Market Expectations, Long Term Risk, and Stochastic Spectral Theory

该项目的重点是学习市场参与者对未来资产回报概率分布的预期，这些资产的期权的当前价格结合经济中潜在风险回报权衡的假设和历史数据。风险评估基于历史数据。 历史方法的局限性在于潜在地低估了在所考虑的历史数据中没有发生的事件的概率。 本研究的目标是通过发展理论和方法来改进市场的概率模型，以校准它们，使其与历史数据之外的其他信息源相匹配。 这将有助于将风险管理和投资决策建立在更坚实的基础上，并将有助于金融服务业和市场监管机构从期权的市场价格中提取隐含的概率分布，以改善风险管理。该项目是跨学科的，借鉴了运筹学，经济学，概率论和数学分析，并将对教育和人力资源开发产生积极影响。该方法是基于罗斯最近的恢复定理的深远扩展，该定理表明，当经济中的所有不确定性（风险）被建模为离散时间不可约的有限时，状态马尔可夫链和随机贴现因子是转移无关的，则存在唯一的从期权价格恢复马尔可夫链的转移概率矩阵。我们的目标是延长恢复方法的一般类的连续时间马尔可夫过程，包括扩散和跳跃扩散。另一方面，我们的目标是放松过渡的独立性假设的基础上，汉森和Scheinkman的长期风险。 我们的方法旨在结合联合收割机结构假设的随机贴现因子从宏观金融文献与联合校准的模型，以目前观察到的市场期权价格与历史时间序列数据的基础资产回报。这将涉及马尔可夫过程的谱理论的分析发展和特定类别模型的恢复的计算实现。

This project focuses on learning expectations of market participants about probability distributions of future asset returns from the current prices of options on those assets combined with assumptions and historical data about the underlying risk-return trade-offs in the economy. Risk assessments are based on historical data. The limitation of the historical approach is in potentially underestimating the probability of events that did not occur in the historical data under consideration. The goal of this research is to improve probability models of markets by developing theory and methods for calibrating them to additional sources of information in addition to historical data. This will help put risk management and investment decision making on a more solid foundation and will aid the financial services industry and market regulators in extracting implied probability distributions from market prices of options to improve risk management. The project is interdisciplinary, drawing on the fields of operations research, economics, probability theory and mathematical analysis and will have a positive impact on education and human resources development.The methodology is based on far-reaching extensions of the recent Recovery Theorem of Ross that shows that when all uncertainty (risk) in the economy is modeled as a discrete-time irreducible finite-state Markov chain and the stochastic discount factor is transition independent, then there exists a unique recovery of the Markov chain's transition probability matrix from options prices. We aim to extend the recovery methodology to general classes of continuous-time Markov processes, including diffusions and jump-diffusions. On the other hand, we aim to relax the transition independence assumption by building on the fundamental work of Hansen and Scheinkman on long term risk. Our approach aims to combine structural assumptions on the stochastic discount factor drawn from the macro-finance literature with the joint calibration of the resulting models to currently observed market options prices together with historical time series data on the underlying asset returns. This will involve analytical development of the spectral theory for Markov processes and computational implementations of recoveries in specific classes of models.