Robust Data-Driven Applications in Finance

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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