pairs trading
TRANSCRIPT
An investigation into Pairs Trading (Statistical Arbitrage)
Strategies
Brian Bannon
i
Abstract
In this report, the author is going to conduct an investigation into widely used trading
strategy in the financial trading industry known as “Pairs Trading’ or Statistical Arbitrage
trading. This is a trading strategy that looks at the mispricing of two highly correlated or co-
integrated stocks in the market and opening and closing positions in the two stocks so that the
stock mean revert and risk free arbitrage opportunity exists to make profit.
This report will investigate the theory behind Pairs Trading, look at some academic research
into the topic while in parallel drawing from some industrial reports conducted on the
strategy, and finally conduct some empirical back testing and results from the methodologies
learned throughout the author’s research.
ii
Table of Contents
1 INTRODUCTION ............................................................................................................ 1
1.1 PAIRS TRADING ........................................................................................................... 1
2 LITERATURE REVIEW ............................................................................................... 2
3 METHODOLOGY .......................................................................................................... 4
3.1 DISCUSSION OF ACADEMIC THEORY ........................................................................... 4
3.1.1 Correlated Pairs Trading ...................................................................................... 4
3.1.2 Co-Integrated Pairs Trading ................................................................................ 5
3.1.3 Augmented Dicky Fuller Test ............................................................................... 5
4 ANALYSIS RESULTS .................................................................................................... 6
4.1 CORRELATED PAIRS .................................................................................................... 6
4.1.1 Allied Irish Bank vs. Bank or Ireland .................................................................. 6
4.1.2 Christian Dior SE vs. Moet Hennessy Louis Vuitton SE .................................... 9
4.2 CO-INTEGRATED PAIRS ............................................................................................. 11
4.2.1 Google US Equity vs. Apple US Equity .............................................................. 11
4.2.2 Nike US Equity vs. Adidas US Equity ................................................................ 14
5 CONCLUSION .............................................................................................................. 16
5.1 SUMMARY OF RESULTS ............................................................................................. 16
6 BIBLIOGRAPHY .......................................................................................................... 17
1
1 Introduction
1.1 Pairs Trading
The pair trading is a market neutral trading strategy enabling traders to profit from virtually
any market conditions. It matches a long position with a short position in a pair of highly
correlated or co-integrated instruments such as two assets. Pairs traders wait for weakness in
the correlation or divergence from the long term equilibrium in co-integration, and then go
long on the under-performer while simultaneously going short on the over-performer, closing
the positions as the relationship returns to its statistical normal. The strategy’s profit is
derived from the difference in price change between the two instruments, rather than from the
direction in which each moves.
Fig(1): Graph taken from authors presentation on pairs trading. Simple example of what pairs trading looks
graphically. At June 06 the trader would short Pepsi and long Coke and at Jun 08 long Pepsi and short Coke.
2
2 Literature Review Pair’ trading has received quite a lot of academic and industrial investigation. Looking for
better ways to make the process more efficient and looking into different statistical tests for
easier implementation. (UBS Investment Research, 2010). And also looking at back testing to
see if this trading strategy performance has been successful or have been overshadowed by
other trading strategies (Do & Faff, 2010)
UBS research division conducted a very concise empirical report on the performance of pairs
trading in the European market along with an empirical test on all the statistical approaches to
determine co-integration, ranking the performance of different tests across an eight year time
period Figure (2). Concluding that pairs trading was a profitable strategy in the European
market during the period, with the success of multiple statistical screens identifying
opportunities. It concluded that there seems to be no decline in European pairs trading
profitability. (UBS Investment Research, 2010)
In Do and Raff’s paper they also examine the performance of pairs trading, restricted to the
U.S. market. They noted a continuous declining trend towards the end of the 2000’s.This was
mainly caused by the competition in the hedge fund industry. They deduced a 70% decline in
the performance of pairs’ strategies due to the worsening arbitrage risks in various pairs’
portfolios. And found during the 2007 financial crisis pairs trading performance was strong
during market downturns. They investigated the importance of homogeneity is pairs
classification with lower divergence risk. Lastly they found pairs trading by industry group
were quite an important factor. They tested by pairs trading the S&P major industry groups,
utilities, financials, transportation and industrials. Profits across all sectors were found but
significant profit was found in Financial and Utility sectors.
