Financial Signal Processing and Machine Learning E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
The modern financial industry has been required to deal with large and diverse portfolios in a variety of asset classes often with limited market data available. Financial Signal Processing and Machine Learning unifies a number of recent advances made in signal processing and machine learning for the design and management of investment portfolios and financial engineering. This book bridges the gap between these disciplines, offering the latest information on key topics including characterizing statistical dependence and correlation in high dimensions, constructing effective and robust risk measures, and their use in portfolio optimization and rebalancing. The book focuses on signal processing approaches to model return, momentum, and mean reversion, addressing theoretical and implementation aspects. It highlights the connections between portfolio theory, sparse learning and compressed sensing, sparse eigen-portfolios, robust optimization, non-Gaussian data-driven risk measures, graphical models, causal analysis through temporal-causal modeling, and large-scale copula-based approaches.
Key features:
- Highlights signal processing and machine learning as key approaches to quantitative finance.
- Offers advanced mathematical tools for high-dimensional portfolio construction, monitoring, and post-trade analysis problems.
- Presents portfolio theory, sparse learning and compressed sensing, sparsity methods for investment portfolios. including eigen-portfolios, model return, momentum, mean reversion and non-Gaussian data-driven risk measures with real-world applications of these techniques.
- Includes contributions from leading researchers and practitioners in both the signal and information processing communities, and the quantitative finance community.
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Veröffentlichungsjahr: 2016
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Table of Contents
Cover
Title Page
Copyright
List of Contributors
Preface
Chapter 1: Overview
1.1 Introduction
1.2 A Bird's-Eye View of Finance
1.3 Overview of the Chapters
1.4 Other Topics in Financial Signal Processing and Machine Learning
References
Chapter 2: Sparse Markowitz Portfolios
2.1 Markowitz Portfolios
2.2 Portfolio Optimization as an Inverse Problem: The Need for Regularization
2.3 Sparse Portfolios
2.4 Empirical Validation
2.5 Variations on the Theme
2.6 Optimal Forecast Combination
Acknowlegments
References
Chapter 3: Mean-Reverting Portfolios
3.1 Introduction
3.2 Proxies for Mean Reversion
3.3 Optimal Baskets
3.4 Semidefinite Relaxations and Sparse Components
3.5 Numerical Experiments
3.6 Conclusion
References
Chapter 4: Temporal Causal Modeling
4.1 Introduction
4.2 TCM
4.3 Causal Strength Modeling
4.4 Quantile TCM (Q-TCM)
4.5 TCM with Regime Change Identification
4.6 Conclusions
References
Chapter 5: Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process
5.1 Introduction
5.2 Mathematical Definitions
5.3 Derivation of Explicit KLT Kernel for a Discrete AR(1) Process
5.4 Sparsity of Eigen Subspace
5.5 Conclusions
References
Chapter 6: Approaches to High-Dimensional Covariance and Precision Matrix Estimations
6.1 Introduction
6.2 Covariance Estimation via Factor Analysis
6.3 Precision Matrix Estimation and Graphical Models
6.4 Financial Applications
6.5 Statistical Inference in Panel Data Models
6.6 Conclusions
References
Chapter 7: Stochastic Volatility
7.1 Introduction
7.2 Asymptotic Regimes and Approximations
7.3 Merton Problem with Stochastic Volatility: Model Coefficient Polynomial Expansions
7.4 Conclusions
Acknowledgements
References
Chapter 8: Statistical Measures of Dependence for Financial Data
8.1 Introduction
8.2 Robust Measures of Correlation and Autocorrelation
8.3 Multivariate Extensions
8.4 Copulas
8.5 Types of Dependence
References
Chapter 9: Correlated Poisson Processes and Their Applications in Financial Modeling
9.1 Introduction
9.2 Poisson Processes and Financial Scenarios
9.3 Common Shock Model and Randomization of Intensities
9.4 Simulation of Poisson Processes
9.5 Extreme Joint Distribution
9.6 Numerical Results
9.7 Backward Simulation of the Poisson–Wiener Process
9.8 Concluding Remarks
Acknowledgments
Appendix A
References
Chapter 10: CVaR Minimizations in Support Vector Machines
10.1 What Is CVaR?
10.2 Support Vector Machines
10.3 -SVMs as CVaR Minimizations
10.4 Duality
10.5 Extensions to Robust Optimization Modelings
10.6 Literature Review
References
Chapter 11: Regression Models in Risk Management
11.1 Introduction
11.2 Error and Deviation Measures
11.3 Risk Envelopes and Risk Identifiers
11.4 Error Decomposition in Regression
11.5 Least-Squares Linear Regression
11.6 Median Regression
11.7 Quantile Regression and Mixed Quantile Regression
11.8 Special Types of Linear Regression
11.9 Robust Regression
References, Further Reading, and Bibliography
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 3: Mean-Reverting Portfolios: Tradeoffs between Sparsity and Volatility
Figure 3.1 Option implied volatility for Apple between January 4, 2004, and December 30, 2010.
Figure 3.2 Three sample trading experiments, using the PCA, sparse PCA, and crossing statistics estimators. (a) Pool of 9 volatility time series selected using our fast PCA selection procedure. (b) Basket weights estimated with in-sample data using the eigenvector of the covariance matrix with the smallest eigenvalue, the smallest eigenvector with a sparsity constraint of , and the crossing statistics estimator with a volatility threshold of , (i.e., a constraint on the basket's variance to be larger than the median variance of all 8 assets). (c) Using these 3 procedures, the time series of the resulting basket price in the in-sample part (c) and out-of-sample parts (d) are displayed. (e) Using the Jurek and Yang (2007) trading strategy results in varying positions (expressed as units of baskets) during the out-sample testing phase. (f) Transaction costs that result from trading the assets to achieve such positions accumulate over time. (g) Taking both trading gains and transaction costs into account, the net wealth of the investor for each strategy can be computed (the Sharpe ratio over the test period is displayed in the legend). Note how both sparsity and volatility constraints translate into portfolios composed of fewer assets, but with a higher variance.
