The framework of online learning with memory naturally captures learning problems with temporal constraints, and was previously studied for the experts setting. In this work we extend the notion of learning with memory to the general Online Convex Optimization (OCO) framework, and present two algorithms that attain low regret. The first algorithm applies to Lipschitz continuous loss functions, obtaining optimal regret bounds for both convex and strongly convex losses. The second algorithm attains the optimal regret bounds and applies more broadly to convex losses without requiring Lipschitz continuity, yet is more complicated to implement. We complement our theoretic results with an application to statistical arbitrage in finance: we devise algorithms for constructing mean-reverting portfolios.
An informative guide to market microstructure and trading strategiesOver the last decade, the financial landscape has undergone a significant transformation, shaped by the forces of technology, globalization, and market innovations to name a few. In order to operate effectively in today's markets, you need more than just the motivation to succeed, you need a firm understanding of how modern financial markets work and what professional trading is really about. Dr. Anatoly Schmidt, who has worked in the financial industry since 1997, and teaches in the Financial Engineering program of Stevens Institute of Technology, puts these topics in perspective with his new book.Divided into three comprehensive parts, this reliable resource offers a balance between the theoretical aspects of market microstructure and trading strategies that may be more relevant for practitioners. Along the way, it skillfully provides an informative overview of modern financial markets as well as an engaging assessment of the methods used in deriving and back-testing trading strategies. Details the modern financial markets for equities, foreign exchange, and fixed income Addresses the basics of market dynamics, including statistical distributions and volatility of returns Offers a summary of approaches used in technical analysis and statistical arbitrage as well as a more detailed description of trading performance criteria and back-testing strategies Includes two appendices that support the main material in the bookIf you're unprepared to enter today's markets you will underperform. But with Financial Markets and Trading as your guide, you'll quickly discover what it takes to make it in this competitive field. This paper analytically solves the portfolio optimization problem of an investor faced with a risky arbitrage opportunity (e.g. relative mispricing in equity pairs). Unlike the extant literature, which typically models mispricings through the Ornstein--Uhlenbeck (OU) process, we introduce a nonlinear generalization of OU which jointly captures several important risk factors inherent in arbitrage trading. While these factors are absent from the standard OU, we show that considering them yields several new insights into the behavior of rational arbitrageurs: Firstly, arbitrageurs recognizing these risk factors exhibit a diminishing propensity to exploit large mispricings. Secondly, optimal investment behavior in light of these risk factors precipitates the gradual unwinding of losing trades far sooner than is entailed in existing approaches including OU. Finally, an empirical application to daily FTSE100 pairs data shows that incorporating these risks renders our model's risk-management capabilities superior to both OU and a simple threshold strategy popular in the literature. These observations are useful in understanding the role of arbitrageurs in enforcing price efficiency. Preface. Acknowledgments. PART ONE: BACKGROUND MATERIAL. Chapter 1. Introduction. The CAPM Model. Market Neutral Strategy. Pairs Trading. Outline. Audience. Chapter 2. Time Series. Overview. Autocorrelation. Time Series Models. Forecasting. Goodness of Fit versus Bias. Model Choice. Modeling Stock Prices. Chapter 3. Factor Models. Introduction. Arbitrage Pricing Theory. The Covariance Matrix. Application: Calculating the Risk on a Portfolio. Application: Calculation of Portfolio Beta. Application: Tracking Basket Design. Sensitivity Analysis. Chapter 4. Kalman Filtering. Introduction. The Kalman Filter. The Scalar Kalman Filter. Filtering the Random Walk. Application: Example with the Standard & Poor Index. PART TWO: STATISTICAL ARBITRAGE. Chapter 5. Overview. History. Motivation. Cointegration. Applying the Model. A Trading Strategy. Road Map for Strategy Design. Chapter 6. Pairs Selection in Equity Markets. Introduction. Common Trends Cointegration Model. Common Trends Model and APT. The Distance Measure. Interpreting the Distance Measure. Reconciling Theory and Practice. Chapter 7. Testing for Tradability. Introduction. The Linear Relationship. Estimating the Linear Relationship: The Multifactor Approach. Estimating the Linear Relationship: The Regression Approach. Testing Residual for Tradability. Chapter 8. Trading Design. Introduction. Band Design for White Noise. Spread Dynamics. Nonparametric Approach. Regularization. Tying Up Loose Ends. PART THREE: RISK ARBITRAGE PAIRS. Chapter 9. Risk Arbitrage Mechanics. Introduction. History. The Deal Process. Transaction Terms. The Deal Spread. Trading Strategy. Quantitative Aspects. Chapter 10. Trade Execution. Introduction. Specifying the Order. Verifying the Execution. Execution During the Pricing Period. Short Selling. Chapter 11. The Market Implied Merger Probability. Introduction. Implied Probabilities and Arrow-Debreu Theory. The Single-Step Model. The Multistep Model. Reconciling Theory and Practice. Risk Management. Chapter 12. Spread Inversion. Introduction. The Prediction Equation. The Observation Equation. Applying the Kalman Filter. Model Selection. Applications to Trading. Index. The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples.If each element of a vector of time series xt first achieves stationarity after differencing, but a linear combination α'xt, is already stationary, the time series xt are said to be co-integrated with co-integrating vector α. There may be several such co-integrating vectors so that α becomes a matrix. Interpreting α'xt, = 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance.The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representations for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application.In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.  We study model-driven statistical arbitrage in US equities. Trading signals are generated in two ways: using Principal Component Analysis (PCA) or regressing stock returns on sector Exchange Traded Funds (ETFs). In both cases, the idiosyncratic returns are modelled as mean-reverting processes, which leads naturally to 'contrarian' strategies. We construct, back-test and compare market-neutral PCA- and ETF-based strategies applied to the broad universe of US equities. After accounting for transaction costs, PCA-based strategies have an average annual Sharpe ratio of 1.44 over the period 1997 to 2007, with stronger performances prior to 2003. During 2003-2007, the average Sharpe ratio of PCA-based strategies was only 0.9. ETF-based strategies had a Sharpe ratio of 1.1 from 1997 to 2007, experiencing a similar degradation since 2002. We also propose signals that account for trading volume, observing significant improvement in performance in the case of ETF-based signals. ETF-strategies with volume information achieved a Sharpe ratio of 1.51 from 2003 to 2007. The paper also relates the performance of mean-reversion statistical arbitrage strategies with the stock market cycle. In particular, we study in detail the performance of the strategies during the liquidity crisis of the summer of 2007, following Khandani and Lo [Social Science Research Network (SSRN) working paper, 2007].