3
Advancements of the traditional pairs trading co-integration strategy on two stocks has also
been adopted by hedge funds in the last couple of years. Hedge funds are growing in the
amount of market neutral hedge fund strategies being implemented by these firms. One such
is the long short hedge strategy applied to portfolios. Traditional strategies do not guarantee
that the tracking error is stationary and thus rebalancing for the hedge to remain tied to the
benchmark. The con-integration strategy, the hedge is mean reverting to the benchmark and
tracking errors are stationary by design, this can be achieved with relatively few stocks and
much lower turnover. (Alexander, Giblin, & Weddington, 2001)
4
3 Methodology
3.1 Discussion of Academic Theory
3.1.1 Correlated Pairs Trading The main theory behind this Bloomberg Spread sheet is to identify pairs that have good
mean reversion which is the basis behind pairs trading.This spreadsheet identifies pairs of
stocks that have this criteria and analyse the trades and performance over the trading period.
The mean reversion technique is used to see if paris of stocks are co-integrated .In correlation
two random stocks may move together and are perfectly correlated but the differnce is not
arbitrary and may drift off from each over time while still being perfectly correlated.Co-
integration is a measure of how well the two stocks are tied together.Two stocks that stay in
the same range as each other and have a long term relationship are co-integrated.Co-
integrated stocks are expected to stay parallel to one another over time, since the
differnce(Residuals) between the two will be corrected over time..
When the price of one or both stocks moves above or below the long term equilibrium
position, a trader can profit by taking a long/short postions on the pair of stocks and wait for
the relative price of the stock to move back into equilibrium.
In this template, we first establish if this long-term co-integration relationship exists by
performing a linear regression of the two times series of the stock.
𝑦! = 𝛽!𝑥! + 𝜀! 𝑖 = 1…𝑛 (1)
Where y is the dependent variable, x the independent variable, 𝛽 is the linear coefficient and
𝜀 is the error variable. Then use these results to find if the current prices deviate from the
equilibrium, when to put a trade on and when to profit from these trades. When fitting the
linear regression it is good to look at the residual spread series, which is the difference
between the fitted value and the actual value. A good residual spread series is that crosses 0
times.
5
3.1.2 Co-Integrated Pairs Trading (Ganapthy, 2004) Again in this section we look into co-integration in pairs to conduct pairs trading but this time
we are going to use a statistical test for co-integration called “Augmented Dicky Fuller Test”
(ADF). There are many statistical tests researchers can perform to test for co-integration such
as Run’s Test, KPSS, IKPSS, and Sum of Squares. Usually in the industry research
departments or traders would perform several different statistical co-integration tests on a pair
of stocks and pick the best 4 out 6, 3 out 5 etc. to increase the statistical testing of the pair.
(UBS Investment Research, 2010)
Fig (2): Performance of the six different co-integration tests ranging from 1992-2010 (UBS Investment
Research, 2010)
3.1.3 Augmented Dicky Fuller Test (Herlemont, 2000) Two non-stationary time series ( I(1) ) {Xt} and {Yt} are co-integrated if some linear
combination aXt + bYt , with a and b being constants, is a stationary series ( I(0) ).The reason
this is stated is because in the ADF test that a variable follows a unit-root process, then a and
b would be co-integrated. The theory behind the test is, perform a linear regression on the
residuals and see what the value of 𝑎 is:
𝑦! = 𝑎𝑦!!! + 𝜀! (2)
Where 𝑦! = 0 and 𝜀~𝑁(0,𝜎!) for an AR (1) process then the residual process can be written
as
Δ𝑦! = 𝛿𝑦!!! + 𝜀! (3)
.We then perform the hypothesis testing that if
6
𝐻!: 𝑎 = 0
𝐻!: 𝑎 ≠ 0
If, 𝐻!: 𝑎 = 0 is true there is no co-integration and the signal contains a unit root, a unit root
means our signal is non-stationary. The test gives a pValue, the lower this number the more
confident we can be that we have found a stationary signal. P-Values less than 0.1 are
considered to be good candidates. (Quant, 2012)
4 Analysis Results
4.1 Correlated Pairs
4.1.1 Allied Irish Bank vs. Bank or Ireland We conducted a correlation test using the Bloomberg correlation template testing from March
2014 to March 2015.
