Subspace portfolios: design and performance comparison

Data processing and engineering techniques enable people to observe and better understand the natural and human-made systems and processes that generate huge amounts of various data types. Data engineers collect data created in almost all fields and formats, such as images, audio, and text streams, biological and financial signals, sensing and many others. They develop and implement state-of-the art machine learning (ML) and artificial intelligence (AI) algorithms using big data to infer valuable information with social and economic value. Furthermore, ML/AI methodologies lead to automate many decision making processes with real-time applications serving people and businesses. As an example, mathematical tools are engineered for analysis of financial data such as prices, trade volumes, and other economic indicators of instruments including stocks, options and futures in order to automate the generation, implementation and maintenance of investment portfolios. Among the techniques, subspace framework and methods are fundamental, and they have been successfully employed in widely used technologies and real-time applications spanning from Internet multimedia to electronic trading of financial products. In this dissertation, the eigendecomposition of empirical correlation matrix created from market data (normalized returns) for a basket of US equities plays a central role. Then, the merit of approximating such an empirical matrix by a Toeplitz matrix, where closed form solutions for its eigenvalues and eigenvectors exist, is investigated. More specifically, the exponential correlation model that populates such a Toeplitz matrix is used to approximate pairwise empirical correlations of asset returns in a portfolio. Hence, the analytically derived eigenvectors of such a random vector process are utilized to design its eigenportfolios. The performances of the model based and the traditional eigenportfolios are studied and compared to validate the proposed portfolio design method. It is shown that the model based designs yield eigenportfolios that track the variations of the market statistics closely and deliver comparable or better performance. The theoretical foundations of information theory and the rate-distortion theory that provide the basis for source coding methods, including transform coding, are revisited in the dissertation. This theoretical inquiry helps to construct the basic question of trade-offs between dimension of the eigensubspace versus the correlation structure of the random vector process it represents. The signal processing literature facilitates developing an efficient subspace partitioning algorithm to design novel portfolios by combining eigenportfolios of partitions for US equities that outperform the existing eigenportfolios (EP), market portfolios (MP), minimum variance portfolios (MVP), and hierarchical risk parity (HRP) portfolios for US equities. Additionally, the pdf-optimized quantizer framework is employed to sparse eigenportfolios in order to reduce the (trading) cost of their maintenance. Then, the concluding remarks are presented in the last section of the Dissertation

Subspace methods for portfolio design

Financial signal processing (FSP) is one of the emerging areas in the field of signal processing. It is comprised of mathematical finance and signal processing. Signal processing engineers consider speech, image, video, and price of a stock as signals of interest for the given application. The information that they will infer from raw data is different for each application. Financial engineers develop new solutions for financial problems using their knowledge base in signal processing. The goal of financial engineers is to process the harvested financial signal to get meaningful information for the purpose. Designing investment portfolios have always been at the center of finance. An investment portfolio is comprised of financial instruments such as stocks, bonds, futures, options, and others. It is designed based on risk limits and return expectations of investors and managed by portfolio managers. Modern Portfolio Theory (MPT) offers a mathematical method for portfolio optimization. It defines the risk as the standard deviation of the portfolio return and provides closed-form solution for the risk optimization problem where asset allocations are derived from. The risk and the return of an investment are the two inseparable performance metrics. Therefore, risk normalized return, called Sharpe ratio, is the most widely used performance metric for financial investments. Subspace methods have been one of the pillars of functional analysis and signal processing. They are used for portfolio design, regression analysis and noise filtering in finance applications. Each subspace has its unique characteristics that may serve requirements of a specific application. For still image and video compression applications, Discrete Cosine Transform (DCT) has been successfully employed in transform coding where Karhunen-Loeve Transform (KLT) is the optimum block transform. In this dissertation, a signal processing framework to design investment portfolios is proposed. Portfolio theory and subspace methods are investigated and jointly treated. First, KLT, also known as eigenanalysis or principal component analysis (PCA) of empirical correlation matrix for a random vector process that statistically represents asset returns in a basket of instruments, is investigated. Auto-regressive, order one, AR(1) discrete process is employed to approximate such an empirical correlation matrix. Eigenvector and eigenvalue kernels of AR(1) process are utilized for closed-form expressions of Sharpe ratios and market exposures of the resulting eigenportfolios. Their performances are evaluated and compared for various statistical scenarios. Then, a novel methodology to design subband/filterbank portfolios for a given empirical correlation matrix by using the theory of optimal filter banks is proposed. It is a natural extension of the celebrated eigenportfolios. Closed-form expressions for Sharpe ratios and market exposures of subband/filterbank portfolios are derived and compared with eigenportfolios. A simple and powerful new method using the rate-distortion theory to sparse eigen-subspaces, called Sparse KLT (SKLT), is developed. The method utilizes varying size mid-tread (zero-zone) pdf-optimized (Lloyd-Max) quantizers created for each eigenvector (or for the entire eigenmatrix) of a given eigen-subspace to achieve the desired cardinality reduction. The sparsity performance comparisons demonstrate the superiority of the proposed SKLT method over the popular sparse representation algorithms reported in the literature

