奈特不确定性下动态投资组合选择模型与算法研究

Knightian uncertainty means that the future has a plurality of natural states, and the probability of each natural state is unknown. Such uncertainties widely exist in the financial decision-making, however, the classical portfolio theory just takes into account the risk factors whose probabiltities are known. What's more, the current research on Knightian uncertainty is mainly based on probability theory, also did not consider the jumps in price impacted by rare events. This project is aimed at these shortcomings. In return-risk analysis framework, firstly, we propose dynamic portfolio selection models based on fuzzy measure. Fuzzy number is used to capture the measurement about the expected return and the volatility of a dynamic portfolio under Knightian uncertainty. Moreover, we consider the proposed models under several constraints, such as market friction factors. Secondly, we propose dynamic portfolio selection models with uncertain jumps, considering the occurrence of rare events and their estimation. The Knightian uncertain jumps in expected return, volatility and mixed financial variables are embeded into the research about dynamic portfolio selection problem, to give more robust asset allocation strategies. Finally, we use the data of China's financial market to do the empirical analysis for the poposed models and algorithms. The research results from this project will deepen and develop the existing portfolio theory, and have a positive guiding significance upon the investment and risk management in reality.

奈特不确定性是指未来具有多个自然状态，而每个自然状况概率是不可知的。这种不确定性广泛存在于金融决策中，但经典投资组合理论大多只考虑可知概率的风险因素，而目前已有对奈特不确定性的研究主要还是在概率角度上，同时未考虑突发事件冲击下的价格跳跃情况。本项目针对这些不足，在收益-风险分析框架下，提出基于模糊测度的动态投资组合选择模型，从模糊数的形式给出了动态投资组合奈特不确定期望收益率和波动率的度量方法，并且考虑市场摩擦因素等约束条件的影响。进一步提出了带奈特不确定跳跃的动态投资组合选择模型，考虑突发事件的发生和估计，把存在奈特不确定性的期望收益率跳跃、波动率跳跃及其不同的组成形成纳入到动态投资组合选择问题研究中，给出更为稳健的配置策略。最后，利用中国金融市场数据对模型和算法进行实证检验。研究结果是对现有投资组合理论的深化和发展，且对现实投资活动和风险管理具有积极的指导意义。

早在1921年，经济学界就提出了有别于风险的不确定性概念，虽被赋予不同的名称，如模糊性、奈特不确定性等，但其本质是相同的，即意识到个体对经济状态的概率分布存在一定的未知性。投资者只有正确识别各种不确定性，承担真正的不确定性才能获得收益。.我们根据实际金融数据，对无风险资产和包括股本权证和股票在内的风险资产的收益率变化模式进行模型选择。无风险资产收益率的实证表明其带有均值回复性和厚尾性的特征，风险资产收益率的实证研究表明金融风险资产的对数收益率之间并非独立的而呈自相似性和长记忆性、收益率的分布函数具有明显的弱衰减性，这些性质说明具有长期相依性以及自相似性的分数布朗运动能够较好地刻画金融资产的这些随机现象，我们进一步用极大似然法和随机逼近方法给出参数估计量。然而，如果单纯地将分数高斯过程引入Black-Scholes模型，将会产生金融套利机会，我们推导出套利组合策略的清晰表达式，进一步提出了消除套利的具体思路。.针对投资者对损失和收益的敏感度和偏好不一样，我们研究了损失规避投资组合选择问题，并覆盖了条件风险值模型。我们设计了一种带退化求解机制的仅基于资产数量规模的低维积极集线性求解算法，并通过实例说明模型和算法的有效性。项目也研究了不确定资产收益率下的动态投资组合选择问题，提出了基于可能性理论多阶段组合收益与风险的模糊测度方法，并给出相应的求解算法。在此基础上，更进一步考虑考虑交易费用、基数约束以及借款约束对多阶段动态资产配置的影响。