Fig (3): Linear regression of AIB and Bank of Ireland
We see from this regression that that the two stocks, while correlated over the long run from
initial testing, are not correlated over the trading period with a very low R2 which is close to
zero.
Next we look at the residual series of the stocks, from this we see that the number of
crossings is quite high (16) which is a good indicator that it has a good mean reversion.
y = -‐0.0395x -‐ 1.3154 R² = 0.00456
-‐1.6
-‐1.4
-‐1.2
-‐1
-‐0.8
-‐0.6
-‐0.4
-‐0.2
0 -‐3 -‐2.5 -‐2 -‐1.5 -‐1 -‐0.5 0
BKIR ID
Equ
ity
ALBK ID Equity
7
Fig (4): Residual spread of the stocks showing the deviation from the statistical mean with indicators of
standard deviations Difference between the regression and the actual fit
Next we look at the strategy itself over the trading period (365 days) with a standard
deviation multiplier for the residual spread of 1.5*σ .We have invested 100 shares in AIB
and the number of shares in Bank of Ireland is slope matching from regression.
Fig (5): Summary of trade entry and exit strategies for the trading period.(Green triangle indicating entry and
red triangle indicating exit. All trades are closed at the end of the trading period)
-‐0.3
-‐0.2
-‐0.1
0
0.1
0.2
0.3
Res
idua
l Spr
ead
Time
Residual spreads Residual Spread +1 Std. Dev. +2 Std Dev. -‐1 Std. Dev. -‐2 Std Dev.
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Res
idua
l Spr
ead
Trades -‐ Entry and Exit Entry Exit Residual Spread
Mean of Residual spreads +1.5 Std. Dev. -‐1.5 Std. Dev.
8
Fig (6). Trade performance of the strategy. Blue line indicating the cumulative profit of the strategy blue diamond’s indicating profit and loss. As you can see this strategy has experienced both.
During this trading period the two stocks were very volatile and there was a very wide spread
in the profit/loss margin over this time period. The maximum loss of any trade exiting at that
point would have been €489.25. Conversely the maximum loss at any trade point could have
been -€247.76. So there was quite a volatile spread during the period.
Table (1): Table of the trades list from this strategy.
-‐400
-‐200
0
200
400
600
800
1000
1200
Pro
fit/L
oss(
Eur
os)
Trade Performance Profit(Loss) Excursion CumulaOve P&L
Trades List
No. of Price (A)
Price (B)
Qty (A) Qty (B) Entry/Exit
Cumulative
Individual
Cumulative
Date Entry/Exit
Trading Days
ALBK ID Equity
BKIR ID Equity
ALBK ID Equity
BKIR ID Equity
Cashflow
Cashflow
P&L P&L
15/0/14
Entry 0.097 0.247 -‐1031
10256
(2,433.)
(2,433)
27/08/2014
Exit 74 0.086 0.305 1031 -‐10256
3,039.41
606.19
606.19
606.19
05/12/2014
Entry 0.094 0.345 1064 -‐7344
2,433.66
3,039.
09/01/2015
Exit 23 0.08 0.292 -‐1064
7344 (2,059.3)
980.53
374.34
980.53
02/03/2015
Entry 0.087 0.347 1149 -‐7297
2,432.10
3,412.2
9
As we can see from Table (1) that the strategy has a positive cumulative profit. This would be
expected as the pairs were not correlated well and thus the spread between the two stocks was
existent and arbitrage opportunities arose. This profit is large mainly because of the volatility
at the time of trading and the trade spread at which we were trading at.
