get the explicit expressionfor the fair spread of credit default swap (CDS) with the counterparty risk under theproposed credit risk model. We also present the numerical solutions for the fair spread.In second part, we investigate the reinsurance strategy in the risk model with thin-ning dependence structure. For the optimization proportional reinsurance problem, weinvestigate the optimal reinsurance strategy that maximizes the adjustment coefcientand minimizes the variance of the surplus under the given expected proft, respectively.We derive the optimal solutions and the numerical illustrations to show the impact ofthe dependence among the classes of business on the optimal reinsurance arrangements.Similarly, for the optimization excess of loss reinsurance problem, we investigate theoptimal reinsurance strategy that maximizes the expected exponential utility and theadjustment coefcient, respectively. Correspondingly, we derive the optimal solutionsand give a numerical example to analyze the impacts of the thinning factor on eachstrategy.