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Machine Learning for Quantitative Finance
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Lab - Machine Learning
Machine Learning has gathered attention and evolved tremendously in recent years, due to its main application, namely artificial intelligence.
In order to navigate the vast domain of Machine Learning we find it useful to recognize the symmetry between two frameworks.
- The Mean-Covariance Framework, where randomness is modeled by only the mean and covariance. This framework is equivalent to the Gaussian framework, but it encompasses a much broader set of distributions. In this framework, transformations are only affine, and concepts such as uncorrelation impose structure.
- The Probabilistic Framework, where randomness is modeled by full distributions. In this framework, transformations are non-linear, and concepts such as independence impose structure.
On the one hand, the Mean-Covariance framework plays a dominant role across all parts of quantitative finance: Pricing (CAPM, APT); Econometrics (spectral analysis, Wold decomposition, filtering); Risk management (duration/Greeks repricing, hedging, risk attribution, stress testing); Portfolio construction (mean-variance optimization, factor investing, Black-Litterman, performance attribution).
On the other hand, all the techniques in Machine Learning are direct probabilistic generalizations of their simpler mean-covariance counterparts.
Recognizing the symmetry between these two frameworks provides an effective learning experience. Accordingly, the Lab's coverage of machine learning is divided into the following portions:
- Mathematical Statistics for Finance: the theoretical foundations behind the symmetry between the Mean-Covariance versus the Probabilistic Framework; and the theoretical approaches to learn from data in both frameworks;
- Linear Mean-Covariance Statistics: practical ways of learning observational and causal models from i.i.d. data samples and taking optimal decisions within the Mean-Covariance Framework;
- Probabilistic Machine Learning: machine learning/artificial intelligence models, which are practical ways of learning observational and causal models from i.i.d. data samples and taking optimal decisions within the Probabilistic Framework;
- Time Series and Reinforcement Learning: the practical ways of learning observational and causal models and taking optimal decisions in both the Mean-Covariance Framework and the Probabilistic Framework, when data is not i.i.d.
The Lab's coverage of machine learning is summarized in the "Machine Learning Map" below.
Mathematical Statistics for Finance
Mathematical Statistics for Finance provides an in-depth discussion of the mathematical topics which lie at foundation of the applications of statistics to finance:
- The roots of the symmetry between the Mean-Covariance versus the Probabilistic Framework;
- The theory to learn from data in both frameworks.
More precisely Mathematical Statistics for Finance consists of the following parts of the "Data Science Map":
- The Probabilistic Framework describes the essential tools to operate in the Probabilistic Ecosystem, where:
- Statistical relationships among variables are modeled by probability distributions;
- Transformations among variables are non-linear;
- Structure is imposed via independence or more generally via conditional independence, which follows from the notion of conditioning.
- The Mean-Covariance Framework describes the essential tools to operate in the Mean-Covariance Ecosystem, where:
- Statistical relationships among variables are modeled by their mean-covariance equivalence classes;
- Transformations among variables are linear or affine;
- Structure is imposed via uncorrelation, or more generally partial uncorrelation, which follows from the notion of L² linear projection.
- Decision theory under risk addresses modeling and optimization of decisions under the assumption that the joint mean-covariance classes, or probabilistic distributions, of all random variables are known.
- Estimation leverages decision theory under uncertainty to learn, from data, relevant features of the joint mean-covariance classes, or probabilistic distributions, when these are not known. Key concepts include elicitability, asymptotic/random matrix theory, hypothesis testing.
- Inference covers how to learn not only from data, but also from subjective opinions, both mean-covariance classes (Black-Litterman) and probability distributions (minimum relative entropy).
Mean-Covariance Learning
Linear Mean-Covariance Statistics represents the linear blueprint for Probabilistic Machine Learning.
It covers practical ways of learning observational and causal models from i.i.d. data samples and taking optimal decisions within the Mean-Covariance Framework.
The key ingredients are linear factor models, which model all mean-covariance structures: supervised (linear regression); unsupervised (principal component and factor analysis); hybrid (canonical correlation, total least squares); and causal (structural equation models).
This part also covers the estimation of linear factor models, namely mean/loadings and (high-dimensional) covariance matrices, in the context of financial applications.
This part covers the below portion of the "Data Science Map".
Probabilistic Machine Learning
Probabilistic Machine Learning discusses machine learning/artificial intelligence models, presented as generalizations of Linear Mean-Covariance Statistics.
It covers practical ways of learning observational and causal models from i.i.d. data samples and taking optimal decisions within the Probabilistic Framework.
The key ingredients are conditional distributions, which model all probabilistic structures: supervised learning (point and probabilistic); unsupervised learning (autoencoders and graphical models); and one-period reinforcement learning (causal Bayesian networks).
This part also covers the estimation of specific conditional distributions in the context of financial applications.
This part covers the below portion of the "Data Science Map".
Time Series and Reinforcement Learning
Time Series and Reinforcement Learning covers the dynamic counterparts of Linear Mean-Covariance Statistics and Probabilistic Machine Learning.
It covers practical ways of learning observational and causal models and taking optimal decisions in both the Mean-Covariance Framework and the Probabilistic Framework, when data is not i.i.d.
As such, this part includes multivariate econometrics, continuous time stochastic processes, and optimal sequential decision making.
This part covers the below portion of the "Data Science Map".