**number of shares**of stocks A and B fixed, in the ratio hA:hB, and short this spread when it is much higher than average, and long this spread when it is much lower. On the other hand, for a stationary log price spread hA*log(yA)-hB*log(yB), we need to keep the

**market values**of stocks A and B fixed, in the ratio hA:hB, which means that at the end of every bar, we need to rebalance the shares of A and B due to price changes.

For most cointegrating pairs that I have studied, both the price spreads and the log price spreads are stationary, so it doesn't matter which one we use for our trading strategy. However, for an unusual pair where its log price spread cointegrates but price spread does not (Hat tip: Adam G. for drawing my attention to one such example), the implication is quite significant. A stationary price spread means that prices differences are mean-reverting, a stationary log price spread means that returns differences are mean-reverting. For example, if stock A typically grows 2 times as fast as B, but has been growing 2.5 times as fast recently, we can expect the growth rate differential to decrease going forward. We would still short A and long B, but we would exit this position when the growth rates of A vs B return to a 2:1 ratio, and not when the price spread of A vs B returns to a historical mean. In fact, the price spread of A vs B should continue to increase over the long term.

This much is easy to understand. But thanks to a reader Ferenc F. who referred me to a paper by Fernholz and Maguire, I realize there is a simple mathematical relationship between stock A and B in order for their log prices to cointegrate.

Let us start with a formula derived by these authors for the change in log market value P of a portfolio of 2 stocks: d(logP) = hA*d(log(yA))+hB*d(log(yB))+gamma*dt.

The gamma in this equation is

gamma=1/2*(hA*varA + hB*varB), where varA is the variance of stock A

__minus__the variance of the portfolio market value, and ditto for varB.

Note that this formula holds for a portfolio of any two stocks, not just when they are cointegrating. But if they are in fact cointegrating, and if hA and hB are the weights which create the stationary portfolio P, we know that d(logP) cannot have a non-zero long term drift term represented by gamma*dt. So gamma must be zero. Now in order for gamma to be zero, the

**covariance**of the two stocks must be positive (no surprise here) and equal to the

**average of the variances**of the two stocks. I invite the reader to verify this conclusion by expressing the variance of the portfolio market value in terms of the variances of the individual stocks and their covariance, and also to extend it to a portfolio with N stocks. This cointegration test for log prices is certainly simpler than the usual CADF or Johansen tests! (The price to pay for this simplicity? We must assume normal distributions of returns.)

===

My online Quantitative Momentum Strategies workshop will be offered on December 2-4. Please visit epchan.com/my-workshops for registration details.

## 47 comments:

Ernie,

You wrote,

"For example, if stock A typically grows 2 times as fast as B, but has been growing 2.5 times as fast recently, we can expect the growth rate differential to decrease going forward. We would still short A and long B, but we would exit this position when the growth rates of A vs B return to a 2:1 ratio, and not when the price spread of A vs B returns to a historical mean. In fact, the price spread of A vs B should continue to increase over the long term."

I believe this explanation is incorrect (no offense!).

Let's assume A and B have fixed long-term growth rates, but that each has an instantaneous growth rate that fluctates randomly around the mean. If stock A typically grows twice as fast as stock B, then the log(A) price series will grow, on average, at twice the linear rate as the log(B) price series. So, viewed in log space, A and B will both tend to rise linearly with some fluctuations around the best fit line, but log(A)'s best fit line will have twice the slope as log(B)'s. Therefore, log(A) will diverge from log(B) over time, and therefore they cannot be cointegrated.

In order for log(A) and log(B) to be cointegrated, A and B must have the same long-term growth rate. Consider the example of two classes of common stock for a single company that trade on different exchanges in different countries. After accounting for forex effects these two stocks must grow at the same rate since they fundamentally represent the same company. Therefore their log-price series will grow at the same rate and their log-prices will be cointegrated. But over a very long period of time both price series should grow exponentially, so their raw price series will diverge because even a small percentage difference between the two will correspond to a large absolute difference compared to their inital values.

- aagold (Adam G.)

