基于跳跃扩散过程的非连续随机模型的资产定价理论及其风险管理应用

As a compound of the diffusion process and jump process, stochastic models based on jump-diffusion process play an important role in asset pricing theory. However, because the jump process is sophisticated in mathematics (especially using mathematical technology in continuous form), jump-diffusion pricing models only have two analytical pricing formulas for two kinds of jump-size distribution, respectively. Innovatively, this project pioneers to explore the non-continuous form of stochastic pricing models based on the jump-diffusion process with the aid of discrete mathematics. ..In the doctoral stage, the applicant studied the discrete-form jump-diffusion model, established the corresponding theoretical framework and obtained the analytical pricing formula for European option (related articles published in MANAGEMENT SCIENCE). On this basis, this project completes asset pricing theory and its risk management application through studies on jump-diffusion models in non-continuous form (not just discrete form): using discrete mathematical techniques to establish the theoretical model and expand its pricing range to American options, etc.; improving the model's parameter estimation and calibration methods and optimizing the efficiency of the algorithm; developing the application of the model in risk management through the translational deformation of asset pricing theory.

作为扩散过程和跳跃过程的复合，基于跳跃扩散过程所建立的随机模型在资产定价理论中占有重要地位。但也正由于跳跃过程在数学处理上(尤其是运用连续形式的数学技术)较为困难，跳跃扩散定价模型仅仅在两种跳幅分布下具有封闭形式的解析定价公式。本项目首举创新，试图通过离散形式数学技术的运用来探索基于跳跃扩散过程的非连续形式的随机定价模型。..申请人在博士阶段，对于离散形式跳幅分布下的跳跃扩散模型进行了研究，建立了相应的理论框架并得到了欧式期权的定价结论(相关文章发表在MANAGEMENT SCIENCE杂志)。本项目拟以此为基础，研究并完善非连续形式(而不只是离散形式)下的跳跃扩散模型的资产定价理论及其风险管理应用：运用离散数学技术进行模型的理论建立、并扩大其理论定价范围至美式期权等；完善模型的估参及校准方法、并优化算法效率；通过资产定价理论的平移变形来研究模型在风险管理中的应用。

本项目计划通过离散形式数学技术的运用来探索基于跳跃扩散过程的非连续形式的随机定价模型的理论和应用，按照执行计划较好完成了研究目标。.主要研究内容和结果有：(1)运用离散数学技术进行跳跃扩散模型的理论建模，(2)推导基于非连续跳跃扩散过程的欧式期权定价解析公式，(3)扩展该模型的理论定价范围至变额年金保险，(4)完善该模型的估参及校准方法推导基于非连续跳跃扩散过程的美式期权定价渐近公式，(5)通过资产定价理论的平移变形完成该模型在供应链信用风险度量中的应用，(6)拓展离散数学技术的应用完成多维指标度量模型的构建和系统重要性风险度量模型等。.本项目产出学术论文三篇，会议论文一篇，工作论文四篇。.本项目支持学术会议多人次，并培养研究生两人。