• Quote
• Problems
• Key Ideas
• Books
• RL
• DL, Online and Bandits
• Classic Portfolio Selection Materials
• Machine Learning based Portfolio Selection
• Canonical Correlation Analysis
• Sample Efficient

One should avoid solving more difficult intermediate problems when solving a target problem. Vladimir Vapnik, Statistical Learning Theory, 1998

Related to stochastic optimal control - Can do model free with simulations RL and Deep RL

1. Merton Problem (Portfolio and consumption)
2. Optimal Execution - Liquidation Problem
3. Optimal Execution - Limit Order Placement
4. Optimal Stopping and Control
5. Optimal Execution for statistical arbitrage
6. Optimal execution targetting volume
7. Market Making problems
8. Pairs trading - optimal entry/ exit
9. Multi-period parametric policies (Brandt in mult-period)
10. Optimal hedging of derivatives with path dependency (JPM - explore the model and more, when is it better than monte carlo and greeks).

Simulation

Stochastic optimal control assumes a model, simulations assume some knowledge of the world (say monte carlo), alternative and more robust simulation methods ? (for example GAN's for time series).

We are looking to allocate to assets or strategies in a manner that is better than the current state of the art and to get RL working in real world finance. Reinforcement learning is a method for solving MDP's in a model free fashion. There are many MDP problems in finance and a whole mathematical methodology such as stochastic optimal control. Applications are myriad and range from investment/ consumption decisions, derivative hedging, algorithmic trading and inventory management. Solutions may have particular value when there is path dependency on an agent's decisions into the future.

In the derivative hedging method of finance, problems are usually solved in a step-wise fashion...often by calculating or adjusting the greeks, or in more awkward cases by monte carlo methods. Recent paper's hint at the ability to directly learn a hedging strategy in a greek free fashion from a simulation of the environment. In other words rather than a 2 step process - model the environment, solve the model, we can go straight from simulation to hedging, including where there are difficult real life problems such as transaction costs and path dependency and indeed complex risk adjusted functions of our final distribution of returns that we wish to maximise.

Allocation decisions within Finance lie within a most difficult environment. It is partially observed, noisy, and non-stationary, there may be outliers and regimes. Time also plays a key role and decisions again may have long term consequences. In contradistinction to standard methods we are not looking to apply single period prediction and then combine these predictions using an optimiser. This is akin to supervised learning, but in the real world our actions may have long term effects and indeed actions taken by our agents may be reacted against by the environment.

Most standard methods are single period and represent a two stage process, this involves two sets of parameters and forecast error is not utility, so we may even be optimising the wrong target. Other works give up upon some of our ability to predict and are thus more heuristic but more practical methods for allocation decisions, albeit pessimistic.

An information bottleneck is created between the supervised forecast error minimisation and the subsequent forecasts which are then used by an optimiser (some argue that this also serves as an error maximiser and indeed has its own parameters to be found). Given the noise inherent within finance and the fact that predictions are either very weak or indeed only exist for small windows of time then this makes the two stage process even more problematic.

Research has been created to address the two stage parameter estimation - Brandt and this enable a more aligned target. However most current academic work applying RL to allocation decisions is either on a very small scale or ignores basic practical realities of markets (such as transaction costs). Moody et al. appear to have been the earliest to understand these issues and attempt to have one set of parameters, a single utility, include transaction costs and directly map from inputs to actions (rather than predict then optimise).

The Moody work was nearly 20 years ago and indeed he left academic in 2003 to set up a successful hedge fund (which continues to be successful).

My goal is to advance this work using the latest in deep reinforcement learning (and potentially deep learning). The goal is to examine the state of the art, and advance it - particularly with a view to practicality, it is the author’s view that the current gap between the state of the art in RL in academia but applied within finance remains impractical. And in both parts of the research I am examining multi-period, path dependent decision making in difficult environments in a direct rather than 2 stage indirect fashion.

It should also be noted that explainability and sensitivity analysis is important in finance, black boxes are not widely trusted and indeed legally there may be cases where explainability is forced. I propose to also examine RL methods within this domain where sensitivity analysis and explainability is enabled.

A further question is if we are seeking to move directly from inputs to actions which in this case will be allocation weights with an objective of maximising say some long run utility, then there are practical questions as regards transaction costs, sparsity and indeed including practical constraints such as a draw down constraint. Also there are questions as regards throwing noisy time series into an RL agent and the best way to do this, for example if we go ‘deep’ do autoencoders have a part to play and should we be seeking to induce sparsity in our agent’s allocations?