Figure 3.3 Average Sharpe ratio for the Jurek and Yang (2007) trading strategy captured over about 922 trading episodes, using different basket estimation approaches. These 922 trading episodes were obtained by considering 7 disjoint time-windows in our market sample, each of a length of about one year. Each time-window was divided into 85% in-sample data to estimate baskets, and 15% outsample to test strategies. On each time-window, the set of 210 tradable assets during that period was clustered using sectorial information, and each cluster screened (in the in-sample part of the time-window) to look for the most promising baskets of size between 8 and 12 in terms of mean reversion, by choosing greedily subsets of stocks that exhibited the smallest minimal eigenvalues in their covariance matrices. For each trading episode, the same universe of stocks was fed to different mean-reversion algorithms. Because volatility time-series are bounded and quite stationary, we consider the PCA approach, which uses the eigenvector with the smallest eigenvalue of the covariance matrix of the time-series to define a cointegrated relationship. Besides standard PCA, we have also consider sparse PCA eigenvectors with minimal eigenvalue, with the size of the support of the eigenvector (the size of the resulting basket) constrained to be 30%, 50% or 70% of the total number of considered assets. We consider also the portmanteau, predictability and crossing stats estimation techniques with variance thresholds of and a support whose size (the number of assets effectively traded) is targeted to be about of the size of the considered universe (itself between 8 and 12). As can be seen in the figure, the sharpe ratios of all trading approaches decrease with an increase in transaction costs. One expects sparse baskets to perform better under the assumption that costs are high, and this is indeed observed here. Because the relationship between sharpe ratios and transaction costs can be efficiently summarized as being a linear one, we propose in the plots displayed in Figure 3.4 a way to summarize the lines above with two numbers each: their intercept (Sharpe level in the quasi-absence of costs) and slope (degradation of Sharpe as costs increase). This visualization is useful to observe how sparsity (basket size) and volatility thresholds influence the robustness to costs of the strategies we propose. This visualization allows us to observe how performance is influenced by these parameter settings.
Figure 3.4 Relationships between Sharpe in a low cost setting (intercept) in the -axis and robustness of Sharpe to costs (slope of Sharpe/costs curve) of a different estimators implemented with varying volatility levels and sparsity levels parameterized as a multiple of the universe size. Each colored square in the Figure above corresponds to the performance of a given estimator (Portmanteau in subFigure , Predictability in subFigure and Crossing Statistics in subFigure ) using different parameters for and . The parameters used for each experiment are displayed using an arrow whose vertical length is proportional to and horizontal length is proportional to .
Chapter 4: Temporal Causal Modeling
Figure 4.1 Causal CSM graphs of ETFs from iShares formed during four different 750-day periods in 2007–2008. Each graph moves the window of data over 50 business days in order to discover the effect of time on the causal networks. The lag used for VAR spans the 5 days (i.e., uses five features) preceding the target day. Each feature is a monthly return computed over the previous 22 business days.
Figure 4.2 Generic TCM algorithm.
Figure 4.3 Method
group OMP
.
Figure 4.4 Output causal structures on one synthetic dataset by the various methods. In this example, the group-based method exactly reconstructs the correct graph, while the nongroup ones fail badly.
Figure 4.5 Method
Quantile group OMP
.
Figure 4.6 Log-returns for ticker IVV (which tracks S&P 500) from April 18, 2005, through April 10, 2008. Outliers introduced on 10/26/2005, 12/14/2007, and 01/16/2008 are represented by red circles.
Figure 4.7 (Left) Output switching path on one synthetic dataset with two Markov states. Transition jumps missing in the estimated Markov path are highlighted in red. (Right) The corresponding output networks: (a) true network at state 1; (b) estimated network at state 1; (c) true network at state 2; and (d) estimated network at state 2. Edges coded in red are the false positives, and those in green are the false negatives.
Figure 4.8 Results of modeling monthly stock observations using MS-TCM. MS-TCM uncovered a regime change after the 19th time step; columns Model 1 and Model 2 contain the coefficients of the corresponding two TCM models. The column Model all gives the coefficients when plain TCM without regime identification is used. The symbols C, KEY, WFC, and JPM are money center banks; SO, DUK, D, HE, and EIX are electrical utilities companies; LUX, CAL, and AMR are major airlines; AMGN, GILD, CELG, GENZ, and BIIB are biotechnology companies; CAT, DE, and HIT are machinery manufacturers; IMO, HES, and YPF are fuel refineries; and X.GPSC is an index.
Chapter 5: Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process
Figure 5.1 (a) Performance of KLT and DCT for an AR(1) process with various values of and ; (b) performance of KLT and DCT as a function of for .
Figure 5.2 Functions and for various values of where , , and .
Figure 5.3 Functions and for the AR(1) process with and various values of , where , , , and , .
Figure 5.4 The roots of the transcendental tangent equation 5.29, , as a function of for .
Figure 5.5 Computation time, in seconds, to calculate and for an AR(1) process with , and different values of () and (Torun and Akansu, 2013).
Figure 5.6 Probability density function of arcsine distribution for and . Loadings of a second PC for an AR(1) signal source with and are fitted to arcsine distribution by finding minimum and maximum values in the PC.
Figure 5.7 Normalized histograms of (a) PC1 and (b) PC2 loadings for an AR(1) signal source with and . The dashed lines in each histogram show the probability that is calculated by integrating an arcsine pdf for each bin interval.
Figure 5.8 Rate (bits)-distortion (SQNR) performance of zero mean and unit variance arcsine pdf-optimized quantizer for bins. The distortion level is increased by combining multiple bins around zero in a larger zero-zone.
Figure 5.9 Orthogonality imperfectness-rate (sparsity) trade-off for sparse eigen subspaces of three AR(1) sources with .
Figure 5.10 (a) Variance loss (VL) measurements of sparsed first PCs generated by SKLT, SPCA, SPC, ST, and DSPCA methods with respect to nonsparsity (NS) for an AR(1) source with and ; (b) NS and VL measurements of sparsed eigenvectors for an AR(1) source with and generated by the SKLT method and SPCA algorithm.
