By Chris Fenaroli, Delaney Mackenzie, and Maxwell Margenot
https://www.quantopian.com/lectures/introduction-to-pairs-trading
https://www.quantopian.com/lectures/mean-reversion-on-futures
This lecture corresponds to the Mean Reversion on Futures lecture, which is part of the Quantopian lecture series. This homework expects you to rely heavily on the code presented in the corresponding lecture. Please copy and paste regularly from that lecture when starting to work on the problems, as trying to do them from scratch will likely be too difficult.
Part of the Quantopian Lecture Series:
In [1]:
# Useful 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
In [2]:
# Useful Libraries
import numpy as np
import pandas as pd
import statsmodels
import statsmodels.api as sm
from statsmodels.tsa.stattools import coint, adfuller
from quantopian.research.experimental import history, continuous_future
# just set the seed for the random number generator
np.random.seed(107)
import matplotlib.pyplot as plt
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.
Determine whether the following two artificial series $A$ and $B$ are cointegrated using the coint() function and a reasonable confidence level.
In [3]:
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
In [4]:
## 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();
In [5]:
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
In [6]:
## 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();
In [7]:
cn = continuous_future('CN', offset = 0, roll = 'calendar', adjustment = 'mul')
sb = continuous_future('SB', offset = 0, roll = 'calendar', adjustment = 'mul')
cn_price = history(cn, 'price', '2015-01-01', '2016-01-01', 'daily')
sb_price = history(sb, 'price', '2015-01-01', '2016-01-01', 'daily')
#Your code goes here
#print history.__doc__
In [8]:
## answer key ##
score, pvalue, _ = coint(cn_price, sb_price)
confidence_level = 0.05
if pvalue < confidence_level:
print ("CN and SB are cointegrated")
print pvalue
else:
print ("CN and SB are not cointegrated")
print pvalue
cn_price.name = "CN"
sb_price.name = "SB"
pd.concat([cn_price, sb_price], axis=1).plot();
In [9]:
cl = continuous_future('CL', offset = 0, roll = 'calendar', adjustment = 'mul')
ho = continuous_future('HO', offset = 0, roll = 'calendar', adjustment = 'mul')
cl_price = history(cl, 'price', '2015-01-01', '2016-01-01', 'daily')
ho_price = history(ho, 'price', '2015-01-01', '2016-01-01', 'daily')
#Your code goes here
In [10]:
## answer key ##
confidence_level = 0.05
score, pvalue, _ = coint(cl_price, ho_price)
if pvalue < confidence_level:
print ("CL and HO are cointegrated")
print pvalue
else:
print ("CL and HO are not cointegrated")
print pvalue
cl_price.name = 'CL'
ho_price.name = 'HO'
pd.concat([cl_price, ho_price.multiply(42)], axis=1).plot();
In [11]:
## answer key ##
results = sm.OLS(cl_price, sm.add_constant(ho_price)).fit()
b = results.params['HO']
print b
spread = cl_price - b * ho_price
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']);
In [12]:
cl_out = get_pricing(cl, fields='price',
start_date='2016-01-01', end_date='2016-07-01')
ho_out = get_pricing(ho, fields='price',
start_date='2016-01-01', end_date='2016-07-01')
#Your code goes here
In [13]:
## answer key ##
spread = cl_out - b * ho_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 less than 0.05 so we conclude that this calculation of
# the spread is stationary out of sample
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:
In [14]:
# No solution provided for extra credit exercises.
Congratulations on completing the Mean Reversion on Futures exercises!
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