Researching a Pairs Trading Strategy¶
By Delaney Granizo-Mackenzie
Part of the Quantopian Lecture Series:
Notebook released under the Creative Commons Attribution 4.0 License.
Pairs trading is a nice example of a strategy based on mathematical analysis. The principle is as follows. Let's say you have a pair of securities X and Y that have some underlying economic link. An example might be two companies that manufacture the same product, or two companies in one supply chain. We'll start by constructing an artificial example.
import numpy as np
import pandas as pd
import statsmodels
from statsmodels.tsa.stattools import coint
# just set the seed for the random number generator
np.random.seed(107)
import matplotlib.pyplot as plt
Explaining the Concept: We start by generating two fake securities.¶
We model X's daily returns by drawing from a normal distribution. Then we perform a cumulative sum to get the value of X on each day.
X_returns = np.random.normal(0, 1, 100) # Generate the daily returns
# sum them and shift all the prices up into a reasonable range
X = pd.Series(np.cumsum(X_returns), name='X') + 50
X.plot()
Now we generate Y. Remember that Y is supposed to have a deep economic link to X, so the price of Y should vary pretty similarly. We model this by taking X, shifting it up and adding some random noise drawn from a normal distribution.
some_noise = np.random.normal(0, 1, 100)
Y = X + 5 + some_noise
Y.name = 'Y'
pd.concat([X, Y], axis=1).plot()
Def: Cointegration¶
We've constructed an example of two cointegrated series. Cointegration is a "different" form of correlation (very loosely speaking). The ratio between two cointegrated timeseries will vary around a mean. The expected value of the ratio over time must converge to the mean for pairs trading work work. Another way to think about this is that cointegrated timeseries might not necessarily follow a similar path to a same destination, but they both end up at this destination on expectation.
For a more formal definition of cointegration, please see the Integration, Cointegration, and Stationarity Lecture from the Quantopian Lecture Series.
We'll plot the ratio between the two now so we can see how this looks.
(Y/X).plot() # Plot the ratio
plt.axhline((Y/X).mean(), color='red', linestyle='--') # Add the mean
plt.xlabel('Time')
plt.legend(['Price Ratio', 'Mean']);
Testing for Cointegration¶
That's an intuitive definition, but how do we test for this statisitcally? There is a convenient test that lives in statsmodels.tsa.stattools. We should see a very low p-value, as we've artifically created two series that are as cointegrated as physically possible.
# compute the p-value of the cointegration test
# will inform us as to whether the ratio between the 2 timeseries is stationary
# around its mean
score, pvalue, _ = coint(X,Y)
print pvalue
Correlation vs. Cointegration¶
Correlation and cointegration, while theoretically similar, are not the same. To demonstrate this, we'll show examples of series that are correlated, but not cointegrated, and vice versa. To start let's check the correlation of the series we just generated.
X.corr(Y)
That's very high, as we would expect. But how would two series that are correlated but not cointegrated look?
Correlation Without Cointegration¶
A simple example is two series that just diverge.
X_returns = np.random.normal(1, 1, 100)
Y_returns = np.random.normal(2, 1, 100)
X_diverging = pd.Series(np.cumsum(X_returns), name='X')
Y_diverging = pd.Series(np.cumsum(Y_returns), name='Y')
pd.concat([X_diverging, Y_diverging], axis=1).plot()
print 'Correlation: ' + str(X_diverging.corr(Y_diverging))
score, pvalue, _ = coint(X_diverging,Y_diverging)
print 'Cointegration test p-value: ' + str(pvalue)
Cointegration Without Correlation¶
A simple example of this case is a normally distributed series and a square wave.
Y2 = pd.Series(np.random.normal(0, 1, 1000), name='Y2') + 20
Y3 = Y2.copy()
# Y2 = Y2 + 10
Y3[0:100] = 30
Y3[100:200] = 10
Y3[200:300] = 30
Y3[300:400] = 10
Y3[400:500] = 30
Y3[500:600] = 10
Y3[600:700] = 30
Y3[700:800] = 10
Y3[800:900] = 30
Y3[900:1000] = 10
Y2.plot()
Y3.plot()
plt.ylim([0, 40])
# correlation is nearly zero
print 'Correlation: ' + str(Y2.corr(Y3))
score, pvalue, _ = coint(Y2,Y3)
print 'Cointegration test p-value: ' + str(pvalue)
Sure enough, the correlation is incredibly low, but the p-value shows perfect cointegration.