Note that allocation problems, may in some cases be reduced to single state bandit problems and note that sometimes a poor model of the environment may be known and the agent may possibly be able to bootstrap from here. Allocations may be to experts, assets, or indeed strategies.

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• (book) Portfolio Construction and Risk Budgeting, Scherer (2015).
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• (book) Fooled by Randomness, Taleb (2007).
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• Deep Reinforcement Learning for Foreign Exchange Trading, Wang (2019).
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• The Four Horsemen of Machine Learning in Finance, Halperin (2019).
• Machine Learning for Trading, Ritter (2017).
• Reinforcement Learning With Continuous States, Ritter (2018).
• The Usefulness of Reinforcement Learning in Finance - Pres, Ritter (2018).
• Generative Bayesian neural network model for risk-neutral pricing of American index options, Jang et al. (2018).
• Reinforcement Learning Applied to Option Pricing - MSc, Martin (2014).
• Quant GANs:Deep Generation of Financial Time Series, Wiese et al. (2019).
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• Deep Learning Approximation for Stochastic Control Problems Han et al. (2016).
• Deep Reinforcement Learning for Trading, Zhang (2019).
• Model-based Reinforcement Learning for Predictions and Control for Limit Order Books, Wei et al. (2019).
• Topics in Dynamic Portfolio Choice Problems -PhD Thesis, Wimonkittiwat, (2013)
• Reinforcement-Learning based Portfolio Management with Augmented Asset Movement Prediction States, Ye et al. (2020).

• Reinforcement Learning for Stochastic Control Problems in Finance, Rao (2018).
• Deep Learning in Finance, Heaton et al. (2015).
• Deep Portfolio Theory, Heaton et al. (2016).
• Deep Learning in Asset Pricing, Feng et al. (2018).
• Adaptive Portfolio Asset Allocation Optimization with Deep Learning Obeidat et al. (2018).
• Portfolio Optimisation in an Uncertain world, de Jong (2017).
• Empirical Asset Pricing via Machine Learning, Kelly et al (2018).
• Real-valued (Medical) Time Series Generation with Recurrent Conditional GANs, Estaban (2017).
• Portfolio Choices with Orthogonal Bandit Learning, Shen et al. (2015)
• Risk-Aware Multi-Armed Bandit Problem with Application to Portfolio Selection, Huo (2017)
• Structured Online Learning with Full and Bandit Information, Johnson PhD Thesis (2016).
• Censored exploration and the dark pool problem, Ganchev et al. (2010).
• Adaptive Execution:Exploration and Learning of Price Impact, Park et al (2012).
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• Conditional time series forecasting with convolutional neural networks, Borovykh et al (2018).
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• Portfolio Selection, Markowitz (1952).
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• Parametric Portfolio Policies:Exploiting Characteristics in the Cross-Section of Equity Returns, Brandt (2004).
• Dynamic Portfolio Selection by Augmenting the Asset Space, Brandt and Santa Clara (2004).
• Testing Portfolio Efficiency with Conditioning Information, Ferson and Siegel (2007).