Figure 5.11 Normalized histogram of eigenmatrix elements for an empirical correlation matrix of end-of-day (EOD) returns for 100 stocks in the NASDAQ-100 index. -day measurement window ending on April 9, 2014.
Figure 5.12 VL measurements of sparsed first PCs generated by SKLT, SPCA, SPC, ST, and DSPCA methods with respect to NS for an empirical correlation matrix of EOD returns for 100 stocks in the NASDAQ-100 index with -day measurement window ending on April 9, 2014.
Figure 5.13 Cumulative explained variance loss with generated daily from an empirical correlation matrix of EOD returns between April 9, 2014, and May 22, 2014, for 100 stocks in the NASDAQ-100 index by using KLT, SKLT, SPCA, and ST methods. NS levels of 85%, 80%, and 75% for all PCs are forced in (a), (b), and (c), respectively, using days.
Figure 5.14 (a) and (b) of sparse eigen subspaces generated daily from an empirical correlation matrix of EOD returns between April 9, 2014, and May 22, 2014, for 100 stocks in the NASDAQ-100 index by using SKLT, SPCA, and ST methods, respectively. NS level of 85% for all PCs is forced with days.
Chapter 6: Approaches to High-Dimensional Covariance and Precision Matrix Estimations
Figure 6.1 Minimum eigenvalue of as a function of for three choices of thresholding rules. Adapted from Fan
et al.
(2013).
Figure 6.2 Averages of (left panel) and (right panel) with known factors (solid red curve), unknown factors (solid blue curve), and sample covariance (dashed curve) over 200 simulations, as a function of the dimensionality . Taken from Fan
et al.
(2013).
Figure 6.3 Boxplots of for 10 stocks. As can be seen, the original data has many outliers, which is addressed by the normal-score transformation on the rescaled data (right).
Figure 6.4 The estimated TIGER graph using the S&P 500 stock data from January 1, 2003, to January 1, 2008.
Figure 6.5 The histogram and normal QQ plots of the marginal expression levels of the gene MECPS. We see the data are not exactly Gaussian distributed. Adapted from Liu and Wang (2012).
Figure 6.6 The estimated gene networks of the
Arabadopsis
dataset. The within-pathway edges are denoted by solid lines, and between-pathway edges are denoted by dashed lines. From Liu and Wang (2012).
Figure 6.7 Dynamics of p-values and selected stocks (, from Fan
et al.
, 2014b).
Figure 6.8 Histograms of -values for , , and PEM (from Fan
et al.
, 2014b).
Chapter 7: Stochastic Volatility
Figure 7.1 Implied volatility from S&P 500 index options on May 25, 2010, plotted as a function of log-moneyness to maturity ratio: . DTM, days to maturity.
Figure 7.2 Exact (solid) and approximate (dashed) implied volatilities in the Heston model. The horizontal axis is -moneyness . Parameters: , , , , .
Chapter 8: Statistical Measures of Dependence for Financial Data
Figure 8.1 Top left: Strong and persistent positive autocorrelation, that is, persistence in local level; top right: moderate volatility clustering, that is, i.e., persistence in local variation. Middle left: Right tail density estimates of Gaussian versus heavy- or thick-tailed data; middle right: sample quantiles of heavy-tailed data versus the corresponding quantiles of the Gaussian distribution. Bottom left: Linear regression line fit to non-Gaussian data; right: corresponding estimated density contours of the normalized sample ranks, which show a positive association that is stronger in the lower left quadrant compared to the upper right.
Figure 8.2 Bank of America (BOA) daily closing stock price. Bottom: Standardized (Fisher's transformation) ACF based on Kendall's tau and Pearson's correlation coefficient for the squared daily stock returns.
Figure 8.3 Realized time-series simulated from each of the three process models discussed in Example 8.1.
Figure 8.4 Tree representation of the fully nested (left) and partially nested (right) Archimedean copula construction. Leaf nodes represent uniform random variables, while the internal and root nodes represent copulas. Edges indicate which variables or copulas are used in the creation of a new copula.
Figure 8.5 Graphical representation of the C-vine (left) and D-vine (right) Archimedean copula construction. Leaf nodes labeled represent uniform random variables, whereas nodes labeled represent the th copula at the th level. Edges indicate which variables or copulas are used in the creation of a new copula.
Chapter 9: Correlated Poisson Processes and Their Applications in Financial Modeling
Figure 9.1 Typical monotone paths.
Figure 9.2 Partitions of the unit interval: .
Figure 9.3 Partitions of the unit interval: .
Figure 9.4 Support of the distribution : .
Figure 9.5 Support of the distribution : .
Figure 9.6 Support of the distribution
:
.
Figure 9.7 Correlation boundaries: .
Figure 9.8 Comparison of correlation boundaries: .
Figure 9.9 Correlation bounds.
Chapter 10: CVaR Minimizations in Support Vector Machines
Figure 10.1 CVaR, VaR, mean, and maximum of distribution. (a, c) The cumulative distribution function (cdf) and the density of a continuous loss distribution; (b, d) the cdf and histogram of a discrete loss distribution. In all four figures, the location of VaR with is indicated by a vertical dashed line. In (c) and (d), the locations of CVaR and the mean of the distributions are indicated with vertical solid and dashed-dotted lines. In (b) and (d), the location of the maximum loss is shown for the discrete case.
Figure 10.2 Convex functions dominating .
Figure 10.3 Illustration of in a discrete distribution on with . This Figure shows how varies depending on (). As approaches 1, approaches the unit simplex . The risk envelope shrinks to the point as decreases to 0.
Figure 10.4 Two separating hyperplanes and their geometric margins. The dataset is said to be
linearly separable
if there exist and such that for all . If the dataset is linearly separable, there are infinitely many hyperplanes separating the dataset. According to generalization theory (Vapnik, 1995), the hyperplane is preferable to . The optimization problem (10.12) (or, equivalently, (10.13)) finds a hyperplane that separates the datasets with the largest margin.
Figure 10.5 -SVC as a CVaR minimization. The Figure on the left shows an optimal separating hyperplane given by -SVC (). The one on the right is a histogram of the optimal distribution of the negative margin, . The locations of the minimized CVaR (solid line) and the corresponding VaR (broken line) are indicated in the histogram.