Def: Hedged Position¶
Because you'd like to protect yourself from bad markets, often times short sales will be used to hedge long investments. Because a short sale makes money if the security sold loses value, and a long purchase will make money if a security gains value, one can long parts of the market and short others. That way if the entire market falls off a cliff, we'll still make money on the shorted securities and hopefully break even. In the case of two securities we'll call it a hedged position when we are long on one security and short on the other.
The Trick: Where it all comes together¶
Because the securities drift towards and apart from each other, there will be times when the distance is high and times when the distance is low. The trick of pairs trading comes from maintaining a hedged position across X and Y. If both securities go down, we neither make nor lose money, and likewise if both go up. We make money on the ratio of the two reverting to the mean. In order to do this we'll watch for when X and Y are far apart, then short Y and long X. Similarly we'll watch for when they're close together, and long Y and short X.
Going Long the Ratio¶
This is when the ratio is small and we expect it to become larger. We place a bet on this by longing Y and shorting X.
Going Short the Ratio¶
This is when the ratio is small and we expect it to become larger. We place a bet on this by shorting Y and longing X.
Specific Bets¶
One important concept here is that we are placing a bet on one specific thing, and trying to reduce our bet's dependency on other factors such as the market.
Finding real securities that behave like this¶
The best way to do this is to start with securities you suspect may be cointegrated and perform a statistical test. If you just run statistical tests over all pairs, you'll fall prey to multiple comparison bias.
Here's a method I wrote to look through a list of securities and test for cointegration between all pairs. It returns a cointegration test score matrix, a p-value matrix, and any pairs for which the p-value was less than 0.05.
WARNING: This will incur a large amount of multiple comparisons bias.¶
For those not familiar with multiple comparisons bias, it is the increased chance to incorrectly generate a significant p-value when many tests are run. If 100 tests are run on random data, we should expect to see 5 p-values below 0.05 on expectation. Because we will perform $n(n-1)/2$ comparisons, we should expect to see many incorrectly significant p-values. For the sake of example will will ignore this and continue. In practice a second verification step would be needed if looking for pairs this way. Another approach is to pick a small number of pairs you have reason to suspect might be cointegrated and test each individually. This will result in less exposure to multiple comparisons bias.
def find_cointegrated_pairs(data):
n = data.shape[1]
score_matrix = np.zeros((n, n))
pvalue_matrix = np.ones((n, n))
keys = data.keys()
pairs = []
for i in range(n):
for j in range(i+1, n):
S1 = data[keys[i]]
S2 = data[keys[j]]
result = coint(S1, S2)
score = result[0]
pvalue = result[1]
score_matrix[i, j] = score
pvalue_matrix[i, j] = pvalue
if pvalue < 0.05:
pairs.append((keys[i], keys[j]))
return score_matrix, pvalue_matrix, pairs
Looking for Cointegrated Pairs of Alternative Energy Securities¶
I'm looking through a set of solar company stocks to see if any of them are cointegrated. We'll start by defining the list of securities we want to look through. Then we'll get the pricing data for each security for the year of 2014.
Note¶
We include the market in our data. This is because the market drives the movement of so many securities that you often times might find two seemingingly cointegrated securities, but in reality they are not cointegrated and just both conintegrated with the market. This is known as a confounding variable and it is important to check for market involvement in any relationship you find.
get_pricing() is a Quantopian method that pulls in stock data, and loads it into a Python Pandas DataPanel object. Available fields are 'price', 'open_price', 'high', 'low', 'volume'. But for this example we will just use 'price' which is the daily closing price of the stock.
symbol_list = ['ABGB', 'ASTI', 'CSUN', 'DQ', 'FSLR','SPY']
prices_df = get_pricing(symbol_list, fields=['price']
, start_date='2014-01-01', end_date='2015-01-01')['price']
prices_df.columns = map(lambda x: x.symbol, prices_df.columns)
Example of how to get all the prices of all the stocks loaded using get_pricing() above in one pandas dataframe object
prices_df.head()
Example of how to get just the prices of a single stock that was loaded using get_pricing() above
prices_df['SPY'].head()
Now we'll run our method on the list and see if any pairs are cointegrated.
# Heatmap to show the p-values of the cointegration test between each pair of
# stocks. Only show the value in the upper-diagonal of the heatmap
scores, pvalues, pairs = find_cointegrated_pairs(prices_df)
import seaborn
seaborn.heatmap(pvalues, xticklabels=symbol_list, yticklabels=symbol_list, cmap='RdYlGn_r'
, mask = (pvalues >= 0.95)
)
print pairs
Looks like 'ABGB' and 'FSLR' are cointegrated. Let's take a look at the prices to make sure there's nothing weird going on.