• Sparse and Stable Markowitz Portfolios, Brodie et al. (2009).
• A generalised approach to Portfolio Optimisation, DeMiguel (2009).
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• Hierarchical Clustering based Asset Allocation, Raffinot (2017).
• Portfolio Selection via Subset Resampling, Shen (2017).
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• Comparison of Penalty Functions for Sparse Canonical Correlation Analysis, Chalise et al. (2012).
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• Kernel Canonical Variate Analysis for Nonlinear Dynamic Process Monitoring, Samuel et al. (2015).
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• Inference for Robust Canonical Variate Analysis, Aest et al.
• Correlation Analysis 1, Tibshirani, (2013).
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• Optimisation of Trading Strategies using Parameterised Decision Rules, Towers et al. (1998).
• On the Equivalence Between Canonical Correlation Analysis and Orthonormalized Partial Least Squares, Sun et al.
• A Least Squares Formulation for Canonical Correlation Analysis, Sun et al.
• A Probabilistic Interpretation of Canonical Correlation Analysis, Bach et al. (2005).
• Moments and Absolute Moments of the Normal Distribution, Winkelbauer, (2014).
• Large Scale Canonical Correlation Analysis with Iterative Least Squares, Lu, (2014).
• Sparse canonical correlation analysis from a predictive point of view, Wilms et al. (2015).
• Variational Analysis of the Ky Fan k-norm, Ding, (2015).
• Canonical correlation analysis of high-dimensional data with very small sample support, Song, (2016).
• FDR-Corrected Sparse Canonical Correlation Analysis with Applications to Imaging Genomics, Gossman, (2018).
• Acoustic Feature Learning via Deep Variational Canonical Correlation Analysis, Tang et al. (2018).
• A Tutorial on Canonical Correlation Methods, Uurtio et al. (2018).
• Change Point Analysis of Correlation in Non-stationary Time Series, Dette, (2018).
• Pricing without martingale measure, Batiste, (2018).
• Canonical Analysis, A review with applications in Ecology, Gittins ,(1986).
• The Estimation of Prediction Error: Covariance Penalties and Cross-Validation, Efron.
• On the Moreau-Yosida regularization of the vector k-norm related functions, Wu et al. (2011).
• Gen-Oja: A Simple and Efficient Algorithm for Streaming Generalized Eigenvector Computation, Bhatia et al. (2018).
• Robust sparse canonical correlation analysis, Wilms et al. (2016).
• The Geometry of Canonical Variant Analysis, Campbell et al. (1981).
• Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Johannsen, (1991).
• Sufficient Canonical Correlation Analysis, Guo et al. (2016).
• Some properties of canonical correlations and variates in infinite dimensions, Cupidon et al. (2007).
• High Dimensional Asymptotic expansions for the distributions of canonical correlations, Fujikoshi el al. (2008).
• A Canonical Correlation Analysis of Intelligence and Executive Functioning, Davis et al. (2011).
• A Closed Testing Procedure for Canonical Correlations, Calinski et al. (2006).
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• A modification of canonical variates analysis to handle highly collinear multivariate data, Norgaard et al. (2006).
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• An analysis of the total least squares problem, Golub, (1980).
• Bayesian Canonical Correlation Analysis, Klami et al. (2013).
• Canonical Variate Analysis and Related Methods with Longitudinal Data, Beaghan PhD thesis (1997).
• Dynamic Portfolio Selection by Augmenting the Asset Space, Brandt, (2006).
• Canonical Correlation Clarified by Singular Value Decomposition, Press, (2011).
• Chapter 8, Multivariate Data Analysis, Anderson et al. (1998).
• Multivariate Data Analysis, Anderson et al. (1998).
• NCSS Stat software explanation
• Canonical Correlation Analysis, Weenink, (2003).
• CCA for Optimal Signal Combination in Algorithmic Trading, Firoozye, (2019).
• Canonical Correlation: A Tutorial, Borga, (2001).
• Convex Estimation of Cointegrated VAR Models by a Nuclear Norm Penalty, Signoretto et al. (2012).
• A modified information Criterion for Cointegration Tests based on a VAR approximation, Qu et al. (2007).
• Convergence analysis of kernel Canonical Correlation Analysis: Theory and practice, Hardoon et al. (2009).
• Canonical Variates Analysis - ch 5 book extract
• Extreme Canonical Correlations and high dimensional cointegration analysis, Onatski et al. (2018).
• DynOpt: Incorporating Dynamics into Mean-Variance Portfolio Optimization, Signoretto.
• The efficient use of conditioning information in Portfolios, Ferson et al. (2001).
• Correlations and canonical forms of bivariate distributions, Lancaster, (1962).
• Convex optimization methods for dimension reduction and coefficient estimation in multivariate linear regression, Lu et al. (2008).
• Statistical Consistency of Kernel Canonical Correlation Analysis, Fukumizu et al. (2007).
• Generalized canonical analysis based on optimizing matrix correlations and a relation with IDIOSCAL, Kiers et al. (1991).
• Determination of vector error correction models in high dimensions, Liang et al. (2019).
• Partial Sparse Canonical Correlation Analysis (PSCCA) for Population Studies in Medical Imagine, Dhillon et al.
• The Convex Algebraic Geometry of rank minimization, Parrilo, (2009).
• Fitting method based on correlation maximization: Applications in space physics, Livadiotis et al. (2012).
• Kernel Canonical Correlation Analysis, Welling.
• Controlling Singular Values with Semidefinite Programming, Kovalsky et al. (2014).
• Lecture 7: Analysis of Factors and Canonical Correlations, Thulin.
• Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization, Recht et al. (2008).
• L1-Regularized Multiway Canonical Correlation Analysis for SSVEP-based BCI, Zhang et al. (2013).
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• Stochastic PCA with L1 and L2 Regularization, Mianji (2018).
• Stochastic PCA with L1 and L2 Regularization - supplement, Mianji (2018).
• Iterative Reweighted Algorithms for Matrix Rank Minimization, Mohan et al.
• A Rank Minimization Heuristic with Application to Minimum Order System Approximation, Fazel et al.
• Ridge-Penalty Regularization for Kernel-CCA, Rieter et al.
• Canonical Correlation Forests, Rainforth, (2015).
• Constrained Singular Value Decomposition, Zollman et al. (2009).
• Large-Scale Convex Minimization with a Low-Rank Constraint, Shalev-Schwartz et al. (2011).
• Sparse Canonical Correlation Analysis: New Formulation and Algorithm, Chu et al.
• Sparse CCA via Precision Adjusted Iterative Thresholding, Choi et al. (2017).
• Generalised Canonical Regression, Estrella, (2007).
• Statistics Methods and Applications - Book, Lewicki.
• Structured Sparse Canonical Correlation Analysis, Chen et al. (2012).
• A singular value thresholding algorithm for matrix completion, Cai et al.
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• Variants of Canonical Correlation Analysis, Choi (2016).
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• On the Variance, Reliability and Importance of Canonical Variates, Momirovic.
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• Comparison of Penalty Functions for Sparse Canonical Correlation Analysis, Chalise et al. (2012).
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• Developing Long/Short ETF Strategies - CCA, Kinlay.
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• Avoiding Backtesting Overfitting by Covariance-Penalties: An Empirical Investigation of the Ordinary and Total Least Squares cases,Koshiyama et al. (2019).
• Deep Canonical Correlation Analysis, Andrew et al. (2013).
• Deep Generalized Canonical Correlation Analysis, Benton et al. (2017).
• Multiview Canonical Correlation Analysis, Rupnik (2016).
• Multiview Canonical Correlation Analysis - Phd Thesis, Rupnik (2016).
• Generative Adversarial Networks for FinancialTrading Strategies Fine-Tuning and Combination, Koshiyama (2019).
• Temporal Kernel CCA and its Application in MultimodalNeuronal Data Analysis, Bierman et al. (2009)

