Exercises: Introduction to Pairs Trading - Answer Key¶
By Chris Fenaroli, Delaney Mackenzie, and Maxwell Margenot
Lecture Link¶
This exercise notebook refers to this lecture. Please use the lecture for explanations and sample code.
https://www.quantopian.com/lectures#Introduction-to-Pairs-Trading
Part of the Quantopian Lecture Series:
Notebook released under the Creative Commons Attribution 4.0 License.
import numpy as np
import pandas as pd
import statsmodels
import statsmodels.api as sm
from statsmodels.tsa.stattools import coint, adfuller
# just set the seed for the random number generator
np.random.seed(107)
import matplotlib.pyplot as plt
Helper Functions¶
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
Exercise 1: Testing Artificial Examples¶
We'll use some artificially generated series first as they are much cleaner and easier to work with. In general when learning or developing a new technique, use simulated data to provide a clean environment. Simulated data also allows you to control the level of noise and difficulty level for your model.
a. Cointegration Test I¶
Determine whether the following two artificial series $A$ and $B$ are cointegrated using the coint() function and a reasonable confidence level.
A_returns = np.random.normal(0, 1, 100)
A = pd.Series(np.cumsum(A_returns), name='X') + 50
some_noise = np.random.exponential(1, 100)
B = A - 7 + some_noise
#Your code goes here
## answer key ##
score, pvalue, _ = coint(A,B)
confidence_level = 0.05
if pvalue < confidence_level:
print ("A and B are cointegrated")
print pvalue
else:
print ("A and B are not cointegrated")
print pvalue
A.name = "A"
B.name = "B"
pd.concat([A, B], axis=1).plot();
b. Cointegration Test II¶
Determine whether the following two artificial series $C$ and $D$ are cointegrated using the coint() function and a reasonable confidence level.
C_returns = np.random.normal(1, 1, 100)
C = pd.Series(np.cumsum(C_returns), name='X') + 100
D_returns = np.random.normal(2, 1, 100)
D = pd.Series(np.cumsum(D_returns), name='X') + 100
#Your code goes here
## answer key ##
score, pvalue, _ = coint(C,D)
confidence_level = 0.05
if pvalue < confidence_level:
print ("C and D are cointegrated")
print pvalue
else:
print ("C and D are not cointegrated")
print pvalue
C.name = "C"
D.name = "D"
pd.concat([C, D], axis=1).plot();
ual = get_pricing('UAL', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
aal = get_pricing('AAL', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
#Your code goes here
## answer key ##
score, pvalue, _ = coint(ual, aal)
confidence_level = 0.05
if pvalue < confidence_level:
print ("UAL and AAL are cointegrated")
print pvalue
else:
print ("UAL and AAL are not cointegrated")
print pvalue
ual.name = "UAL"
aal.name = "AAL"
pd.concat([ual, aal], axis=1).plot();
b. Real Cointegration Test II¶
Determine whether the following two assets FCAU and HMC were cointegrated during 2015 using the coint() function and a reasonable confidence level.
fcau = get_pricing('FCAU', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
hmc = get_pricing('HMC', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
#Your code goes here
## answer key ##
confidence_level = 0.05
score, pvalue, _ = coint(fcau, hmc)
if pvalue < confidence_level:
print ("FCAU and HMC are cointegrated")
print pvalue
else:
print ("FCAU and HMC are not cointegrated")
print pvalue
fcau.name = "FCAU"
hmc.name = "HMC"
pd.concat([fcau, hmc], axis=1).plot();
Exercise 3: Searching for Cointegrated Pairs¶
Use the find_cointegrated_pairs function, defined in the "Helper Functions" section above, to find any cointegrated pairs among a set of metal and mining securities.
symbol_list = ['MTRN', 'CMP', 'TRQ', 'SCCO', 'HCLP','SPY']
prices_df = get_pricing(symbol_list, fields=['price']
, start_date='2015-01-01', end_date='2016-01-01')['price']
prices_df.columns = map(lambda x: x.symbol, prices_df.columns)
#Your code goes here
## answer key ##
scores, pvalues, pairs = find_cointegrated_pairs(prices_df)
import seaborn
seaborn.heatmap(pvalues, xticklabels=symbol_list, yticklabels=symbol_list, cmap='RdYlGn_r'
, mask = (pvalues >= 0.99)
)
print pairs
S1 = prices_df['MTRN']
S2 = prices_df['SCCO']
#Your code goes here
## answer key ##
S1 = sm.add_constant(S1)
results = sm.OLS(S2, S1).fit()
b = results.params['MTRN']
S1 = S1['MTRN']
print b
spread = S2 - b * S1
print "p-value for in-sample stationarity: ", adfuller(spread)[1]
# The p-value is less than 0.05 so we conclude that this spread calculation is stationary in sample
spread.plot()
plt.axhline(spread.mean(), color='black')
plt.legend(['Spread']);
b. Testing the Coefficient¶
Use your coefficient from part a to plot the weighted spread using prices from the first half of 2016, and check whether the result is still stationary.
S1_out = get_pricing('MTRN', fields=['price'],
start_date='2016-01-01', end_date='2016-07-01')['price']
S2_out = get_pricing('SCCO', fields=['price'],
start_date='2016-01-01', end_date='2016-07-01')['price']
#Your code goes here
## answer key ##
spread = S2_out - b * S1_out
spread.plot()
plt.axhline(spread.mean(), color='black')
plt.legend(['Spread']);
print "p-value for spread stationarity: ", adfuller(spread)[1]
# Our p-value is greater than 0.05 so we conclude that this calculation of
# the spread is non-stationary out of sample
Extra Credit Exercise: Hurst Exponent¶
This exercise is more difficult and we will not provide initial structure.
The Hurst exponent is a statistic between 0 and 1 that provides information about how much a time series is trending or mean reverting. We want our spread time series to be mean reverting, so we can use the Hurst exponent to monitor whether our pair is going out of cointegration. Effectively as a means of process control to know when our pair is no longer good to trade.
Please find either an existing Python library that computes, or compute yourself, the Hurst exponent. Then plot it over time for the spread on the above pair of stocks.
These links may be helpful:
# No solution provided for extra credit exercises.
This presentation is for informational purposes only and does not constitute an offer to sell, a solic itation to buy, or a recommendation for any security; nor does it constitute an offer to provide investment advisory or other services by Quantopian, Inc. ("Quantopian"). Nothing contained herein constitutes investment advice or offers any opinion with respect to the suitability of any security, and any views expressed herein should not be taken as advice to buy, sell, or hold any security or as an endorsement of any security or company. In preparing the information contained herein, Quantopian, Inc. has not taken into account the investment needs, objectives, and financial circumstances of any particular investor. Any views expressed and data illustrated herein were prepared based upon information, believed to be reliable, available to Quantopian, Inc. at the time of publication. Quantopian makes no guarantees as to their accuracy or completeness. All information is subject to change and may quickly become unreliable for various reasons, including changes in market conditions or economic circumstances.