Figure 10.6 Minimized CVaR and corresponding VaR with respect to . CVaR indicates the optimal value of E-SVC (10.26) for binary classification. is the value of at which the optimal value becomes zero. For , E-SVC (10.26) reduces to -SVC (10.25). For , -SVC (10.25) results in a trivial solution, while E-SVC (10.26) still attains a nontrivial solution with the positive optimal value.
Figure 10.7 Relations among four classification formulations. The two formulations on the left are equivalent to the standard -SVC (10.16), while those on the right are equivalent to E-SVC (10.18). By resolving the nonconvexity issues that arise from the equality constraint, E-SVC provides a classifier that cannot be attained by -SVC.
Figure 10.8 -SVR as a CVaR minimization. The left Figure shows the regression model given by -SVR (). The right one shows the histogram of the optimal distribution of the residual . The locations of the minimized CVaR (green solid line) and the corresponding VaR (red dashed line) are indicated in the histogram.
Figure 10.9 Two-dimensional examples of reduced convex hulls. Here, ‘+’ and ‘’ represent the data samples. As increases, the size of each reduced convex hull shrinks. The reduced convex hull is a single point for , whereas it is equal to the convex hull for sufficiently close to 0. For linearly inseparable datasets, the corresponding convex hulls (or the reduced convex hulls for a small ) intersect, and the primal formulation (10.25)0 results in a trivial solution satisfying .
Figure 10.10 Convex hull of the union of risk envelopes ().
List of Tables
Chapter 4: Temporal Causal Modeling
Table 4.1 Results of TCM modeling on an ETF that tracks the S&P 500 between 2005 and 2015 that depicts the causal strength values for the three strongest relationships are given for each time period
Table 4.2 The accuracy () and standard error in identifying the correct model of the two nongrouped TCM methods, compared to those of the grouped TCM methods on synthetic data
Table 4.3 Time-series selected for IVV, which tracks S&P 500, using Q-TCM and TCM on noisy data. The correct features are South Korea, Japan, and China, which are discovered by Q-TCM
Table 4.4 MSE on test period for Q-TCM and TCM models for IVV on noisy data
Table 4.5 Accuracy of comparison methods in identifying the correct Bayesian networks measured by the average Rand index and score on synthetic data with a varying number of Markov states (K) and lags (L). The numbers in the parentheses are standard errors
Chapter 5: Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process
Table 5.1 Relevant parameter values of SKLT example for the first 16 PCs of an AR(1) source with and . They explain 68.28% of the total variance
Chapter 6: Approaches to High-Dimensional Covariance and Precision Matrix Estimations
Table 6.1 Mean and covariance matrix used to generate
Table 6.2 Parameters of
generating process
Table 6.3 Variable descriptive statistics for the Fama–French three-factor model (Adapted from Fan
et al.
, 2014b)
Table 6.4 Three interesting choices of the weight matrix
Table 6.5 Canonical correlations for simulation study (from Bai and Liao, 2013)
Table 6.6 Method comparison for the panel data with interactive effects (from Bai and Liao, 2013)
Chapter 8: Statistical Measures of Dependence for Financial Data
Table 8.1 Percentage of tests failing to reject of no lag-1 correlation
Table 8.2 A Table of common Archimedean copulas
Financial Signal Processing and Machine Learning
Edited by
Ali N. Akansu
New Jersey Institute of Technology, USA
Sanjeev R. Kulkarni
Princeton University, USA
Dmitry Malioutov
IBM T.J. Watson Research Center, USA
This edition first published 2016
© 2016 John Wiley & Sons, Ltd
First Edition published in 2016
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List of Contributors
Ali N. Akansu
New Jersey Institute of Technology, USA
Marco Cuturi
Kyoto University, Japan
Alexandre d'Aspremont
CNRS - Ecole Normale supérieure, France
Christine De Mol
Université Libre de Bruxelles, Belgium
Jianqing Fan
Princeton University, USA
Jun-ya Gotoh
Chuo University, Japan
Nicholas A. James
Cornell University, USA
Prabhanjan Kambadur
Bloomberg L.P., USA
Alexander Kreinin
Risk Analytics, IBM, Canada
Sanjeev R. Kulkarni
Princeton University, USA
Yuan Liao
University of Maryland, USA
Han Liu
Princeton University, USA
Matthew Lorig
University of Washington, USA
Aurélie C. Lozano
IBM T.J. Watson Research Center, USA
Ronny Luss
IBM T.J. Watson Research Center, USA
Dmitry Malioutov
IBM T.J. Watson Research Center, USA
David S. Matteson
Cornell University, USA
William B. Nicholson
Cornell University, USA
Ronnie Sircar
Princeton University, USA
Akiko Takeda
The University of Tokyo, Japan
Mustafa U. Torun
New Jersey Institute of Technology, USA
Stan Uryasev
University of Florida, USA
Onur Yilmaz
New Jersey Institute of Technology, USA
Preface
This edited volume collects and unifies a number of recent advances in the signal-processing and machine-learning literature with significant applications in financial risk and portfolio management. The topics in the volume include characterizing statistical dependence and correlation in high dimensions, constructing effective and robust risk measures, and using these notions of risk in portfolio optimization and rebalancing through the lens of convex optimization. It also presents signal-processing approaches to model return, momentum, and mean reversion, including both theoretical and implementation aspects. Modern finance has become global and highly interconnected. Hence, these topics are of great importance in portfolio management and trading, where the financial industry is forced to deal with large and diverse portfolios in a variety of asset classes. The investment universe now includes tens of thousands of international equities and corporate bonds, and a wide variety of other interest rate and derivative products-often with limited, sparse, and noisy market data.