• Compressed Conditional Mean Embeddings for Model-Based Reinforcement Learning, Lever et al. (2016).
• ACCME: Actively Compressed Conditional Mean Embeddings for Model-Based Reinforcement Learning, Stafford et al. (2018).
• Kernel Mean Embedding of Distributions: A Review and Beyond, Muandet et al. (2016).
• Learning via Hilbert Space Embedding of Distributions - PhD Thesis, Le Song (2008).
• Kernel Embeddings of Conditional Distributions, Le Song et al, (2013).
• Modelling Policies in MDPs in Reproducing Kernel Hilbert Space, Lever et al (2015).
• Kernel-Based Reinforcement Learning, Ormoneit et al. (2002).
• Practical Kernel-Based Reinforcement Learning, Barreto et al. (2014).
• Reinforcement Learning using Kernel-Based Stochastic Factorization, Barreto et al. (2011).
• Conditional mean embeddings as regressors, Grunewalder et al. (2012).
• Modelling transition dynamics in mdps with rkhs embeddings, Grunewalder et al. (2012).
• Kernel-Based Reinforcement Learning on Representative States, Kveton et al. (2012).
• Least-Squares Policy Iteration, Lagoudakis et al. (2003).
• An Analysis of Linear Models, Linear Value-Function Approximation, and Feature Selection for Reinforcement Learning, Parr et al. (2008).
• Pseudo-MDPs and Factored Linear Action Models, Yao et al. (2008).
• Approximate policy iteration: A survey and some new methods, Bertsekas (2012).
• Continuous Deep Q-Learning with Model-based Acceleration,Gu et al. (2016).
• Learning of Non-Parametric Control Policies with High-Dimensional State Features, Van Hoof et al. (2015).
• Learning Transition Dynamics in MDPs with Online Regression and Greedy Feature Selection, Lever et al. (2015).