Robust Data-Driven Applications in Finance

资产管理的一个基本方面不仅围绕着寻找最佳资产进行投资，而且围绕着他们应该如何交易这些资产并管理相关风险。随后，自哈里•马科维茨（Harry Markowitz） 1952年发表《现代投资组合理论》（Modern portfolio Theory）的开创性论文以来，对投资组合优化进行了详尽的研究。传统的方法包括固定一类概率参数模型（例如Black-Scholes），并根据经验价格过程校准模型的参数。虽然经过充分研究的模型为从业者提供了有用的基准，但许多参数模型未能捕捉到有关经验价格的风格化事实，例如非平稳性、重尾和波动性聚类。这在处理多个资产时（例如在投资组合优化中）是最重要的，因为依赖结构是高度非线性的，在实践中经常出现趋势和均值回归模式。这导致许多研究人员探索数据驱动的无模型框架。虽然这些方法可以捕获数据中的复杂结构，但随着金融市场不断从一种制度切换到另一种制度，它们容易出现参数误估和对过去时间序列数据的过拟合。因此，我的目标是在本研究中为交易策略和投资组合分配开发一个数据驱动的无模型框架的健壮扩展。在投资组合分配问题中，我们的任务是找到最优函数，该函数以过去的经验价格路径作为输入，并根据代理人的偏好输出最优投资组合分配。例如，传统上，如果我们想要最大化预期收益并最小化投资组合的收益方差，我们可以从过去的资产收益中校准参数，并根据该概率模型选择最佳的投资组合分配。如果我们想要扩展这个模型来捕捉数据中的时间依赖性，如均值回归和领先滞后模式，那么我们将需要一个更加数据驱动的路径方法。然而，在路径空间上定义函数是随机分析中一个众所周知的问题，因为它是无限维的。无模型金融建立在粗糙路径理论工具的基础上，通过利用签名变换直接在路径空间上定义优化，签名变换是一种受欢迎的工具，由于其几个代数性质。期望特征不仅可以完全表征随机过程的规律，而且我们可以将其用作路径的有限维表示。在这项研究中，我将利用粗糙路径理论的力量，以及现代机器学习工具，如生成对抗网络（GANs），来获得对错误估计、信息缺失和测量变化具有鲁棒性的解决方案。未来的工作还包括用强化学习（RL）来解决这些问题，作为对抗网络结构的一部分，确保鲁棒模型与原始参考模型不会太远。稳健金融的应用不仅限于寻找最佳投资，对稳健市场模拟器的需求也在不断增长，这些模拟器不会过度拟合过去的时间序列，并且可以适应市场机制的变化。该项目的成果将是为投资者提供新的框架和工具集，这些框架和工具集独立于建模假设，同时确保它们对未来市场变化状态的稳健和弹性

A fundamental aspect of asset management revolves around not only finding optimal assets to invest in but how they should trade such assets and manage the associated risk. Subsequently, portfolio optimisation has been exhaustively researched ever since the seminal paper of Modern Portfolio Theory by Harry Markowitz in 1952. Traditional methods consist of fixing a class of probabilistic parametric models (e.g Black-Scholes) and calibrating the models' parameters with respect to the empirical price process. While well-studied models provide a useful benchmark for practitioners, many parametric models fail to capture stylised facts about empirical prices, such as non-stationarity, heavy tails and volatility clustering. This is most important when working with multiple assets (such as in portfolio optimisation) since dependence structures are highly non-linear with trend and mean-reversion patterns occurring often in practice. This has led many researchers to explore data-driven model-free frameworks. While these methods can capture complex structure within the data, they are prone to parameter misestimation and overfitting to past time series data as financial markets continuously switch from one regime to another. Therefore, I aim to develop a robust extension to data-driven model-free frameworks for trading strategies and portfolio allocation in this research.In portfolio allocation problems, we are tasked with finding the optimal function that takes the past empirical price path as an input and outputs the optimal portfolio allocation with respect to the agents' preferences. For example, traditionally if we want to maximise the expected returns and minimise the variance of returns of a portfolio, we can calibrate the parameters from the past asset returns and choose the optimal portfolio allocation with respect to that probabilistic model. If we want to extend this model to capture temporal dependencies such as mean-reversion and lead-lag patterns within data then we will require a much more data-driven pathwise approach. However, defining a function on path space is a well known problem in stochastic analysis due to its infinite dimensionality. Model-free finance builds upon the tools of rough path theory to instead define an optimisation directly on path space by utilising the signature transform, a popular tool due to several of its algebraic properties. Not only can the expected signature fully characterise the law of the stochastic process, but we can use it as a finite-dimensional representation of the path. In this research I will leveraging both the power of rough path theory, as well as modern machine learning tools such as Generative Adversarial Networks (GANs) to obtain solutions that are robust to misestimation, missing information and change of measure. Future works also include solving such problems with reinforcement learning (RL), as part of an adversarial network structure that ensures the robust model is not too far from the original reference model. Applications to robust finance are not only limited to finding optimal investments either, there is a growing demand for robust market simulators that again do not overfit too much to the past time series and can adapt to market regime shifts. The outcomes for this project will be new frameworks and toolsets for investors that are independent of modelling assumptions whilst ensuring they are robust and resilient to changing states of future markets