Using traditional risk measures and return forecasting (such as historical sample covariance and sample means in Markowitz theory) in high-dimensional settings is fraught with peril for portfolio optimization, as widely recognized by practitioners. Tools from high-dimensional statistics, such as factor models, eigen-analysis, and various forms of regularization that are widely used in real-time risk measurement of massive portfolios and for designing a variety of trading strategies including statistical arbitrage, are highlighted in the book. The dramatic improvements in computational power and special-purpose hardware such as field programmable gate arrays (FPGAs) and graphics processing units (GPUs) along with low-latency data communications facilitate the realization of these sophisticated financial algorithms that not long ago were “hard to implement.”
The book covers a number of topics that have been popular recently in machine learning and signal processing to solve problems with large portfolios. In particular, the connections between the portfolio theory and sparse learning and compressed sensing, robust optimization, non-Gaussian data-driven risk measures, graphical models, causal analysis through temporal-causal modeling, and large-scale copula-based approaches are highlighted in the book.
Although some of these techniques already have been used in finance and reported in journals and conferences of different disciplines, this book attempts to give a unified treatment from a common mathematical perspective of high-dimensional statistics and convex optimization. Traditionally, the academic quantitative finance community did not have much overlap with the signal and information-processing communities. However, the fields are seeing more interaction, and this trend is accelerating due to the paradigm in the financial sector which has embraced state-of-the-art, high-performance computing and signal-processing technologies. Thus, engineers play an important role in this financial ecosystem. The goal of this edited volume is to help to bridge the divide, and to highlight machine learning and signal processing as disciplines that may help drive innovations in quantitative finance and electronic trading, including high-frequency trading.
The reader is assumed to have graduate-level knowledge in linear algebra, probability, and statistics, and an appreciation for the key concepts in optimization. Each chapter provides a list of references for readers who would like to pursue the topic in more depth. The book, complemented with a primer in financial engineering, may serve as the main textbook for a graduate course in financial signal processing.
We would like to thank all the authors who contributed to this volume as well as all of the anonymous reviewers who provided valuable feedback on the chapters in this book. We also gratefully acknowledge the editors and staff at Wiley for their efforts in bringing this project to fruition.
Chapter 1OverviewFinancial Signal Processing and Machine Learning
Ali N. Akansu1, Sanjeev R. Kulkarni2 and Dmitry Malioutov3
1New Jersey Institute of Technology, USA
2Princeton University, USA
3IBM T.J. Watson Research Center, USA
1.1 Introduction
In the last decade, we have seen dramatic growth in applications for signal-processing and machine-learning techniques in many enterprise and industrial settings. Advertising, real estate, healthcare, e-commerce, and many other industries have been radically transformed by new processes and practices relying on collecting and analyzing data about operations, customers, competitors, new opportunities, and other aspects of business. The financial industry has been one of the early adopters, with a long history of applying sophisticated methods and models to analyze relevant data and make intelligent decisions – ranging from the quadratic programming formulation in Markowitz portfolio selection (Markowitz, 1952), factor analysis for equity modeling (Fama and French, 1993), stochastic differential equations for option pricing (Black and Scholes, 1973), stochastic volatility models in risk management (Engle, 1982; Hull and White, 1987), reinforcement learning for optimal trade execution (Bertsimas and Lo, 1998), and many other examples. While there is a great deal of overlap among techniques in machine learning, signal processing and financial econometrics, historically, there has been rather limited awareness and slow permeation of new ideas among these areas of research. For example, the ideas of stochastic volatility and copula modeling, which are quite central in financial econometrics, are less known in the signal-processing literature, and the concepts of sparse modeling and optimization that have had a transformative impact on signal processing and statistics have only started to propagate slowly into financial applications. The aim of this book is to raise awareness of possible synergies and interactions among these disciplines, present some recent developments in signal processing and machine learning with applications in finance, and also facilitate interested experts in signal processing to learn more about applications and tools that have been developed and widely used by the financial community.
We start this chapter with a brief summary of basic concepts in finance and risk management that appear throughout the rest of the book. We present the underlying technical themes, including sparse learning, convex optimization, and non-Gaussian modeling, followed by brief overviews of the chapters in the book. Finally, we mention a number of highly relevant topics that have not been included in the volume due to lack of space.
1.2 A Bird's-Eye View of Finance
The financial ecosystem and markets have been transformed with the advent of new technologies where almost any financial product can be traded in the globally interconnected cyberspace of financial exchanges by anyone, anywhere, and anytime. This systemic change has placed real-time data acquisition and handling, low-latency communications technologies and services, and high-performance processing and automated decision making at the core of such complex systems. The industry has already coined the term big data finance, and it is interesting to see that technology is leading the financial industry as it has been in other sectors like e-commerce, internet multimedia, and wireless communications. In contrast, the knowledge base and exposure of the engineering community to the financial sector and its relevant activity have been quite limited. Recently, there have been an increasing number of publications by the engineering community in the finance literature, including A Primer for Financial Engineering (Akansu and Torun, 2015) and research contributions like Akansu et al., (2012) and Pollak et al., (2011). This volume facilitates that trend, and it is composed of chapter contributions on selected topics written by prominent researchers in quantitative finance and financial engineering.
We start by sketching a very broad-stroke view of the field of finance, its objectives, and its participants to put the chapters into context for readers with engineering expertise. Finance broadly deals with all aspects of money management, including borrowing and lending, transfer of money across continents, investment and price discovery, and asset and liability management by governments, corporations, and individuals. We focus specifically on trading where the main participants may be roughly classified into hedgers, investors, speculators, and market makers (and other intermediaries). Despite their different goals, all participants try to balance the two basic objectives in trading: to maximize future expected rewards (returns) and to minimize the risk of potential losses.
