Model Uncertainty and Optimal Transport

最近，模型不确定性问题在金融数学中受到了广泛的关注。在应用方面，这是由于最近的金融危机，对模型的过度自信发挥了重要作用。在数学方面，原因是出现了大量与其他领域的有趣联系；特别是最优运输、随机分析、非线性偏微分方程、Skorokhod嵌入、非线性期望、决策理论和拟确定性分析。在这个项目中，研究人员研究了模型不确定性如何影响金融市场定价、对冲和投资的基本任务。学生被包括在这个项目的工作中。本课题的第一部分是关于模型不确定性下金融衍生品的定价和套期保值问题。如果标的证券和流动性期权都被用作对冲工具，超级复制原理将为衍生品价格产生与市场数据一致的尖锐而稳健的边界，而不会对模型动态做出过强的假设。此外，这种方法产生了一种稳健的对冲策略来管理相关风险。在一种温和的理想化中，可以交易一系列看涨期权，超复制与Monge-Kantorovich最优运输问题密切相关，即证券边际定律之间的运输。金融学中的无套利原理将概率结构强加于这种运输方式，这就导致了所谓的最优运输问题，也就是这一课题中重点研究的问题。这项研究导致了最坏情况的计算和套期保值策略，也是概率论和分析的重要内容。该项目的第二部分致力于了解模型不确定性对退休基金等投资者的最优投资组合选择的影响。第三部分同样与衍生品定价有关，研究金融工程的一个问题：如何通过算法构建与金融市场中观察到的期权报价一致的市场模型？更准确地说，该项目展示了如何构建一个风险中性模型，该模型针对一套给定的流动性交易工具进行校准，不一定是普通的香草型工具。学生被包括在这个项目的工作中。

The problem of model uncertainty has recently received widespread attention in financial mathematics. On the application side, this is due to the recent financial crisis where overconfidence in models played an important role. On the mathematical side, the reason is that an abundance of interesting connections to other areas have emerged; in particular, optimal transport, stochastic analysis, nonlinear partial differential equations, Skorokhod embeddings, nonlinear expectations, decision theory, and quasi-sure analysis. In this project, the investigator studies how model uncertainty influences the fundamental tasks of pricing, hedging and investment in financial markets. Students are included in the work of the project. The first part of this project is concerned with the pricing and hedging of financial derivatives under model uncertainty. If both the underlying security and liquid options are used as hedging instruments, the superreplication principle yields sharp and robust bounds for derivatives prices that are consistent with the market data, without making excessively strong assumptions about model dynamics. Moreover, this approach yields a robust hedging strategy to manage the associated risk. In a mild idealization where a continuum of call options can be traded, superreplication is intimately linked to a Monge-Kantorovich optimal transport problem, namely, a transport between the marginal laws of the security. The no-arbitrage principle of finance imposes a probabilistic structure on this transport, which leads to the so-called martingale optimal transport problem that is studied vigorously in this project. The investigation leads to the computation of worst-case scenarios and hedging strategies, and is also of great interest to probability theory and analysis. A second part of the project is dedicated to understanding the impact of model uncertainty on the optimal portfolio choice for an investor such as a retirement fund. A third part, again related to the pricing of derivatives, studies a problem of financial engineering: How does one algorithmically construct market models that are consistent with quoted option prices as observed in financial markets? More precisely, the project shows how to build a risk-neutral model that is calibrated to a given set of liquidly traded instruments, not necessarily of plain vanilla type. Students are included in the work of the project.