Systematic Trading: Calibration Advances through Machine Learning

Systematic trading in finance uses computer models to define trade goals, risk controls and rules that can execute trade orders in a methodical way. This thesis investigates how performance in systematic trading can be crucially enhanced by both i) persistently reducing the bid-offer spread quoted by the trader through optimized and realistically backtested strategies and ii) improving the out-of-sample robustness of the strategy selected through the injection of theory into the typically data-driven calibration processes. While doing so it brings to the foreground sound scientific reasons that, for the first time to my knowledge, technically underpin popular academic observations about the recent nature of the financial markets. The thesis conducts consecutive experiments across strategies within the three important building blocks of systematic trading: a) execution, b) quoting and c) risk-reward allowing me to progressively generate more complex and accurate backtested scenarios as recently demanded in the literature (Cahan et al. (2010)). The three experiments conducted are: 1. Execution: an execution model based on support vector machines. The first experiment is deployed to improve the realism of the other two. It analyses a popular model of execution: the volume weighted average price (VWAP). The VWAP algorithm targets to split the size of an order along the trading session according to the expected intraday volume's profile since the activity in the markets typically resembles convex seasonality – with more activity around the open and the closing auctions than along the rest of the day. In doing so, the main challenge is to provide the model with a reasonable expected profile. After proving in my data sample that two simple static approaches to the profile overcome the PCA-ARMA from Bialkowski et al. (2008) (a popular two-fold model composed by a dynamic component around an unsupervised learning structure) a further combination of both through an index based on supervised learning is proposed. The Sample Sensitivity Index hence successfully allows estimating the expected volume's profile more accurately by selecting those ranges of time where the model shall be less sensitive to past data through the identification of patterns via support vector machines. Only once the intraday execution risk has been defined can the quoting policy of a mid-frequency (in general, up to a week) hedging strategy be accurately analysed. 2. Quoting: a quoting model built upon particle swarm optimization. The second experiment analyses for the first time to my knowledge how to achieve the disruptive 50% bid-offer spread discount observed in Menkveld (2013) without increasing the risk profile of a trading agent. The experiment depends crucially on a series of variables of which market impact and slippage are typically the most difficult to estimate. By adapting the market impact model in Almgren et al. (2005) to the VWAP developed in the previous experiment and by estimating its slippage through its errors' distribution a framework within which the bid-offer spread can be assessed is generated. First, a full-replication spread, (that set out following the strict definition of a product in order to hedge it completely) is calculated and fixed as a benchmark. Then, by allowing benefiting from a lower market impact at the cost of assuming deviation risk (tracking error and tail risk) a non-full-replication spread is calibrated through particle swarm optimization (PSO) as in Diez et al. (2012) and compared with the benchmark. Finally, it is shown that the latter can reach a discount of a 50% with respect to the benchmark if a certain number of trades is granted. This typically occurs on the most liquid securities. This result not only underpins Menkveld's observations but also points out that there is room for further reductions. When seeking additional performance, once the quoting policy has been defined, a further layer with a calibrated risk-reward policy shall be deployed. 3. Risk-Reward: a calibration model defined within a Q-learning framework. The third experiment analyses how the calibration process of a risk-reward policy can be enhanced to achieve a more robust out-of-sample performance – a cornerstone in quantitative trading. It successfully gives a response to the literature that recently focusses on the detrimental role of overfitting (Bailey et al. (2013a)). The experiment was motivated by the assumption that the techniques underpinned by financial theory shall show a better behaviour (a lower deviation between in-sample and out-of-sample performance) than the classical data-driven only processes. As such, both approaches are compared within a framework of active trading upon a novel indicator. The indicator, called the Expectations' Shift, is rooted on the expectations of the markets' evolution embedded in the dynamics of the prices. The crucial challenge of the experiment is the injection of theory within the calibration process. This is achieved through the usage of reinforcement learning (RL). RL is an area of ML inspired by behaviourist psychology concerned with how software agents take decisions in an specific environment incentivised by a policy of rewards. By analysing the Q-learning matrix that collects the set of state/actions learnt by the agent within the environment, defined by each combination of parameters considered within the calibration universe, the rationale that an autonomous agent would have learnt in terms of risk management can be generated. Finally, by then selecting the combination of parameters whose attached rationale is closest to that of the portfolio manager a data-driven solution that converges to the theory-driven solution can be found and this is shown to successfully outperform out-of-sample the classical approaches followed in Finance. The thesis contributes to science by addressing what techniques could underpin recent academic findings about the nature of the trading industry for which a scientific explanation was not yet given: • A novel agent-based approach that allows for a robust out-of-sampkle performance by crucially providing the trader with a way to inject financial insights into the generally data-driven only calibration processes. It this way benefits from surpassing the generic model limitations present in the literature (Bailey et al. (2013b), Schorfheid and Wolpin (2012), Van Belle and Kerr (2012) or Weiss and Kulikowski (1991)) by finding a point where theory-driven patterns (the trader's priors tend to enhance out-of-sample robustness) merge with data-driven ones (those that allow to exploit latent information). • The provision of a technique that, to the best of my knowledge, explains for the first time how to reduce the bid-offer spread quoted by a traditional trader without modifying her risk appetite. A reduction not previously addressed in the literature in spite of the fact that the increasing regulation against the assumption of risk by market makers (e.g. Dodd–Frank Wall Street Reform and Consumer Protection Act) does yet coincide with the aggressive discounts observed by Menkveld (2013). As a result, this thesis could further contribute to science by serving as a framework to conduct future analyses in the context of systematic trading. • The completion of a mid-frequency trading experiment with high frequency execution information. It is shown how the latter can have a significant effect on the former not only through the erosion of its performance but, more subtly, by changing its entire strategic design (both, optimal composition and parameterization). This tends to be highly disregarded by the financial literature. More importantly, the methodologies disclosed herein have been crucial to underpin the setup of a new unit in the industry, BBVA's Global Strategies & Data Science. This disruptive, global and cross-asset team gives an enhanced role to science by successfully becoming the main responsible for the risk management of the Bank's strategies both in electronic trading and electronic commerce. Other contributions include: the provision of a novel risk measure (flowVaR); the proposal of a novel trading indicator (Expectations’ Shift); and the definition of a novel index that allows to improve the estimation of the intraday volume’s profile (Sample Sensitivity Index)