Naturally, one desires to buy a product cheap and sell it at a higher price in order to achieve the ultimate goal of profiting from this trading activity. Therefore, the expected return of an investment over any holding time (horizon) is one of the two fundamental performance metrics of a trade. The complementary metric is its variation, often measured as the standard deviation over a time window, and called investment risk or market risk.1 Return and risk are two typically conflicting but interwoven measures, and risk-normalized return (Sharpe ratio) finds its common use in many areas of finance. Portfolio optimization involves balancing risk and reward to achieve investment objectives by optimally combining multiple financial instruments into a portfolio. The critical ingredient in forming portfolios is to characterize the statistical dependence between prices of various financial instruments in the portfolio. The celebrated Markowitz portfolio formulation (Markowitz, 1952) was the first principled mathematical framework to balance risk and reward based on the covariance matrix (also known as the variance-covariance or VCV matrix in finance) of returns (or log-returns) of financial instruments as a measure of statistical dependence. Portfolio management is a rich and active field, and many other formulations have been proposed, including risk parity portfolios (Roncalli, 2013), Black–Litterman portfolios (Black and Litterman, 1992), log-optimal portfolios (Cover and Ordentlich, 1996), and conditional value at risk (cVaR) and coherent risk measures for portfolios (Rockafellar and Uryasev, 2000) that address various aspects ranging from the difficulty of estimating the risk and return for large portfolios to the non-Gaussian nature of financial time series, and to more complex utility functions of investors.
The recognition of a price inefficiency is one of the crucial pieces of information to trade that product. If the price is deemed to be low based on some analysis (e.g. fundamental or statistical), an investor would like to buy it with the expectation that the price will go up in time. Similarly, one would shortsell it (borrow the product from a lender with some fee and sell it at the current market price) when its price is forecast to be higher than what it should be. Then, the investor would later buy to cover it (buy from the market and return the borrowed product back to the lender) when the price goes down. This set of transactions is the building block of any sophisticated financial trading activity. The main challenge is to identify price inefficiencies, also called alpha of a product, and swiftly act upon it for the purpose of making a profit from the trade. The efficient market hypothesis (EMH) stipulates that the market instantaneously aggregates and reflects all of the relevant information to price various securities; hence, it is impossible to beat the market. However, violations of the EMH assumptions abound: unequal availability of information, access to high-speed infrastructure, and various frictions and regulations in the market have fostered a vast and thriving trading industry.
Fundamental investors find alpha (i.e., predict the expected return) based on their knowledge of enterprise strategy, competitive advantage, aptitude of its leadership, economic and political developments, and future outlook. Traders often find inefficiencies that arise due to the complexity of market operations. Inefficiencies come from various sources such as market regulations, complexity of exchange operations, varying latency, private sources of information, and complex statistical considerations. An arbitrage is a typically short-lived market anomaly where the same financial instrument can be bought at one venue (exchange) for a lower price than it can be simultaneously sold at another venue. Relative value strategies recognize that similar instruments can exhibit significant (unjustified) price differences. Statistical trading strategies, including statistical arbitrage, find patterns and correlations in historical trading data using machine-learning methods and tools like factor models, and attempt to exploit them hoping that these relations will persist in the future. Some market inefficiencies arise due to unequal access to information, or the speed of dissemination of this information. The various sources of market inefficiencies give rise to trading strategies at different frequencies, from high-frequency traders who hold their positions on the order of milliseconds, to midfrequency trading that ranges from intraday (holding no overnight position) to a span of a few days, and to long-term trading ranging from a few weeks to years. High-frequency trading requires state-of-the-art computing, network communications, and trading infrastructure: a large number of trades are made where each position is held for a very short time period and typically produces a small return with very little risk. Longer term strategies are less dependent on latency and sophisticated technology, but individual positions are typically held for a longer time horizon and can pose substantial risk.
1.2.1 Trading and Exchanges
There is a vast array of financial instruments ranging from stocks and bonds to a variety of more sophisticated products like futures, exchange-traded funds (ETFs), swaps, collateralized debt obligations (CDOs), and exotic options (Hull, 2011). Each product is structured to serve certain needs of the investment community. Portfolio managers create investment portfolios for their clients based on the risk appetite and desired return. Since prices, expected returns, and even correlations of products in financial markets naturally fluctuate, it is the portfolio manager's task to measure the performance of a portfolio and maintain (rebalance) it in order to deliver the expected return.
The market for a security is formed by its buyers (bidding) and sellers (asking) with defined price and order types that describe the conditions for trades to happen. Such markets for various financial instruments are created and maintained by exchanges (e.g., the New York Stock Exchange, NASDAQ, London Stock Exchange, and Chicago Mercantile Exchange), and they must be compliant with existing trading rules and regulations. Other venues where trading occurs include dark pools, and over-the-counter or interbank trading. An order book is like a look-up table populated by the desired price and quantity (volume) information of traders willing to trade a financial instrument. It is created and maintained by an exchange. Certain securities may be simultaneously traded at multiple exchanges. It is a common practice that an exchange assigns one or several market makers for each security in order to maintain the robustness of its market.
The health (or liquidity) of an order book for a particular financial product is related to the bid–ask spread, which is defined as the difference between the lowest price of sell orders and the highest price of buy orders. A robust order book has a low bid–ask spread supported with large quantities at many price levels on both sides of the book. This implies that there are many buyers and sellers with high aggregated volumes on both sides of the book for that product. Buying and selling such an instrument at any time are easy, and it is classified as a high-liquidity (liquid) product in the market. Trades for a security happen whenever a buyer–seller match happens and their orders are filled by the exchange(s). Trades of a product create synchronous price and volume signals and are viewed as discrete time with irregular sampling intervals due to the random arrival times of orders at the market. Exchanges charge traders commissions (a transaction cost) for their matching and fulfillment services. Market-makers are offered some privileges in exchange for their market-making responsibilities to always maintain a two-sided order book.
The intricacies of exchange operations, order books, and microscale price formation is the study of market microstructure (Harris, 2002; O'Hara, 1995). Even defining the price for a security becomes rather complicated, with irregular time intervals characterized by the random arrivals of limit and market orders, multiple definitions of prices (highest bid price, lowest ask price, midmarket price, quantity-weighted prices, etc.), and the price movements occurring at discrete price levels (ticks). This kind of fine granularity is required for designing high-frequency trading strategies. Lower frequency strategies may view prices as regular discrete-time time series (daily or hourly) with a definition of price that abstracts away the details of market microstructure and instead considers some notion of aggregate transaction costs. Portfolio allocation strategies usually operate at this low-frequency granularity with prices viewed as real-valued stochastic processes.
