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Functional Calculus for Pathwise Hedging

路径对冲的函数微积分

基本信息

批准号：
EP/W007215/1
负责人：
Johannes Muhle-Karbe
金额：
$ 10.23万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW007215%2F1
关键词：
Functional Calculus Pathwise Hedging

项目摘要

Uncertainty about the underlying probabilistic dynamics is a key problem in finance. Indeed, in this context, every model is at best a useful but rough approximation of reality. It is therefore crucial to discern which results depend delicately on the chosen model assumptions, and which ones are robust, in that they can be deduced from broad qualitative properties. Accordingly, the analysis of model uncertainty and how to take it into account via the robust pricing and hedging of financial derivatives are key directions of current research. From a mathematical perspective, this naturally leads to deep questions about what parts of stochastic calculus can be developed in a purely pathwise manner. The present research project will make profound contributions at this intersection of stochastic analysis and its financial applications. The key tool in this context is "functional calculus", which describes the actions of functionals on general, path-dependent random systems. In a financial context, this allows to link "superhedging strategies" (that completely hedge a given financial risk) to path-dependent optimality equations. These in turn lead to explicit solutions in some concrete examples and generally open the door to the deployment of efficient numerical methods. The present project explores this approach in a number of practically important settings, e.g., the case where a complex financial derivative is not only hedged with the underlying asset that determines its payoff, but also by continuously readjusting a position in simpler derivatives. Such risk management strategies are routinely used in practice, but the underlying theory is not well understood - a gap in the literature that will be filled in this project using functional calculus. In addition to these financial applications, the proposed research will also further develop the general theory of functional calculus in a number of fundamental ways, e.g., to functionals that do not evolve continuously in time.

潜在的概率动力学的不确定性是金融中的一个关键问题。事实上，在这种情况下，每一个模型充其量都是对现实的有用但粗略的近似。因此，关键是要辨别哪些结果微妙地依赖于所选择的模型假设，哪些是稳健的，因为它们可以从广泛的定性属性中推导出来。因此，分析模型的不确定性以及如何通过金融衍生品的稳健定价和套期保值来考虑模型的不确定性是当前研究的重点方向。从数学的角度来看，这自然会导致关于随机微积分的哪些部分可以以纯粹的路径方式发展的深层次问题。目前的研究项目将在随机分析及其金融应用的交叉点做出深远的贡献。在这方面的关键工具是“功能演算”，它描述了一般的，路径依赖的随机系统的泛函的行动。在金融背景下，这允许将“超级对冲策略”（完全对冲给定的金融风险）与路径依赖最优方程联系起来。这些反过来又导致显式的解决方案，在一些具体的例子，一般打开大门，部署有效的数值方法。本项目在一些实际重要的环境中探索这种方法，例如，复杂的金融衍生工具不仅通过决定其收益的基础资产进行套期保值，而且还通过不断调整简单衍生工具的头寸进行套期保值。这样的风险管理策略在实践中经常使用，但基本的理论并没有很好地理解-在文献中的空白，将在本项目中使用功能演算来填补。除了这些金融应用外，拟议的研究还将以一些基本方式进一步发展函数微积分的一般理论，例如，不随时间连续演化的泛函。