Adam,

For 2 stocks with growth rates in the ratio of 2:1, we merely have to keep the ratio of their market values to be 1:2 so that their positions will have the same long-term growth rate.

As I wrote, if log prices are cointegrating, we need to constantly rebalance these positions so that their market values are always in 1:2 ratio.

Ernie

Ernie,

Let's separate the discussion of a trading strategy from the discussion of cointegration definition. Let's see if we can agree on the following definitions.

1) If stocks A and B are cointegrated in raw price space with hedge ratio h, then the difference A - h*B will fluctuate randomly around 0 with no drift.

2) If stocks A and B are cointegrated in log price space with hedge ratio h, then the difference log(A) - log(h*B) will fluctuate randomly around 0 with no drift.

My claim is that any two stocks A & B which satisfy definition #2 must have the same long-term growth rate. Consider this example: A=exp(alpha0*t) and B=exp(alpha1*t). The difference in their logs is (alpha0 - alpha1)*t, which has no drift only when alpha0 = alpha1 (i.e., same growth rate).

- Adam

Adam,

I believe your definition of cointegration of log prices is incorrect.

The spread in this case is defined as log(A)-h*log(B), not log(A)-log(h*B) as you wrote.

Ernie

Ernie,

Ok, I see your point. Defined this way, stocks with different growth rates can be cointegrated in log space. The hedge ratio h compensates for the different growth rates.

However, I think the most interesting real life examples where log prices are cointegrated, but raw prices are not, occur when h=1 (i.e., same growth rate). At least that's the case for any real-world examples I can think of.

Regards,

Adam

Ernie,

Sorry if I'm beating this to death, but in the example you cited with A and B the hedge ratio would be 2. So the log spread would be log(A) - 2*log(B).

You wrote we would exit this position when the growth rate of A reverts to 2 times the growth rate of B, but I think this is incorrect. We should exit the position when the *ratio* A/B^2 reverts to its historical mean (which is equivalent to the log spread returning to its historical mean).

The analagous statement for raw prices is, we would exit the position when the *difference* A - 2*B reverts to its historical mean.

- Adam

Adam,

You are right that I was being imprecise when I said entry/exit signals should be based on differential "growth rates". By growth rates, I don't mean the instantaneous growth rates d(log(P))/dt, but the average growth rate log(P)/t where t is the time since some distant past at the beginning of our backtest period. Since t is the same for both stocks, difference in average growth rates are essentially the same as the difference in log prices.

Ernie

Ernie,

ON the topic of Kelly leverage. I'm following the example in your book (Quantitative Trading pp. 99) and for SPY I calculate a leverage of 21.08 usinfg the last 252 returns. Is that also what you get?

Anon,

Yes, that's about what I get too.

Ernie

Anon,

I certainly hope you're not planning on investing on SPY with a leverage factor of 21.08! You know that would be absolutely insane, right? Even half-Kelly at 10.5 would be insane.

It might be ok to estimate future variance using the past 252 daily returns, but it's certainly not correct to estimate future expected returns that way.

I use a half-Kelly model to determine how much stock market exposure I should have with my real-life portfolio, and right now it's saying I should be 70% in the US stock market and 30% in cash. My estimate of the market's future daily mean return is 6.05% per year, annualized standard deviation is 15.4%, and risk-free interest rate is 2.71%.

- Adan

Hi Ernie,

It seems Yahoo Finance starts to provide real-time tick data (no delay).

Do you hear any good news or comments about that?

Hi Anon,

Indeed Yahoo Finance now offers real-time data, but only from Nasdaq. So if a stock like IBM is primarily traded on NYSE, the Nasdaq price may be slightly different.

Ernie

Hi Ernie,

Is that ok to use IB real-time data stream to do pairs trading in US markets?

Hi Ernie

After reading your books, I find that a pair of futures contracts traded on Shnaghai Futures exchange that is great to extract roll return form. the question is if one is in backwardation and the other is in contango, how do you determine the hedge ratio between them. Are we suppose to belance their spot return fluctuation? Right now I optimize the backtesting sharp ratio,to determine the ratio, any advice?