S1 = prices_df['ABGB']
S2 = prices_df['FSLR']
score, pvalue, _ = coint(S1, S2)
pvalue
We'll plot the ratio of the two series.
ratios = S1 / S2
ratios.plot()
plt.axhline(ratios.mean(), color='black')
plt.legend(['Price Ratio']);
The absolute ratio isn't very useful in statistical terms. It is more helpful to normalize our signal by treating it as a z-score.
WARNING¶
In practice this is usually done to try to give some scale to the data, but this assumes some underlying distribution. Usually normal. Under a normal distribution, we would know that approximately 84% of all ratio values will be smaller. However, much financial data is not normally distributed, and one must be very careful not to assume normality, nor any specific distribution when generating statistics. It could be the case that the true distribution of ratios was very fat-tailed and prone to extreme values. This could mess up our model and result in large losses.
def zscore(series):
return (series - series.mean()) / np.std(series)
zscore(ratios).plot()
plt.axhline(zscore(ratios).mean(), color='black')
plt.axhline(1.0, color='red', linestyle='--')
plt.axhline(-1.0, color='green', linestyle='--')
plt.legend(['Ratio z-score', 'Mean', '+1', '-1']);
Simple Strategy:¶
- Go "Long" the ratio whenever the z-score is below -1.0
- Go "Short" the ratio when the z-score is above 1.0
- Exit positions when the z-score approaches zero
This is just the tip of the iceberg, and only a very simplistic example to illustrate the concepts. In practice you would want to compute a more optimal weighting for how many shares to hold for S1 and S2. Some additional resources on pair trading are listed at the end of this notebook
Trading using constantly updating statistics¶
In general taking a statistic over your whole sample size can be bad. For example, if the market is moving up, and both securities with it, then your average price over the last 3 years may not be representative of today. For this reason traders often use statistics that rely on rolling windows of the most recent data.
Def: Moving Average¶
A moving average is just an average over the last $n$ datapoints for each given time. It will be undefined for the first $n$ datapoints in our series. Shorter moving averages will be more jumpy and less reliable, but respond to new information quickly. Longer moving averages will be smoother, but take more time to incorporate new information.
# Get the price ratio between the 2 stocks
ratios = S1 / S2
ratios.name = 'ratio'
# Get the 10 day moving average of the price ratio
ratios_mavg10 = pd.rolling_mean(ratios, window=10)
ratios_mavg10.name = 'ratio 10d mavg'
# Get the 60 day moving average
ratios_mavg60 = pd.rolling_mean(ratios, window=60)
ratios_mavg60.name = 'ratio 60d mavg'
plt.plot(ratios_mavg10.index, ratios_mavg10.values)
plt.plot(ratios_mavg60.index, ratios_mavg60.values)
plt.legend(['10 Day Ratio MAVG', '60 Day Ratio MAVG'])
plt.ylabel('Ratio');
We can use the moving averages to compute the z-score of the ratio at each given time. This will tell us how extreme the ratio is and whether it's a good idea to enter a position at this time. Let's take a look at the z-score now.
# Take a rolling 60 day standard deviation
std_60 = pd.rolling_std(ratios, window=60)
std_60.name = 'std 60d'
# Compute the z score for each day
zscore_60_10 = (ratios_mavg10 - ratios_mavg60)/std_60
zscore_60_10.name = 'z-score'
zscore_60_10.plot()
plt.axhline(0, color='black')
plt.axhline(1.0, color='red', linestyle='--')
plt.axhline(-1.0, color='green', linestyle='--')
plt.legend(['Rolling Ratio z-Score', 'Mean', '+1', '-1']);
The z-score doesn't mean much out of context, let's plot it next to the prices to get an idea of what it looks like. We'll take the negative of the z-score because the ratios were all negative and that's kinda confusing.
# Plot the prices scaled down along with the negative z-score
# just divide the stock prices by 10 to make viewing it on the plot easier
plt.plot(S1.index, S1.values/10)
plt.plot(S2.index, S2.values/10)
plt.plot(zscore_60_10.index, zscore_60_10.values)
plt.legend(['S1 Price / 10', 'S2 Price / 10', 'Price Ratio Rolling z-Score']);
This notebook contained some simple introductory approaches. In practice one should use more sophisticated statistics, some of which are listed here.¶
- Augmented-Dickey Fuller test
- Hurst exponent
- Half-life of mean reversion inferred from an Ornstein–Uhlenbeck process
- Kalman filters
(this is not an endorsement) But, a very good practical resource for learning more about pair trading is Dr. Ernie Chan's book: Algorithmic Trading: Winning Strategies and Their Rationale http://www.amazon.com/Algorithmic-Trading-Winning-Strategies-Rationale/dp/1118460146/