Model Uncertainty and Optimal Transport

模型不确定性问题最近受到金融数学领域的广泛关注。 在应用方面，这是由于最近的金融危机，对模型的过度自信起了重要作用。 在数学方面，原因是与其他领域的大量有趣的联系已经出现。特别是最优传输、随机分析、非线性偏微分方程、Skorokhod 嵌入、非线性期望、决策理论和准确定分析。 在这个项目中，研究者研究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。 学生参与该项目的工作。 该项目的第一部分涉及模型不确定性下的金融衍生品的定价和对冲。 如果基础证券和流动性期权都被用作对冲工具，超级复制原理就会产生与市场数据一致的衍生品价格的尖锐而稳健的界限，而无需对模型动态做出过分强烈的假设。 此外，这种方法产生了强大的对冲策略来管理相关风险。 在可以交易看涨期权连续体的温和理想化中，超级复制与 Monge-Kantorovich 最优传输问题（即证券边际法则之间的传输）密切相关。 金融的无套利原则对这种运输施加了概率结构，这导致了本项目中大力研究的所谓鞅最优运输问题。 这项研究导致了最坏情况情景和对冲策略的计算，并且对概率论和分析也很有意义。 该项目的第二部分致力于了解模型不确定性对投资者（例如退休基金）最佳投资组合选择的影响。 第三部分同样与衍生品定价相关，研究金融工程问题：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，该项目展示了如何建立一个风险中性模型，该模型针对一组给定的流动性交易工具（不一定是普通类型）进行校准。 学生参与该项目的工作。

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.