4.1.2 Christian Dior SE vs. Moet Hennessy Louis Vuitton SE The next pair for test of correlation is Christian Dior and Moet Hennessy Louis Vuitton. I
choose this pair because over the 10 year period they were 99% correlated on the Euronext
Paris. Again we had the same conditions of above initial investment standard deviation again
etc. From this we see that the number of crossings is high which is unusual considering its
correlation. So the series is slightly mean reverting. The residual spread initially stays within
2std dev. in the spread chart until it goes outside near the end (Figure.8).Because of the
spread within the two series the trader would enter and exit the strategy a lot more as the
spread narrows and returns to equilibrium as show in Figure.9.
Fig(7): Linear regression of the Dior and Moet as we can see from the R2 is very high indicating it is a good
correlation fit over the trading period.
y = 1.1019x -‐ 0.4587 R² = 0.97372
4.7
4.8
4.9
5
5.1
5.2
5.3
4.65 4.7 4.75 4.8 4.85 4.9 4.95 5 5.05 5.1 5.15 5.2
CDI FP Equity
MC FP Equity
10
Fig (8): Residual spread of the stocks showing the deviation from the statistical mean with indicators of
standard deviations Difference between the regression and the actual fit
The spread between the max loss and max profit at any trade point was just under €10. So
there was not much volatility in the trade positions. Thus because of this small spread
throughout the whole trading period, returns would be low thus the cumulative profit over
the whole trading period is €11.77.Which is poor if your take into account trading costs and
commission.
-‐0.06
-‐0.04
-‐0.02
0
0.02
0.04
0.06
0.08
Res
idua
l Spr
eads
Residual spreads
Residual Spread +1 Std. Dev. +2 Std Dev. -‐1 Std. Dev. -‐2 Std Dev.
11
Fig(9): Summary of trade entry and exit strategies for the trading period.(Green triangle indicating entry and
red triangle indicating exit. All trades are closed at the end of the trading period)
Fig(10). Trade performance of the strategy. Blue line indicating the cumulative profit of the strategy blue
diamond’s indicating profit and loss. As you can see this strategy has experienced both.
4.2 Co-integrated Pairs
4.2.1 Google US Equity vs. Apple US Equity
From implementing the ADF test we find that the ADF t-statistic from both stocks is quite
low indicating good performance from the ADF test result on both stocks. I did not provide
these graphs as they have not much connotation behind the strategy other to present that the
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Res
idua
l Spr
ead
Trades -‐ Entry and Exit Entry Exit Residual Spread
Mean of Residual spreads +1.5 Std. Dev. -‐1.5 Std. Dev.
-‐8 -‐6 -‐4 -‐2 0 2 4 6 8
10 12 14
Pro
fit/L
oss(
Eur
os)
Trade Performance Profit(Loss) Excursion CumulaOve P&L
12
two stocks are non-stationary. From looking at the OLS regression we find that the R2 is
0.0095 showing that there is not much correlation which is good for this approach. We look
at the residual plot which we are interested in. The p-value is quite low and is statistically
significant relating that the residual plot is stationary.
Fig(11): Summary of trade entry and exit strategies for the trading period(Yellow triangle indicating long entry
and red triangle indicating short entry and green square is exit position.)
We have a max lag of 10 lags on and a standard deviation of 1.5*, note that this is just the
threshold range of when to place trades with 2*σ being the max due to the properties of the
normal distribution. Further research can be done into changing this for different thresholds
but we are going to keep it constant for the purpose of this report.
-‐0.2
-‐0.15
-‐0.1
-‐0.05
0
0.05
0.1
0.15
0.2
27/03/2014 16/05/2014 05/07/2014 24/08/2014 13/10/2014 02/12/2014 21/01/2015 12/03/2015
Res
idua
l Spr
ead
Residual Residual Mean +1STD -‐1STD +2STD
-‐2STD ShortEntry LongEntry ExitPoint
13
Fig(12): Profit and loss chart indicates unrealised profit while positions are open with indications of the
strategy entry and exit points
Fig(13): this is a threshold analysis graph it show the max number of trades that should be executed when the
threshold reaches a certain point to maximise profit opportunities Max number of trades at 0.24*σ is 11 and the
min number of trade at1.84σ is 1
From looking at cumulative profit/loss summary table we can see that the cumulative profit
for this strategy is $4713.84 with an execution of 4 trades with an average holding period of
-‐1500
-‐1000
-‐500
0
500
1000
1500
27/03/2014 16/05/2014 05/07/2014 24/08/2014 13/10/2014 02/12/2014 21/01/2015 12/03/2015
Pro
fit/L
oss(
$)
Time
P&L CumulaOve P&L ShortEntry LongEntry ExitPoint
0
2
4
6
8
10
12
14
16
18
0 0.5 1 1.5 2 2.5
Num
ber o
f Trade
s
Threshold of Standard DeviaOon
14
40 days with the long threshold being -0.089σ and the short threshold 0.089σ.Overall this is a
very profitable pairs trading strategy.
