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.
Statistical arbitrage and in particular pairs trading strategies initially took place in the mid 80's @cite_5 . Since then, a great deal of work has been done on the problem of assembling mean reverting portfolios, mostly using cointegration techniques (see @cite_3 for more comprehensive information). In order to quantify the amount of mean reversion in various portfolios, different proxies are often suggested such as zero-crossing and predictability @cite_7 @cite_10 . In this work, we consider a new proxy for mean reversion which is aimed at maximizing fluctuation, as well as keeping the mean close to zero. Furthermore, whereas classical cointegration techniques require training period before applying a trading strategy (see for instance @cite_11 @cite_1 ), the online approach does not require that, in addition to providing a performance guarantee against the best mean reverting portfolio in hindsight.
@cite_7 @cite_1 @cite_3 @cite_5 @cite_10 @cite_11
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We test a Wall Street investment strategy known as "pairs trading" with daily data over the period 1962 through 1997. Stocks are matched into pairs according to minimum distance in historical normalized price space. We test the profitability of several trading rules with six-month trading periods over the 1962-1997 period, and find average annualized excess returns of up to 12 percent for a number of self-financing portfolios of top pairs. Part of these profits may be due to market microstructure effects. Nevertheless, our historical trading profits exceed a conservative estimate of transaction costs through most of the period. We bootstrap random pairs in order to distinguish pairs trading from pure mean-reversion strategies. The bootstrap results suggest that the ?pairs? effect differs from previously documented mean reversion profits. 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]. 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. 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.