Norwegian School of Economics Bergen, Spring, 2014 Statistical Arbitrage: High Frequency Pairs Trading Ruben Joakim Gundersen Supervisor: Associate Professor Michael Kisser NORWEGIAN SCHOOL OF ECONOMICS Master Thesis, MSc. Economics and Business Administration Financial Economics This thesis was written as a part of the Master of Science in Economics and Business Administration at NHH. Please note that neither the institution nor the examiners are responsible − through the approval of this thesis − for the theories and methods used, or results and conclusions drawn in this work. Abstract In this thesis we examine the performance of a relative value strategy called Pairs Trading. Pairs Trading is one of several strategies collectively referred to as Statistical Arbitrage strategies. Candidate pairs are formed by matching stocks with similar historical price paths. The pairs, once matched, are automatically traded based on a set of trading rules. We conduct an empirical analysis using high frequency intraday data from the first quarter of 2014. Our findings indicate that the strategy is able to generate positive risk adjusted returns, even after controlling for moderate transaction costs and placing constraints on the speed of order execution. i Preface This thesis marks the end of my master studies at NHH. The work has at times been challenging, but at the same time it has also been very rewarding. I would like to thank my supervisor, Michael Kisser, for valuable input and suggestions during the writing process. ii Contents 1. Introduction ............................................................................................................ 1 2. The concept of pairs trading .................................................................................... 3 2.1 Normalization of stock prices .............................................................................. 5 3. Pairs trading in previous literature ......................................................................... 7 4. Different approaches to pairs trading .................................................................... 11 4.1 The Cointegration approach ...............................................................................11 4.2 The Distance approach ......................................................................................15 4.3 The Stochastic approach ....................................................................................16 5. Simulation testing and choice of approach ............................................................ 19 5.1 Model one - The granger representation theorem ..............................................20 5.1.1 Results ..........................................................................................................22 5.2 Model two - The Stock & Watson Common trends model ..................................23 5.2.1 Results ..........................................................................................................27 5.3 Consequences of inaccurate cointegration coefficient estimates .......................27 5.4 Choice of method .................................................................................................28 6 Testing a high frequency pairs trading strategy .................................................... 29 6.1 Data .....................................................................................................................29 6.2 Formation and trading periods ...........................................................................31 6.3 Practical Implementation of the distance approach ..........................................32 6.3.1 Pairs formation ............................................................................................32 6.3.2 Pairs Trading ...............................................................................................32 6.4 Return computation ............................................................................................34 6.5 An algorithmic representation of the test setup ................................................35 6.6 Results .................................................................................................................36 6.6.1 Unrestricted pairs matching .......................................................................37 6.6.2 Restricted case – Pairs from corresponding industries ...............................39 6.7 Impact of transaction costs and timing constraints ...........................................42 6.7.1 Commissions and short fees ........................................................................43 6.7.2 Speed of execution ........................................................................................45 6.7.3 Concluding remarks commissions and execution speed .............................47 6.6.4 Trade slippage .............................................................................................48 iii 6.8 Exposure to market risk ....................................................................................49 6.9 Is pairs trading a masked mean reversion strategy? .........................................51 7. Conclusion ............................................................................................................. 53 References .................................................................................................................... 53 Appendices ................................................................................................................... 55 iv Statistical Arbitrage: High Frequency Pairs Trading 1. Introduction In this paper we examine a popular quantitative investment strategy commonly referred to as “pairs trading”. The basic concept of pairs trading is remarkably simple; one identifies a pair of stocks that exhibit historical co–movement in prices. Subsequently, if significant deviations from the historical relationship are observed, a position is opened. The position is formed by simultaneously selling short the relative winner and buying long the relative looser. When the prices eventually converge the position is closed and a profit is made. The strategy builds upon the notion that the relative prices in a market are in equilibrium, and that deviations from this equilibrium eventually will be corrected. Applying a pairs trading strategy is therefore an attempt to profit from temporary deviations from this equilibrium. According to Gatev, Goetzmann & Rouwenhorst (2006) pairs trading strategies have been used by practitioners on Wall Street in various forms since the mid– 1980s. The strategy is often said to have originated within Morgan Stanley in a group led by Nunzio Tartaglia. The focus of the group was to develop quantitative trading strategies by employing advanced statistical models and information technology. The group sought to “mechanize” the investment process by developing trading rules that could be automated. Pairs trading was one of the resulting strategies. The group used this strategy with great success in 1987 – when the group is said to have generated a profit of $50 million – but were dissolved in 1989 after a period of poor performance. In the last decades, as technology has become more accessible, the strategy has been increasingly popular with investors. Pairs trading is often placed in a group of quantitative trading approaches collectively referred to as statistical arbitrage strategies. The arbitrage part in this context is somewhat misleading