Exercises: Mean Reversion on Futures - Answer Key

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

IMPORTANT NOTE:

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:


Key Concepts


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

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.


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();


A and B are cointegrated
6.96867595624e-15

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.


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();


C and D are not cointegrated
0.487261538359

Exercise 2: Testing Real Examples

a. Real Cointegration Test I

Determine whether the following two assets CN and SB were cointegrated during 2015 using the coint() function and a reasonable confidence level.


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();


CN and SB are not cointegrated
0.605183628116

b. Real Cointegration Test II

Determine whether the following two underlyings CL and HO were cointegrated during 2015 using the coint() function and a reasonable confidence level.


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();


CL and HO are cointegrated
0.0470273399855

Exercise 3: Out of Sample Validation

a. Calculating the Spread

Using pricing data from 2015, construct a linear regression to find a coefficient for the linear combination of CL and HO that makes their spread stationary.


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']);


30.3502596624
p-value for in-sample stationarity:  0.0124116865765

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.


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


p-value for spread stationarity:  0.00434011120184

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:


In [14]:
# No solution provided for extra credit exercises.

Congratulations on completing the Mean Reversion on Futures exercises!

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