几类相依风险模型下的资产定价与随机控制问题研究

The financial crisis merged around 2007-2008 has vividly demonstrated the significance of interaction between default and liquidity in financial markets. As a consequence, researchers have realized that the correlation among individual risks should be considered theoretically. In this project, the above concerns would be addressed according to a three-step scheme. The first initiative is to take the contagious characteristics of risks into the reduced-form intensity based model. To achieve that, default indicators of all the references will be introduced into the default intensity processes, thus any default in the market would cause variations in the prices and default probabilities of other references. After that, the portfolio optimization problem, in which default contagion risks are included, will be figured out. The primal problem would be converted to an equivalent dual problem, therefore an explicit-formed optimal investment strategy could be obtained by exploiting the inherent dual relation. In the second stage, liquidity risk will be further combined into the model，while a Hawkes process, which exhibits self-excitation and cross-excitation behaviors, would be used to illustrates the positive default-liquidity spiral phenomenon between the credit risk and liquidity risk, i.e., model the correlations between them. After obtaining the price of a corporate bond with credit risk in an illiquidity market, the two factors respectively related to the credit risk and the liquidity risk in the yield spread of the bond would be disentangled, which will be followed by exploring the hedging strategies of these two risks. Finally, the modeling of stock price affected by its volatility with common market factor will be established, in which a Markov-switching jump diffusion process will be introduced. Based upon this, the portfolio optimization problem related to volatility derivatives and stocks will also be studied. In summary, the anticipated results will further extend the theories in risk forecasting and asset pricing, whilst play an increased role in quantifying the financial risks and designing investment programs.

在投资市场中，合理控制风险对稳定收益的实现十分重要，然现有研究大多重视各类风险各自的表现，对其之间相关性的研究却不够深入。本项目针对市场中的几类相依风险建模，并以此分析相关产品定价与最优投资策略问题。首先在简约模型框架内考虑多主体信用风险模型，将所有主体的违约状态引入违约强度过程，探讨信用风险在整个市场中的传染扩散；基于此，研究具有传染性信用风险资产的最优投资组合问题。另一方面，针对信用风险和流动性风险间的相关性进行建模，并研究对市场上受流动性风险影响的可违约债券进行定价。以此为基础，将债券超额收益中的信用溢差和流动性溢差进行分离，并研究不同风险的管理和对冲策略。最后，引入马氏链表示的市场公共因子并对股票价格过程进行建模，进而研究波动率衍生产品的定价和最优投资组合问题。本项目的研究将拓展金融资产风险预测与定价模型的理论及方法，所得结果将对量化市场风险和构建投资方案具有重要价值。

在投资市场中，合理控制风险对稳定收益的实现十分重要，然现有研究大多重视各类风险各自的表现，对其之间相关性的研究却不够深入。本项目针对市场中的几类相依风险建模，并以此分析相关产品定价、风险管理以及最优投资策略问题。.考虑股票价格过程与其波动率之间的相关性，方法一是用随机波动率模型，包括跳扩散过程以及马氏链调节的随机过程。首先引入马氏链表示的市场公共因子并对股票价格过程进行建模，进而研究波动率衍生产品的定价和最优投资组合问题。另外，用扩散过程对波动率建模，并且波动和资产价格本身存在相关性，同时考虑到期权发行方的违约风险，定价了欧式期权和亚式期权并且做了数值模拟，讨论了不同期权价格对于各参数的敏感性，以此指导投资。方法二用GARCH模型刻画波动率的聚集现象，同时用波动率随机扰动项与资产过程扰动项的相关性来刻画风险的相依性，在此模型下定价各种奇异期权和管理层股票期权。.市场上的多个金融资产不仅价格变化有相关性，其信用状况也会相互影响。在简约模型框架内考虑多主体信用风险模型，将所有主体的违约状态引入违约强度过程，探讨信用风险在整个市场中的传染扩散；基于此，研究具有传染性信用风险资产的最优投资组合问题。另外，大量的场外期权以具有相关性的资产为标的物，我们研究了多个资产的最大（或最小）值为参考价值的复杂期权定价，对于无法一些得到精确解的情况，我们也给出了精度比较高的数值解。.本项目的研究将拓展金融资产风险预测与定价模型的理论及方法，所得结果将对量化市场风险和构建投资方案具有重要价值。