as arbitrage implies a risk free profit opportunity at zero upfront cost. A pairs trading strategy is by no means risk free. There is no guarantee that the stocks in a pair will converge. They could even continue to 1 Statistical Arbitrage: High Frequency Pairs Trading diverge, resulting in significant losses. Furthermore, the strategy is also often claimed to be “market–neutral”, meaning that the investor is unexposed to the general market risk. However, while it certainly is possible to create market neutral pairs, the total market risk of a position depends on the amount of capital placed in each stock and the sensitivity of the stocks to such risk. * In the first part of this thesis we explore the background and the theoretical basis for a pairs trading strategy. In addition we compare the performance of two existing pairs trading methods by applying them to sets of simulated data. In the latter part of the paper we conduct an empirical analysis of a concrete pairs trading strategy. Through this analysis, we seek to determine if a pairs trading strategy delivers returns that are superior when compared to a buy–and–hold strategy. We use high frequency data for stocks listed on the Oslo Stock Exchange. The obtained results indicate that it is possible to generate positive risk–adjusted returns by following a pairs trading strategy. The results are robust after controlling for transaction costs, and placing restrictions on the execution speed. Specifically, we report annualized returns as high as 12 after costs %. In addition, the standard deviations of the returns are low. This combination leads to an impressive Sharpe ratio exceeding 3. We find that the constructed portfolios have close to zero exposure to market risk. 2 Statistical Arbitrage: High Frequency Pairs Trading 2. The concept of pairs trading The pairs trading strategy is based on the concept of relative pricing. If two securities have identical payoffs in all states their price should also be identical. This is a variant of the principle commonly referred to as the Law of One price (LOP). Lamont and Thaler (2003, 191) defines the LOP as follows “[…] identical goods must have identical prices”. It is important to note that the prices do not need to be “correct”, from an economical point of view, for the LOP to be valid. The LOP simply asserts that stocks yielding identical payoffs should have the same current price. The law is therefore applicable to the relative pricing of the stocks in a market, even if the pricing is economically incorrect (Gatev et al., 2008). We can further extend the example with identical payoffs to a situation where the payoffs are very similar but not identical. In such a situation the prices of the securities should also be similar. If a temporary deviation from this relative pricing relationship occurs it should be possible to exploit this by taking a position that generates a profit when the deviation is corrected. Pairs trading is one example of a strategy aiming to profit from such temporary deviations. Before a pairs trading strategy can be implemented on a practical level we need to address some fundamental questions: What pairs of stocks are suitable? When should a position be opened or closed? How should one determine the amount of capital placed in the individual long/short positions? As we will see in section 4, there are multiple approaches to pairs trading, all offering different answers to these questions. Even so, the basic structure of a pairs trading strategy is common for all approaches. The first step involves identifying a pair of stocks whose prices appear to move together according to some fixed relationship. The period of time used to establish such a relationship is referred to as the formation period. After the suitable pairs are identified we enter the trading period. In this period we continue to observe the spread. If a significant deviation from the relationship is observed a position is opened. The investor then buys long some quantity of the relative looser and sells short some quantity of the relative winner. 3 Statistical Arbitrage: High Frequency Pairs Trading The following figure graphically illustrates the concept of the pairs trading strategy. Stock A Stock B Position 1,2 5 1 4 0,8 3 0,6 2 0,4 1 0,2 0 0 50 100 150 200 250 300 Time Figure 2.1 – A pairs trading example The figure shows two simulated stock prices on the left scale. In addition a dummy variable (right scale) indicates if a position is open or closed . A position is opened if the spread exceeds a previously calculated entry–threshold value. The position is closed at the next crossing of the prices. In this specific example, a position is opened at , and later closed when the prices cross at . At a position is again opened. This position stays open until . An intuitive way to understand the payoffs that would result from a trade is to think of the spread between the two stocks as a synthetic asset. When a position is opened the trader is effectively selling the spread short, speculating that it will decrease. When the stock prices later cross the value of the spread is zero. The trader then closes the position, and earns a profit equal to the value of the spread at the time the position was entered. Since pairs trading is a relative value strategy, a framework for assessing the relative value development in a pair is essential. In the hypothetical example above the two stock price series both start at unity. This makes calculating the relative changes in values simple. At any given point in time the cumulative returns to the series are directly observable. Any return differences between the stocks are therefore easily calculated. 4 Statistical Arbitrage: High Frequency Pairs Trading Obviously, in a real situation the stock prices will not be as well behaved as in this example, but instead start at values that vary widely. This makes the comparison more complicated. In addition, Do, Faff & Hamza (2006,4) point out that the raw price spread between two stocks is not expected to stay at a constant level, even if the stocks yield identical returns1. This makes the raw price spread unsuitable as an indicator of when a position should be opened or closed. In order to overcome these issues we need to apply a transformation to the series. By transforming the price series we achieve price level independency and we are able to consistently assess the relative value development in the stocks. 2.1 Normalization of stock prices In previous academic literature (Engelberg, Gao & Jagannathan, 2009; Gatev et al., 2006), a common transformation to achieve price level independency is to construct cumulative return indexes for the stocks. These indexes reflect the total return since the beginning of a period, adjusted for dividends and splits. The indexes are then rebased to some constant common for all stocks considered. In the literature this transformation is usually referred to as normalization of the stock prices. Example – Normalized price series As the concept of normalization is central to this thesis we will provide a concrete practical example of the procedure. In the example we will consider the intraday development of the two stocks Seadrill and Fred Olsen Energy on January 6th 2014. The figure on the next page shows the raw price series. 1 To see this, think of the spread between two stocks A and B currently trading at 15 and 20 NOK respectively. The spread at time 0 is equal to . Now we assume a 100 % return in period 1. The spread then also doubles because . 5