A reader recently asked me whether setting a stop loss for a trading strategy is a good idea. I am a big fan of setting stop loss, but there are certainly myriad views on this.

One of my former bosses didn't believe in stop loss: his argument is that the market does not care about your personal entry price, so your stop price may be somebody else’s entry point. So stop loss, to him, is irrational. Since he is running a portfolio with hundreds of positions, he doesn’t regard preserving capital in just one or a few specific positions to be important. Of course, if you are an individual trader with fewer than a hundred positions, preservation of capital becomes a lot more important, and so does stop loss.

Even if you are highly diversified and preservation of capital in specific positions is not important, are there situations where stop loss is rational? I certainly think that applies to trend-following strategies. Whenever you incur a big loss when you have a trend-following position, it ususally means that the latest entry signal is opposite to your original entry signal. In this case, better admit your mistake, close your position, and maybe even enter into the opposite side. (Sometimes I wish our politicians think this way.) On the other hand, if you employ a mean-reverting strategy, and instead of reverting, the market sticks to its original direction and causes you to lose money, does it mean you are wrong? Not necessarily: you could simply be too early. Indeed, many traders in this case will double up their position, since the latest entry signal in this case is in the same direction as the original one. This raises a question though: if incurring a big loss is not a good enough reason to surrender to the market, how would you ever decide if your mean-reverting model is wrong? Here I propose a stop loss criterion that looks at another dimension: time.

The simplest model one can apply to a mean-reverting process is the Ornstein-Uhlenbeck formula. As a concrete example, I will apply this model to the commodity ETF spreads I discussed before that I believe are mean-reverting (XLE-CL, GDX-GLD, EEM-IGE, and EWC-IGE). It is a simple model that says the next change in the spread is opposite in sign to the deviation of the spread from its long-term mean, with a magnitude that is proportional to the deviation. In our case, this proportionality constant θ can be estimated from a linear regression of the daily change of the spread versus the spread itself. Most importantly for us, if we solve this equation, we will find that the deviation from the mean exhibits an exponential decay towards zero, with the half-life of the decay equals ln(2)/θ. This half-life is an important number: it gives us an estimate of how long we should expect the spread to remain far from zero. If we enter into a mean-reverting position, and 3 or 4 half-life’s later the spread still has not reverted to zero, we have reason to believe that maybe the regime has changed, and our mean-reverting model may not be valid anymore (or at least, the spread may have acquired a new long-term mean.)

Let’s now apply this formula to our spreads and see what their half-life’s are. Fitting the daily change in spreads to the spread itself gives us:

These numbers do confirm my experience that the GDX-GLD spread is the best one for traders, as it reverts the fastest, while the XLE-CL spread is the most trying. If we arbitrarily decide that we will exit a spread once we have held it for 3 times the half-life, we have to hold the XLE-CL spread almost a calendar year before giving up. (Note that the half-life count only trading days.) And indeed, while I have entered and exited (profitably) the GDX-GLD spread several times since last summer, I am holding the XLE - QM (substituting QM for CL) spread for the 104^{th} day!

## 44 comments:

Wondering if you could break down how the half life is calculated...?

Hi,

See the Ornstein-Uhlenbeck formula at http://en.wikipedia.org/wiki/Ornstein-Uhlenbeck_process

If you believe the spread is mean-reverting, this formula will describe its time-evolution. You will notice that the time-evolution is governed by an exponential decay -- hence the notion of half-life.

Ernie

Good article. One question: How do you use the estimated "theta" to trade? What is your trading strategy after the estimation? Do you still apply threshold rule to trade? Thanks

Dear Volat,

The estimated half-life can be used to determine your maximum holding period. Of course, you can exit earlier if the spread exceeds your profit-cap.

Ernie

I assume half-life means 1/2 of the time for the spread to revert to its mean. Therefore a half-life a 16 days means that it takes 32 days for the spread to revert to its mean (correct me if I am wrong). And with this number, you basically open the position at day 0 and close the position at day 32, then open the position at day 64, and then close the position at day 96...Is that right?

Volat: Yes, that's right. But as I said, you can exit early due to profit cap.

Ernie

Hi - first I want to say thank you for publishing your OT book - excellent writing.

In this article (as well as in the book), you said "...linear regression of the daily change of the spread versus

the spread itself"...by looking the the formula and Matlab example, should this be a more accurate sentance:"...linear regression of the daily change of the spread vesus

the spread's deviation from mean"?

Hi Anonymous,

Thanks for your compliments. Actually, whether you subtract the mean of the spread or not will yield the same regression coefficient.

Ernie

So when you run the regression, which regression coefficient is the theta?

