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2 Pair Trading

We used to think that ifwe knew one, We knew two, because one and one are two. We are finding that we must learn A great deal more about and.

Sir Arthur Eddington

Mathematical Maxims andMinims

Pair trading is a market-neutral strategy where a long position in one stock and a short position in another stock are initiated simultaneously. The profit principle ofthe trade is based on mean reversion, i.e., two stocks that normally trade in tiie same direction become temporarily uncorrelated andeventuallywiU revert to the mean; tiiis technique is also known as statistical arbitrage.

Most of the published work on pair trading pertains to positions held over several days [24] or as much as several months, e.g., a typical arbitrage where an acquiring companys stock is shorted and the target stock is bought. However, recent changes in margin requirements circa 2001 give tiie day trader access to as much as 4:1 intraday buying power, perfectly suited for intraday pair trading.

This chapter presents a complete strategy for trading stock pairs inlraday, although the technique can be extended to positions of several days or more. First, a definition for the spread is presented along with a visual TradeStation indicator. Then, the spread bands are calculated to determine when a pair trade is initiated; a trade triggers only in the area outside tiie upper and lower bands. Finally, the complete entry and exit rules for the pair trading system are defined, followed by several examples.

The allure of pair trading is that it is a strategy witii little risk. Furtiier, tiie trader does not really care about the direction ofthe market and does not have to worry about nagging issues such as the S&P futures or the reaction to economic reports. However, no stock is immune to the risk of a trailing halt or an earnings warning, Before trailing begins each day, review each of the stocks for specific company news: upcoming earnings reports, conference calls, and upgrades and

downgrades. Be aware that news wUl frequently create a spread opportunity, but a stock with major news may demand a reverse spread strategy.

2.1 The Spread

The Spreaelistbe difference between two stock ratios sharing a common anchor point in time, for example, the closing price of tod compared to the closing price ofyesterday. For a stock pair A-B, if Stock A closed tod at 21 and yesterday at 20, then its closmg ratio is 21 / 20 = 1.05. Similaiiy, if StockB closed today at 42 and yesterday at 40, then its closing ratio is 42 / 40 = 1.05. Thus, the spread is 1.05 minus 1.05 equals zero. Here, in this narrow instance, the stocks are moving in sjnchronization, i.e., they are correlated

Consider the stock pair A-B again. If Stock A closed tod at 20 and yesterday at 20, then its closing ratio is 20 / 20 = 1.0. If Stock cksed today at 42 and yesterday at 40, then its closing ratio is 42 / 10 = 1.05. Here, the spread is 1.0 minus 1.05 equals -.05, and Stock A is undent-alued relative to Stock If the spread were apositive number, then StockAwouMbe overvalued relative to StockB.

For a daHy spread sjstem, the anchor point couM be the number of dajs ago, e.g., the close of today compared to the close five ds ago. For an intraday sjstem, we compare the last price on an intraday chart to either the closing price ofyesterday or the opening price oftod. The difference is whether or not the trader wants to fector gaps into the spread calculation. If so, then the closing price ofyesterday is chosen.


We calculate the Spread with Equation 3.1, using the variable Last to indicate a real-time price. Divide the last real-time trade price of Stock A by its closing price yesterday, and do the same for StockB. Subtract the difference to obtain the current Spread.

The spread is displayed in a separate plot below the charts ofthe stock pair. As the charts update in real-time, the spread is plotted as a line within two channel lines known as spread bcnids. In Figure 2.1, the parallel line running across the top ofthe bottom panel is the upper spread band, and the bottom line is the lower spread band. The spread bands are computed at the beginning of eachdayusingyesterdayshistoricalvolatilities and correlation (seebelowl.

When the spread line touches the upper band, the stock in the top panel (Stock A) becomes overvalued relative to the stock in the lower panel (Stock B), a condition indicating that Stock A should be shorted and Stock should be bought. When tin- spread touches the lower band. Stock A becomes underval

ued relative to Stock In this case. Stock A should be bought and Stock should be shorted at the same time.

