对于期权定价问题，拉氏变换方法是非常有效的方法之一，然而该类方法对于某些复杂模型不适用，而且该类算法缺少收敛阶理论。本项目拟研究一类新型拉氏变换方法（拉氏变换方法与差分方法混合），该类方法能够解决更一般模型下的衍生品定价问题，同时算法非常简单有效，并且可以保持算法的精度。本项目拟研究该类新型拉氏变换方法求解跳跃模型、状态转换模型、波动率模型、分数阶模型等复杂模型下的美式期权定价和奇异期权定价问题；研究该算法的收敛阶理论；应用该类算法求解股票抵押贷款产品的定价问题。

Laplace transform method is one of the effective methods for option pricing problems. But this class of Laplace transform methods cannot be applied to some complex models and lack of convergencer rate theory. The aim of this project is to develop a new class of Laplace transform methods (hybrid of Laplace transform methods and finite difference methods). This new method can solve more general model derivative pricing problems and its implementation is easy and accurate. In this project, we study the new Laplace transform methods for American option pricing and exotic option pricing problems under complex models (jump diffusion models, regime-switching models, volatility models and fractional order models etc). We study the theory of convergence rates for this new method and apply it to the extended derivative pricing problems - stock loans.