Ruan Xun

Hi Anon,

I find the real-time feed of IB too noisy for pair trading stocks. I recommend IQFeed or even Yahoo Realtime instead.

Ernie

Hi Ruan,

You can use linear regression on their prices (or log prices) to determine the optimal hedge ratio.

You will always be long the one in backwardation, and short the one in contango if you want to extract roll return.

Ernie

Hi Ernie

So should I use continuious futures prices of these 2 contracts,or should I use the 2 underlying spot prices instead?

Hi Ruan,

You can use linear regression on their prices (or log prices) to determine the optimal hedge ratio.

You will always be long the one in backwardation, and short the one in contango if you want to extract roll return.

Ernie

Ruan,

You should not use continuous futures, nor spot prices.

You should be using individual futures contracts to test for roll returns.

Ernie

Ernie,

I tested the price spread you talked about in your book A - h*B for stationarity and I find that for a lot of pairs , the stationarity keeps fluctuating from true to false , and vice versa if I retest the pair for stationarity every month (verying lookback windows). Even a stable pair like GLD-GDX that you mentioned isn't very stationary month to month. How does one interpret this result ?

Anon,

Stationarity tests should involve at least 1 year of daily data. How are saying that even with 1 year of lookback, the test statistic changes greatly from month-to-month?

Ernie

Yes. Using a 1 year lookback & testing for stationarity after every month, I notice that stationary flips from True to False and vice versa on majority of the pairs.

Is this something you notice too ?

Anon,

Usually stationarity test is more stable than that. Perhaps increasing your lookback to 3 years would help. If not, it simply indicates those pairs are not really stationary.

Ernie

Hi Ernie

do you use fundamental data such as PE and ROE etc? Any good service/api recommendation? thanks

Paul

Hi Paul,

I don't currently use fundamental data. But I believe you can scrape the Yahoo Finance website for such info. Also, IB's API also provides such data.

Ernie

Have you tried Quantum Mechanics trading?!

http://arxiv.org/abs/1307.6727

Hi Anon,

Thanks for the article. I find this article lacking in empirical support.

Ernie

Hi Erine

I just read the paper of

"Optimal Pairs Trading: A Stochastic Control Approach"

http://www.nt.ntnu.no/users/skoge/prost/proceedings/acc08/data/papers/0479.pdf

As I m not familiar of Ornstein–Uhlenbeck process and its application on pairs trading, so I would like to seek your opinion,

Isn't it true that the OU process can model the spread and the mean reverting behaviour in continuous time and dynamic way but the cointegration approach cannot ,but the weakness of the OU process is it does not tell us what is the weightage of each stock in a pair. Thus,We have to use the stochastic control approach to get this weightage, but we will have to set a final period T when we close the position. Whereas for the cointegration approach, it explicitly shows the weightage of each stock in a pair.

Hi Anon,

Indeed the OU process does not tell you the optimal hedge ratio. It is a model of one mean-reverting time series, not the cointegration of several series. The only use for an OU model for me is to extract the halflife of mean reversion from the regression coefficient.

Ernie

Hi Erine

When you do simulation or backtesting for the pairs, you use tick data or just daily High, Low, Open and Close Price

Thx

James

Hi James,

That depends on whether the mean reverting strategy is daily, or intraday. In either case, you need bid-ask quotes: trade prices will inflate results.

Ernie

Hi Ernie,

I do not quite understand the concept of cointegration trading with log prices.

Assuming we have a cointegration relationship

log(A) - 0.5 log(B) = 0

If log(A) - 0.5 log(B) > 0, we will short A and long B.

I do not understand why we have to short A. Our assumption here is the growth rate of A may decrease, and this does not mean that the price of A will drop. Hence, we will make a loss if we short A, as the growth rate of A is still increasing.

In pair trading, whether using raw prices or log prices, we should expect one side to lose while the other side profit.

In your example, hopefully the gain in B will be more than enough to offset the loss in A.

Ernie

Dr Ernie,

For stationary test on USDCAD, you do a logarithmn on it. I read that if we use prices like USDCAD, we should use the difference and hence use a random walk model. If we use returns, we should log it and an exponential random walk model is used. Does it matter if we log or dont log it?

