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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=714466
• 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.

定量投资策略在现代金融市场中起决定性作用。他们确定金融机构如何处理他们的频繁交易以及养老基金如何管理我们的储蓄。这些策略基于金融市场的理论和经验模型。许多策略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，可能会发生灾难性事件。因此，非常需要研究强大的投资策略，即成功的策略并不取决于特定模型的假设。 该研究计划使用概率和几何形状的新工具开发了强大的投资组合理论。同时，我们推进了发现数据科学中重要应用的数学理论。财务思想来自随机投资组合理论和普遍投资组合理论。这些理论基于金融市场的持续特征和结合方式提供了强大的投资策略。我们建议在更现实的设置下结合这些方法。从数学上讲，这些策略与最佳运输有关，这涉及以总体成本效益的方式分配市场场景以资产分配。如果我们将不断发展的市场视为在高维空间中旅行的点，几何形状将进入图片。我们以合适的几何形状赋予该空间，以使市场移动的方向和幅度直接影响投资组合。可以使用最佳传输和信息几何形状研究这种几何形状。我们建议从这种几何观点研究市场的经验特性。通过采用多学科方法，我们希望通过包括算法和软件包在内的实用应用来开发新颖的数学理论。特别是，我们旨在通过研究新的成本功能来概括古典的Wasserstein几何形状。 研究计划的结果将提供强大的工具来管理与市场集中度和波动性有关的投资组合和风险。他们还期望他们对我们对最佳运输和信息几何形状的理解，它们的连接开始引起很多关注。预计开发的数学结果和算法将在定量交易之外有用，并将导致进一步的理论发展以及统计和机器学习中的应用。