4.2.2 Nike US Equity vs. Adidas US Equity In this case we are going to look at two stocks that should be correlated and thus co-
integration should not be present for profit. From looking at the looking at the p value and
ADF t-statistic it looks like the plot of the residuals is stationary.
Fig(14): Summary of trade entry and exit strategies for the trading period(Yellow triangle indicating long
entry and red triangle indicating short entry and green square is exit position.)
But if we look at the OLS regression we see that the R2 ≈0.5 showing forms of correlation.
-‐0.2
-‐0.15
-‐0.1
-‐0.05
0
0.05
0.1
0.15
0.2
02/06/2014 22/07/2014 10/09/2014 30/10/2014 19/12/2014 07/02/2015
Res
idua
l Spr
ead
Residual Residual Mean +1STD -‐1STD +2STD
-‐2STD ShortEntry LongEntry ExitPoint
15
Fig(15): Profit and loss chart indicates unrealised profit while positions are open with indications of the
strategy entry and exit points
Fig(16): this is a threshold analysis graph it show the max number of trades that should be executed when the
threshold reaches a certain point to maximise profit opportunities Max number of trades at 0.19*σ is 17 and the
min number of trade at1.85σ is 1
-‐1500
-‐1000
-‐500
0
500
1000
1500
02/06/2014 22/07/2014 10/09/2014 30/10/2014 19/12/2014 07/02/2015
Pro
fit/L
oss(
$)
P&L CumulaOve P&L ShortEntry LongEntry ExitPoint
0
2
4
6
8
10
12
14
16
18
0 0.5 1 1.5 2 2.5
Num
ber o
f Trade
s
Standard DeviaOon
16
From the P&L summary we see an execution of 2 trades with a loss of -$807.50. Although
the residuals do look stationary and that is fine for co-integration. I feel the stocks may be too
correlated to make a profit opportunity and that is why we see negative returns from this
strategy.
5 Conclusion
5.1 Summary of results In this report the concept of Pairs Trading has been introduced. The report has given a broad
overview of the trading topic and a look at the academic and industrial research that is still
on-going in this field of statistical arbitrage. Some theory about how to test for correlation
and co-integration has been presented and discussed through concise theory.
Lastly we looked at an approach to analysing pairs trading through the powerful Bloomberg
database tool. From this approach my results coincided with my academic approach giving
the reader a clear indication of the methodology and the expected outcome from each of the
strategy’s approaches.
17
6 Bibliography
Alexander, C., Giblin, I., & Weddington, W. (2001). Conintegration and Asset Allocation: A
new Hedge Fund Strategy. ISMA Centre Discussion Papers in Finance 2001-03.
Do, B., & Faff, R. (2010). Does Simple Pairs Trading Still Work? Financial Analysts
Journal, 66(4), 83-95.
Ganapthy, V. (2004). Pairs Trading: Quantitative Methods and Analysis . New Jersey: Wiley
& Sons Inc.
Gatev, E., Goetzmann, W., & K.G., R. (2006). Pairs Trading:Performance of a Relative-
Value Arbitrage Rule . Review of Financial Studies , 19, 797-827.
Herlemont, D. (2000). Pairs Trading, convergence trading, cointegration. YATS Finances &
Technologies .
Quant, G. (2012, December 17). Gekko Quant -Quantative Finance Blog. Retrieved from
http://gekkoquant.com/2012/12/17/statistical-arbitrage-testing-for-cointegration-
augmented-dicky-fuller/
UBS Investment Research. (2010). Understanding Pairs Trading. UBS, Global Equity
Research. UBS.