1.2.2 Technical Themes in the Book
Although the scope of financial signal processing and machine learning is very wide, in this book, we have chosen to focus on a well-selected set of topics revolving around the concepts of high-dimensional covariance estimation, applications of sparse learning in risk management and statistical arbitrage, and non-Gaussian and heavy-tailed measures of dependence.2
A unifying challenge for many applications of signal processing and machine learning is the high-dimensional nature of the data, and the need to exploit the inherent structure in those data. The field of finance is, of course, no exception; there, thousands of domestic equities and tens of thousands of international equities, tens of thousands of bonds, and even more options contracts with various strikes and expirations provide a very rich source of data. Modeling the dependence among these instruments is especially challenging, as the number of pairwise relationships (e.g., correlations) is quadratic in the number of instruments. Simple traditional tools like the sample covariance estimate are not applicable in high-dimensional settings where the number of data points is small or comparable to the dimension of the space (El Karoui, 2013). A variety of approaches have been devised to tackle this challenge – ranging from simple dimensionality reduction techniques like principal component analysis and factor analysis, to Markov random fields (or sparse covariance selection models), and several others. They rely on exploiting additional structure in the data (sparsity or low-rank, or Markov structure) in order to reduce the sheer number of parameters in covariance estimation. Chapter 1.3.5 provides a comprehensive overview of high-dimensional covariance estimation. Chapter 1.3.4 derives an explicit eigen-analysis for the covariance matrices of AR processes, and investigates their sparsity.
The sparse modeling paradigm that has been highly influential in signal processing is based on the premise that in many settings with a large number of variables, only a small subset of these variables are active or important. The dimensionality of the problem can thus be reduced by focusing on these variables. The challenge is, of course, that the identity of these key variables may not be known, and the crux of the problem involves identifying this subset. The discovery of efficient approaches based on convex relaxations and greedy methods with theoretical guarantees has opened an explosive interest in theory and applications of these methods in various disciplines spanning from compressed sensing to computational biology (Chen et al., 1998; Mallat and Zhang, 1993; Tibshirani, 1996). We explore a few exciting applications of sparse modeling in finance. Chapter 1.3.1 presents sparse Markowitz portfolios where, in addition to balancing risk and expected returns, a new objective is imposed requiring the portfolio to be sparse. The sparse Markowitz framework has a number of benefits, including better statistical out-of-sample performance, better control of transaction costs, and allowing portfolio managers and traders to focus on a small subset of financial instruments. Chapter 1.3.2 introduces a formulation to find sparse eigenvectors (and generalized eigenvectors) that can be used to design sparse mean-reverting portfolios, with applications to statistical arbitrage strategies. In Chapter 1.3.3, another variation of sparsity, the so-called group sparsity, is used in the context of causal modeling of high-dimensional time series. In group sparsity, the variables belong to a number of groups, where only a small number of groups is selected to be active, while the variables within the groups need not be sparse. In the context of temporal causal modeling, the lagged variables at different lags are used as a group to discover influences among the time series.
Another dominating theme in the book is the focus on non-Gaussian, non-stationary and heavy-tailed distributions, which are critical for realistic modeling of financial data. The measure of risk based on variance (or standard deviation) that relies on the covariance matrix among the financial instruments has been widely used in finance due to its theoretical elegance and computational tractability. There is a significant interest in developing computational and modeling approaches for more flexible risk measures. A very potent alternative is the cVaR, which measures the expected loss below a certain quantile of the loss distribution (Rockafellar and Uryasev, 2000). It provides a very practical alternative to the value at risk (VaR) measure, which is simply the quantile of the loss distribution. VaR has a number of problems such as lack of coherence, and it is very difficult to optimize in portfolio settings. Both of these shortcomings are addressed by the cVaR formulation. cVaR is indeed coherent, and can be optimized by convex optimization (namely, linear programming). Chapter 1.3.9 describes the very intriguing close connections between the cVaR measure of risk and support vector regression in machine learning, which allows the authors to establish out-of-sample results for cVaR portfolio selection based on statistical learning theory. Chapter 1.3.9 provides an overview of a number of regression formulations with applications in finance that rely on different loss functions, including quantile regression and the cVaR metric as a loss measure.
The issue of characterizing statistical dependence and the inadequacy of jointly Gaussian models has been of central interest in finance. A number of approaches based on elliptical distributions, robust measures of correlation and tail dependence, and the copula-modeling framework have been introduced in the financial econometrics literature as potential solutions (McNeil et al., 2015). Chapter 1.3.7 provides a thorough overview of these ideas. Modeling correlated events (e.g., defaults or jumps) requires an entirely different set of tools. An approach based on correlated Poisson processes is presented in Chapter 1.3.8. Another critical aspect of modeling financial data is the handling of non-stationarity. Chapter 1.3.6 describes the problem of modeling the non-stationarity in volatility (i.e. stochastic volatility). An alternative framework based on autoregressive conditional heteroskedasticity models (ARCH and GARCH) is described in Chapter 1.3.7.
1.3 Overview of the Chapters
1.3.1 Chapter 2: “Sparse Markowitz Portfolios” by Christine De Mol
Sparse Markowitz portfolios impose an additional requirement of sparsity to the objectives of risk and expected return in traditional Markowitz portfolios. The chapter starts with an overview of the Markowitz portfolio formulation and describes its fragility in high-dimensional settings. The author argues that sparsity of the portfolio can alleviate many of the shortcomings, and presents an optimization formulation based on convex relaxations. Other related problems, including sparse portfolio rebalancing and combining multiple forecasts, are also introduced in the chapter.
1.3.2 Chapter 3: “Mean-Reverting Portfolios: Tradeoffs between Sparsity and Volatility” by Marco Cuturi and Alexandre d'Aspremont
Statistical arbitrage strategies attempt to find portfolios that exhibit mean reversion. A common econometric tool to find mean reverting portfolios is based on co-integration. The authors argue that sparsity and high volatility are other crucial considerations for statistical arbitrage, and describe a formulation to balance these objectives using semidefinite programming (SDP) relaxations.
