Pairs Trading

MGMT 638: Data-Driven Investments: Equity

Kerry Back, Rice University

Overview

• Find stock pairs that usually track together
• When the relationship is broken:
  • Buy the stock that is cheap compared to the usual relationship
  • Short sell the stock that is expensive compared to the usual relationship
• Hope the usual relationship is restored soon.

Model

• $P_1/P_2 \approx \text{constant}$
• When the ratio goes above the constant, it tends to come down.
• When the ratio goes below the constant, it tends to go up.

$$ \Delta P_1/P_2 = \begin{cases} & + \;\text{when $P_1/P_2 <$ constant}\\ & - \;\text{when $P_1/P_2 >$ constant} \end{cases} $$

-Assume the change is larger when the ratio is further from the constant as $$\Delta P_1/P_2 = k(\text{constant} - P_1/P_2)$$ for a constant $k>0$.

• The model is equivalent to

$$\Delta P_1/P_2 = a + bP_1/P_2$$

where $a=k \times \text{constant}$, $b=-k$.

• Estimate $a$ and $b$ by linear regression.
• Should get $a>0$, $b<0$.
• If so, $\text{constant} =-a/b$.
• Hold asset 1 and short 2 when $P_1/P_2 < -a/b - \text{threshold}$.
• Hold asset 2 and short 1 when $P_1/P_2 > -a/b + \text{threshold}$.

Example

• Chevron (CVX) and Conoco-Phillips (COP) from 2000 on
• Adjusted closing prices from Yahoo Finance
• Compute the price ratio: CVX / COP

data.ratio.plot(label="CVX/COP")
plt.hlines(
    y=-a/b, 
    xmin = data.index[0], 
    xmax=data.index[-1],
    color="red",
    label="constant"
)
plt.legend(loc="lower right")
plt.show()

Returns

• $-a/b = 1.66$
• Set threshold = 0.2 as an example
• Buy COP and short CVX when CVX / COP is above $1.86$
• Buy CVX and short COP when CVX / COP is below $1.46$

Market Neutrality

• The pairs strategy is an example of a market neutral strategy, meaning its market beta should be approximately zero.
• If it has a return above the risk-free rate, then adding some of it to the market portfolio can improve performance relative to holding the market.
• This is the same as saying that the strategy has a positive alpha.
• It is also the same as saying

$$ \text{Sharpe ratio of strategy} > \text{Sharpe ratio of market} \times \text{correlation with market}$$

• Get the market return from Ken French's data library.

print(f"mean return of pairs strategy = {252*data.ret.mean():.2%} annualized")
print(f"correlation of pairs strategy with market = {data.ret.corr(data.mkt):.2%}")

mean return of pairs strategy = 5.19% annualized
correlation of pairs strategy with market = 4.94%

Avoid Look-Ahead Bias

• Compute the parameter of the strategy (the constant $-a/b$) from data through 2015
• Test the strategy from 2015 on.

print(f"mean return of pairs strategy = {252*future.ret.mean():.2%} annualized")
print(f"correlation of pairs strategy with market = {future.ret.corr(future.mkt):.2%}")

mean return of pairs strategy = 5.48% annualized
correlation of pairs strategy with market = 10.74%

Alpha and Beta

• beta = corr with market excess return x std dev of strategy / std dev of market
• alpha = mean return - beta * mean market excess return

beta = future.ret.corr(future.mkt) * future.ret.std() / future.mkt.std()
alpha = future.ret.mean() - beta * future.mkt.mean()

print(f"beta is {beta:.4f}")
print(f"annualized alpha is {252*alpha:.2%}")

beta is 0.0011
annualized alpha is 4.19%

Regressions in python

• use statsmodels.formula.api
• smf.ols("model", data).fit().summary()

import statsmodels.formula.api as smf
smf.ols("ret~mkt", future).fit().summary()

OLS Regression Results

Dep. Variable:	ret	R-squared:	0.012
Model:	OLS	Adj. R-squared:	0.011
Method:	Least Squares	F-statistic:	25.15
Date:	Sat, 28 Oct 2023	Prob (F-statistic):	5.75e-07
Time:	11:42:18	Log-Likelihood:	6431.3
No. Observations:	2157	AIC:	-1.286e+04
Df Residuals:	2155	BIC:	-1.285e+04
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	0.0002	0.000	0.758	0.448	-0.000	0.001
mkt	0.0011	0.000	5.015	0.000	0.001	0.002

Omnibus:	438.977	Durbin-Watson:	1.951
Prob(Omnibus):	0.000	Jarque-Bera (JB):	11512.640
Skew:	0.267	Prob(JB):	0.00
Kurtosis:	14.305	Cond. No.	1.20

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.