Thank you

Leo

Dr Ernie,

I am trying to see the variance ratio test on your data of USDCAD (1min) for different time frame. If I want to look at 60minute time-frame, how do I choose the number of periods for the numerator and denominator based on the below formula? Is it sampling at 60 points for the top and 1 point for the bottom?

VR(k) =Variance(rk t)/k Variance(rt)

Thank you

Leo

Hi Leo,

No, you should just input the 60-min bars into the vratiotest function. This is a hypothesis test, so you can't just compute the ratio and see if it is 1.

Ernie

Dr Ernie,

This is because we can see the variance ratio <1 , =1, >1 and so know the "state" if we use the formula. But if we use vratio from matlab, it can only as you said reject RW with a probability. No way to form the equation with that 1min data for 60min time frame?

Thank you

Leo

maybe I'm missing something in this discussion, but for me a pair is cointegrated if there is a linear combo that is stationary. Stationarity is not the same as driftless. driftless can be non stationary and stationary processes can have drift, i.e., that which is implied by the OU process.

technically, if log(X) and log(Y) cointegrated, one gets a fairly nonlinear expression for the innovation of prices, owing to the fact that aLog(X)-bLog(Y)=Log(X^a/Y^b), then do OU process and express innovation of X in terms of Y. I get that it isn't as simple as what Ernie says in his opening comments.

Marc,

You are right that stationary processes can have polynomial dependence on t. We are only concerned with whether the residual is a bounded function in time.

And you are also right that a driftless process, such as a random walk, is not stationary, since the variance can increase without bound.

When we test for cointegration of log prices, what we are really testing for is whether the returns per period have a linear relationship to each other.

Ernie

Oh, yes, interesting. thanks Ernie Logs is looking nice for a number of reasons now....

Hi Leo,

The variance ratio is a t-statistic. You need critical values of the t-statistic to test if it is significant. vratiotest provides that (hypothesis) test.

Ernie

Dr Ernie,

In your book, you need holding period and look back period to check for trending. Any statistics to test for trending using lookback period? like the way variance ratio test?

Thank you very much

Leo

Hi Ernie,

I have read the post in detail and find it very interesting.

I just wanted to clarify something... can you please define exactly what you mean by "growth rate" of a stock. And also "growth rate differential".

Thanks.

We assume that stock prices grow exponentially. Thus the growth rate is the exponent. This is also ordinarily known as the CAGR (compounded annual growth rate) in financial literature.

Growth rate differentials is the difference of this exponent for 2 different stocks.

Ernie

Hello Ernie, I'm building a mean reversion algorithm based on H1 data and depends which dates I use to check the cointegration I have good or bad results. So, in live trading I have to check each new candle if the pairs are still cointegrated or not? I'm a little bit confused with this topic because maybe you get 1000 candles and the pairs is cointegrated but then I check 1500 and it is not.

I think that in GLD-GDX case it happens something like this, that the cointegration was broken and then continue.

Another question is that if I have to recalculate my parameters each candle or maybe it is better once a day/week?

Thanks.

Greetings

Hi Carlos,

Typically, cointegration need to be determined only using daily data. It doesn't help to use intraday data, unless you want to liquidate your positions at the close each day.

Even if you were to use daily data, there is no need to update your cointegration test daily. Monthly update is good enough. You can run that on the most recent 3 years of data.

Ernie

Hi Ernie,

In Pairs Trading by Vidyamurthy on page 83, the author describes an elementary example of trading with log prices. However, he seems to use the cointegration coefficient to indicate the ratio of

sharesto hold rather than to indicate the relative market value of positions (as you state above). As I'm sure you have read this book, can you reconcile this discrepancy?From the book, with a cointegration coefficient of 1.5, he states "At time t, buy shares of A and short shares of B in the ratio 1:1.5" on page 83. Would very much appreciate your input.

Best,

Flapjacks

Hi Flapjacks,

I have not read that book. But based on your description, I can only say that I disagree with his interpretation. For a mathematical justification of my interpretation, please read p. 65 of my book Algorithmic Trading.

Ernie

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