Applications of Random Matrix Theory to Portfolio Management and Financial Networks

This thesis is an application of Random Matrix Theory (RMT) to portfolio management and financial networks. From a portfolio management perspective, we apply the RMT approach to clean measurement noise from correlation matrices constructed for large portfolios of stocks of the FTSE 100. We apply this methodology to a number of correlation estimators, i.e., the sample correlation matrix, the Constant Conditional Correlation Model (CCC) of Bollerslev (1990), the Dynamic Conditional Correlation (DCC) Model of Engle (2002) and the Regime-Switching Beta CAPM Correlation Model, based on Ang and Bekaert (2004). For these estimators, we find that the RMT- filtering delivers portfolios with the lowest realised risk, the best prediction accuracy and also the highest cumulated returns and Sharpe Ratios. The gains from using the RMT-filtering, in terms of cumulated wealth, range from 65%, for the sample correlation matrix to 30%, for the regime-dependent correlation estimator. In the case of regime switching CAPM models, we find that the regime switching correlation matrices, in the high volatility regime are found to be a good filter which makes further RMT- filtering to be redundant. This establishes the validity of using regime sensitive portfolio management to deal with asymmetric asset correlations during high and low volatility regimes. From a financial network perspective, we assess the stability of a global banking network built from bilateral exposures of 18 BIS reporting banking systems to net debtor countries. For this, we applied the eigen-pair method of Markose (2012), which is based on the work of May (1972, 1974) for random networks. We use a stability condition based on the maximum eigenvalue (λmax) of a matrix of net bilateral exposures relative to equity capital as a systemic risk index (SRI). We provide evidence of the early warning capabilities of λmax, when this surpasses a prespecified threshold. We use the right and left associated eigenvectors as a gauge for systemic importance and systemic vulnerability, respectively. The λmax SRI was found to be superior in terms of early warning when compared to the standard SRIs based on market price data, viz. the DCC-MES of Acharya et al. (2010), the SRISK of Acharya et al. (2012) and the DCC-ΔCoVaR of Adrian and Brunnermeier (2011)