传染性风险：模型拓展、模拟算法及其在金融与保险中的应用

In this research proposal, we consider first introducing a variety of new stochastic processes for modeling contagion risk by extending the classical Hawkes' self-exiting jump process. The extensions of contagion models are based on the further generalization to the intensity process of the Hawke point process by adding external jumps or independent diffusion components. The generalized model hence would be able to capture not only the endogenous factors of the underlying system for contagion effects but also small and large exogenous factors outside the system. The extensions to multi-dimensional versions are also proposed. The theoretical distributional properties of these models will be investigated, such as the Laplace transforms of the intensity processes and the probability generating functions of the point processes. These results then would help us to develop numerical algorithms for exactly simulating these newly-introduced contagion models. We will combine several simulation approaches together: distributional decomposition, probability transformation and the acceptance-rejection method, all in analytic forms. Our algorithm would be pretty accurate, efficient and flexible: it has no bias or truncation error, and does not involve any numerical inversion; the stationary condition for Hawkes process is not required; both the cases with stationary and non-stationary intensities can be easily generated. Some numerical examples will be provided as illustrations and for validation. Then, we apply these new models to finance and insurance for modeling the contagion effects. For applications to finance, we mainly focus on option pricing. We have observed that there are contagious jumps in the asset returns in the finical markets, and these jumps are modeled by our new models of point processes, in order to capture the contagion effect of the arrival of these jumps. Vanilla options such European call and option will be re-priced under our new models, and the comparisons with other existing models will be provided. The classical problems of option pricing such as "volatility smile" will be reinvestigated. For applications to insurance, we focus on the ruin problem - the probability of bankruptcy of the insurance company. The arrival of loss claims is modeled by our new point processes, as the loss claims in the real world often occurs contagiously or with clustering. The ruin probability is estimated under the new models via efficient simulations with change of probability measures.

本课题提出一系列新的理论模型，并应用于模拟金融与保险业中的“传染性”风险。研究主要内容为：（1）基于经典传染模型Hawkes随机点过程进行理论拓展；在Hawkes模型原有“自激式”跳跃项的基础上再加入“外激式”跳跃项和扩散过程，这是为了可以同时捕捉到内生风险因素以及较大幅度和较小幅度的外生风险因素；我们也会考虑多维的模型拓展；然后我们对这些不同模型的概率分布特征进行理论推导。（2）在这些理论基础上，研究如何对它们进行高效、准确的蒙特卡罗模拟的理论算法。（3）最后，我们将这些新建立的传染理论模型及模拟算法应用到金融和保险中；在金融的应用，我们将集中研究在资产回报存在传染性跳跃的环境下，如何确定期权价格，以及如何做好相应的对冲策略和风险管理；在保险精算方面的应用，我们则研究在保险赔偿事件存在传染性的情况下，如何估算保险公司的破产概率，以及相应的风险管理措施。

本课题主要提出一系列新的理论模型用于更有效地模拟金融与保险业中出现的“传染性”风险。首先，我们基于经典传染模型Hawkes随机点过程进行模型的理论拓展，其拓展将集中在点过程的强度过程上：在Hawkes模型原有的“自激式”跳跃项的基础之上再加入“外激式”跳跃项以及扩散过程，这样就可以同时分别捕捉到内生风险因素以及较大幅度和较小幅度的外生风险因素。我们同时会考虑将这个一维的模型拓展到多维的情况。在这些新的模型框架下，我们将对它们具体不同的分布特征进行理论推导。再在这些理论的基础上，研究如何对它们进行高效、准确的蒙特卡罗模拟的理论算法。最后，我们将这些新引进的传染理论模型及其模拟算法应用到金融和保险中。