2.1. The Spread

2.2 Spread Bands

A SpreadBand (SB) is a standard deviation-based imit ofthe normal probahilitj curve. The SB factors in the combined volatilitjand correlation ofa stockpairto derive an estimate of where a pair trade should be initiated. The trader then determines how many standard deviations are appropriate for any given pair. For example, if one standard deviation is selected, then when the spreadhits the SB, the stock pair has a 68° chance of reverting to the mean. If two standard deviations are selected, thenthe pairhas a 95°oprobabilitj [29]. However, practice is different than theory-these calculations presume the absence ofnews and other external factors that affect stockprices.

First, we review the factors in the SB calculation. The Historical Volatility (1 TV) was discussed in Chapter 1, so the 30-day HV must be calculated for each stock in the pair: HVa and HVb. The next factor in the Spread Band equation is the Cocnicient, also known as Coepicient R. The R-value correlates the movement of one stock price with another. A correlation of +1 means that the two stocks move synchronously, while a correlation of -1 means that the two stocks move in opposite directions. R is the second fector in the SB equation.

ftTVI-Daly NA3DQ



rlii, i.l-l-ii

tl Il.llll


ConBlaHan(ClDseofDat8l.ClaseoTData2.30,.7v%7) 0,33 0.70 4.70



Figure 2.2. Correlation Coefficient

Now, we can calculate the Spread Bands for a stock pair for one trading day, as shown in Equation 3.2. The formula assumes 365 days in a year, so the square root of 1 / 365 is taken to get the constant .0523, represented by A. The v-ariable SZ)represents the number ofstandard deviations. The variables UIa and/ are the historic volatilities of Stock Aand StockB, respectively.

The relationship between the Spread Bands and R is an inverse relationship. As the correlation R increases, the bands narrow and vice versa. Consequentiy, the volatilities are multiplied by the factor (1 R).


The formula for the Spread and is as fohows:

SB = kXSD ( + ) X(1-R)


Here is an example oftwo correlated video game software stocks. Our Stock Ais Activision(ATV:Nasdaq), and StockB is THQIncorporated (THQI:Nasdaq). The 30-day volatilities and correlation are as fohows:

□ HVio of ATVI is 0.643

□ HV3oofTHQlisO.S22

□ R3ois0.33

The Spread Band for a standard deviation for the ATVI-THQl Pair is:

0.0523 X 1 X (0.643 + 0.822) X (1 - 0.33) = 0.0513

The final factor to consider is the number of standard deviations required for generating a pair-trading signal. We have selected two standard deviations as the default value because prices have at least a 95% chance ofreverting to the mean, assuming a normal probabihty curve. Therefore, we multiply the Spread Band value (0.0513) by the number ofstandard deviatbns (2.0) to obtain the upper and lower Spread and values, +0.1025 and-0.1025 (Figure 2.3).

»TV1-Sinin NASDQ




...Jl..."...,. , "--.. ....,



KcnMSprc>dClosBorData3,CloseorDat»),Z,30) 0,0000 0,1025 0.0000 .0.1025

10 25 11:05 1145 12:2S 1305 13 45 14 25 15.05 7/20 1D:35 11 IS

riK»it-2..1. Spi.Md

The chart below (Figure 2.2) shows a dahy chart oftwo correlated stocks with a ciuTentR30of0.33, towards the low end ofthe estabhshed correlation range

The Acme P System is based on highly correlated stock pairs with an R of at least +0.3. However, it is perfectly possible to reverse the signals to trade non-correlated pairs and use the area outside ofthe spread bands. Use an R-value of -0.3 or less for non-correlated stock pairs.

For further information on equity correlations, go to the Market Topologj Web site at http: At this site, a stock symbol query wih retum ahst ofstocks that are most correlated to a reference stock. The site provides equity maps for domestic and foreign market indices such as the S&P 500 showing the correlations among all ofthe stocks composing the index.

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