1.3.3 Chapter 4: “Temporal Causal Modeling” by Prabhanjan Kambadur, Aurélie C. Lozano, and Ronny Luss
This chapter revisits the old maxim that correlation is not causation, and extends the definition of Granger causality to high-dimensional multivariate time series by defining graphical Granger causality as a tool for temporal causal modeling (TCM). After discussing computational and statistical issues, the authors extend TCM to robust quantile loss functions and consider regime changes using a Markov switching framework.
1.3.4 Chapter 5: “Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process” by Mustafa U. Torun, Onur Yilmaz and Ali N. Akansu
The closed-form kernel expressions for the eigenvectors and eigenvalues of the AR(1) discrete process are derived in this chapter. The sparsity of its eigen subspace is investigated. Then, a new method based on rate-distortion theory to find a sparse subspace is introduced. Its superior performance over a few well-known sparsity methods is shown for the AR(1) source as well as for the empirical correlation matrix of stock returns in the NASDAQ-100 index.
1.3.5 Chapter 6: “Approaches to High-Dimensional Covariance and Precision Matrix Estimation” by Jianqing Fan, Yuan Liao, and Han Liu
Covariance estimation presents significant challenges in high-dimensional settings. The authors provide an overview of a variety of powerful approaches for covariance estimation based on approximate factor models, sparse covariance, and sparse precision matrix models. Applications to large-scale portfolio management and testing mean-variance efficiency are considered.
1.3.6 Chapter 7: “Stochastic Volatility: Modeling and Asymptotic Approaches to Option Pricing and Portfolio Selection” by Matthew Lorig and Ronnie Sircar
The dynamic and uncertain nature of market volatility is one of the important incarnations of nonstationarity in financial time series. This chapter starts by reviewing the Black–Scholes formulation and the notion of implied volatility, and discusses local and stochastic models of volatility and their asymptotic analysis. The authors discuss implications of stochastic volatility models for option pricing and investment strategies.
1.3.7 Chapter 8: “Statistical Measures of Dependence for Financial Data” by David S. Matteson, Nicholas A. James, and William B. Nicholson
Idealized models such as jointly Gaussian distributions are rarely appropriate for real financial time series. This chapter describes a variety of more realistic statistical models to capture cross-sectional and temporal dependence in financial time series. Starting with robust measures of correlation and autocorrelation, the authors move on to describe scalar and vector models for serial correlation and heteroscedasticity, and then introduce copula models, tail dependence, and multivariate copula models based on vines.
1.3.8 Chapter 9: “Correlated Poisson Processes and Their Applications in Financial Modeling” by Alexander Kreinin
Jump-diffusion processes have been popular among practitioners as models for equity derivatives and other financial instruments. Modeling the dependence of jump-diffusion processes is considerably more challenging than that of jointly Gaussian diffusion models where the positive-definiteness of the covariance matrix is the only requirement. This chapter introduces a framework for modeling correlated Poisson processes that relies on extreme joint distributions and backward simulation, and discusses its application to financial risk management.
1.3.9 Chapter 10: “CVaR Minimizations in Support Vector Machines” by Junya Gotoh and Akiko Takeda
This chapter establishes intriguing connections between the literature on cVaR optimization in finance, and the support vector machine formulation for regularized empirical risk minimization from the machine-learning literature. Among other insights, this connection allows the establishment of out-of-sample bounds on cVaR risk forecasts. The authors further discuss robust extensions of the cVaR formulation.
1.3.10 Chapter 11: “Regression Models in Risk Management” by Stan Uryasev
Regression models are one of the most widely used tools in quantitative finance. This chapter presents a general framework for linear regression based on minimizing a rich class of error measures for regression residuals subject to constraints on regression coefficients. The discussion starts with least squares linear regression, and includes many important variants such as median regression, quantile regression, mixed quantile regression, and robust regression as special cases. A number of applications are considered such as financial index tracking, sparse signal reconstruction, mutual fund return-based style classification, and mortgage pipeline hedging, among others.
1.4 Other Topics in Financial Signal Processing and Machine Learning
We have left out a number of very interesting topics that all could fit very well within the scope of this book. Here, we briefly provide the reader some pointers for further study.
In practice, the expected returns and the covariance matrices used in portfolio strategies are typically estimated based on recent windows of historical data and, hence, pose significant uncertainty. It behooves a careful portfolio manager to be cognizant of the sensitivity of portfolio allocation strategies to these estimation errors. The field of robust portfolio optimization attempts to characterize this sensitivity and propose strategies that are more stable with respect to modeling errors (Goldfarb and Iyengar, 2003).
The study of market microstructure and the development of high-frequency trading strategies and aggressive directional and market-making strategies rely on short-term predictions of prices and market activity. A recent overview in Kearns and Nevmyvaka (2013) describes many of the issues involved.
Managers of large portfolios such as pension funds and mutual funds often need to execute very large trades that cannot be traded instantaneously in the market without causing a dramatic market impact. The field of optimal order execution studies how to split a large order into a sequence of carefully timed small orders in order to minimize the market impact but still execute the order in a timely manner (Almgren and Chriss, 2001; Bertsimas and Lo, 1998). The solutions for such a problem involve ideas from stochastic optimal control.
Various financial instruments exhibit specific structures that require dedicated mathematical models. For example, fixed income instruments depend on the movements of various interest-rate curves at different ratings (Brigo and Mercurio, 2007), options prices depend on volatility surfaces (Gatheral, 2011), and foreign exchange rates are traded via a graph of currency pairs. Stocks do not have such a rich mathematical structure, but they can be modeled by their industry, style, and other common characteristics. This gives rise to fundamental or statistical factor models (Darolles et al., 2013).
A critical driver for market activity is the release of news, reflecting developments in the industry, economic, and political sectors that affect the price of a security. Traditionally, traders act upon this information after reading an article and evaluating its significance and impact on their portfolio. With the availability of large amounts of information online, the advent of natural language processing, and the need for rapid decision making, many financial institutions have already started to explore automated decision-making and trading strategies based on computer interpretation of relevant news (Bollen et al., 2011; Luss and d'Aspremont, 2008) ranging from simple sentiment analysis to deeper semantic analysis and entity extraction.
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