Portfolio theory, optimal transport and information geometry

投资组合理论、最优传输和信息几何

• 批准号: RGPIN-2019-04419
• 负责人: Wong, TingKamLeonard
• 金额: $ 1.68万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=765693
• 关键词: Portfolio; theory; optimal; transport; information

Quantitative investment strategies play a decisive role in modern financial markets. They determine how financial institutions deal with their frequent tradings and how pension funds manage our savings. These strategies are based on theoretical and empirical models of financial markets. Many strategies are highly model-specific: if the model is wrong, or when market conditions change, disastrous events may happen. Thus, there is a strong need to study robust investment strategies, i.e., strategies whose success do not depend on the assumptions of a specific model. This research program develops robust portfolio theory using novel tools from probability and geometry. Simultaneously, we advance the mathematical theories which have found important applications in data science. The financial ideas come from stochastic portfolio theory and universal portfolio theory. These theories provide robust investment strategies based on persistent features of financial markets and ways to combine them; we propose to combine these approach under more realistic settings. Mathematically, these strategies are related to optimal transport which is about assigning market scenarios to asset allocations in an overall cost-efficient way. Geometry enters the picture if we think of the evolving market as a point traveling in a high dimensional space. We endow the space with a suitable geometry such that the directions and magnitudes of market movements have direct impacts on the portfolio. This geometry can be studied using optimal transport and information geometry. We propose to study the empirical properties of the market from this geometric viewpoint. By adopting a multi-disciplinary approach, we hope to develop novel mathematical theories with practical applications including algorithms and software packages. In particular, we aim to generalize the classical Wasserstein geometry by studying new cost functions. The outcomes of the research program will provide robust tools to manage portfolios and risks related to market concentration and volatility. They are also expected to improve our understanding about optimal transport and information geometry whose connections have started to attract a lot of attention. The mathematical results and algorithms developed are expected to be useful beyond quantitative trading and will lead to further theoretical development and applications in statistics and machine learning.

量化投资策略在现代金融市场中起着决定性的作用。它们决定了金融机构如何处理频繁的交易，以及养老基金如何管理我们的储蓄。这些策略是基于金融市场的理论和经验模型。许多策略都是高度特定于模型的：如果模型是错误的，或者当市场条件发生变化时，灾难性的事件可能会发生。因此，迫切需要研究稳健的投资战略，即，成功不依赖于特定模型假设的策略。该研究计划使用概率和几何学的新工具开发稳健的投资组合理论。同时，我们提出了在数据科学中有重要应用的数学理论。金融思想来源于随机投资组合理论和泛投资组合理论。这些理论提供了稳健的投资策略，基于金融市场的持久性特征和联合收割机的方法，我们建议在更现实的设置下联合收割机这些方法。数学上，这些策略与最优运输有关，最优运输是以整体成本效益的方式将市场情景分配给资产配置。如果我们把不断演变的市场看作是在高维空间中行进的一个点，那么几何学就出现了。我们赋予空间一个合适的几何形状，使市场运动的方向和幅度对投资组合产生直接影响。这种几何可以用最优运输几何和信息几何来研究，我们建议从这种几何的观点来研究市场的经验性质。通过采用多学科的方法，我们希望开发新的数学理论与实际应用，包括算法和软件包。特别是，我们的目标是通过研究新的成本函数来推广经典的Wasserstein几何。研究计划的成果将提供强大的工具来管理与市场集中度和波动性相关的投资组合和风险。它们还有望提高我们对最优传输和信息几何的理解，它们之间的联系已经开始吸引很多关注。开发的数学结果和算法预计将在定量交易之外发挥作用，并将导致统计学和机器学习的进一步理论发展和应用。