If you regress the change in spread against the spread itself, the resulting regression coefficient is theta.

Ernie

Ah, makes a lot of sense. Of course. I also looked over the code again and saw at the end that you tell the computer that OLS beta = theta. Thanks.

Do you use the adjusted cointegration coefficient as the hedge ratio or the normalized. And if you do a cointegration test on the actual instruments and you get little chance of no conintegration coefficients and an over 55% chance of at most one; can you use that, or is it always better to run the test on first differences where there is always a high probability of two coefficients (using eviews output)?

Once I find cointegration is confirmed, I actually performed my own regression to find the hedge ratio.

Ok, that makes sense. What do you think about playing with the hedge ratios with the upper and lower bound being the Beta from a regression and Beta from a cointegration regression and seeing which number in between gives you the most stationary series?

By the way, thank you for being a fantastic resource and answering questions. I find that unless you are a math major (which I am not) some of the statistical arbitrage literature is impossible to get through. Once properly explained it seems fairly simple.

That is not a bad idea. In reality, however, I am not too concerned about the precise value of the hedge ratio. The optimal hedge ratio going forward is likely to differ from the optimal in a backtest period.

Ernie

Pardon my ignorance but I was under the impression that Brownian Motion (dWt) and Ornstein Uhlenbeck were both modeling processes to simulate expectations...not to analyze past data. If I am wrong please correct me and explain: a) the length of period to use for the mean and s.d. b) how should I calculate dWt using past data (the formulas I see use a random function to generate data points to use...should I simply replace the randomly generated points with the actual historicals?)

I guess my real confusion goes back to my assumption that BM and O-U are forward modeling tools...

Also, once you have calculated an half-life, how should it be applied? Should my calculations produce a new (smaller) half life everyday as the price reverts to the mean? Or will my calculations produce a static (somewhat static) number for the half life and I must then figure out where in the mean reversion process to start counting from?

William,

If you assume your price process follows a mean-reverting random walk, then you can fit the parameters of the O-U equation based on historical data.

The fitting procedure is a simple regression fit, described in details in my book, as well as explained in the previous comments on this blog post.

Half-life should be fairly constant over time. It should not decrease as the prices mean-revert. You can use any reasonable number of time periods to calculate the half-life. Typically 3 months to a year are suitable.

Hope I answered your question?

Ernie

Perhaps, it isn't necessarily the case that the shortest half life leads to the best trade?

Would a metric like the following:

sdev(spread)/half_life(spread) not be a useful method of ranking spreads? Here, sdev() would need to be expressed as a % of the mean. Alternatively, the percent excursion from a moving average might be used, since that metric is representative of the expected gain?

Hi geegaw,

Thanks for your suggestion. It is an interesting idea. However, the ultimate performance measure for trading a pair is the Sharpe ratio, regardless of holding period.

Ernie

Ernie, how do you calculate Z score for the Ornstein Uhlenbeck process? I could not find it in your book. Thanks

Hi Anonymous,

Ornstein-Uhlenbeck process does not generate zscores. It is used to calculate half-life of mean-reversion. Zscore is simply the value of a spread divided by its standard deviation.

Ernie

Hi Dr Chan,

Thanks for this nice article.

I was initially thinking about fitting AR(1) to the spread and then calculate the half life.

But your method seems more robust.

I have a question about the result you have posted on this article.

What time frame do you use for the GLD-GDX pairs half life calculation?

Also,when I calculate with the data(GLD1,GDX1) posted on your premium content,I get -ln(2)/theta equal to 8.

Thanks,

Vishal

Hi Vishal,

I don't recall what time frame I used for this half-life calculation. But in my book, the same example uses 20060523-20071221, and the half-life obtained is 10 days.

Ernie

Hi Vishal,

I don't recall what time frame I used for this half-life calculation. But in my book, the same example uses 20060523-20071221, and the half-life obtained is 10 days.

Ernie

Hey Ernie,

I am trying to plug in the numbers in the ornstein formula but I can't seem to get a number. Also the number varies wildly. I tried regressing the changes in the spread against the spread itself but it makes no sense the results. Is there a place where the formula is applied so I can check on how to use it?

Thanks,

J

Hi J,

You can look up Example 7.5 in my book on halflife calculation.

Ernie

Thats where I am trying to understand it. But it is written in matlab code. I don't have Matlab. I am trying to do it in an excel spreadsheet.

i have 101 cells of data.

i make a=yesterdaysclose-avgoflast100yesterdaysclose;

b=today-yesterdaysclose;

then regress a on b for last 100 bars. I get a weird answer all the time.

Hi Anon,

You regression formula appears correct. But when you mentioned yesterday's "close", are you expecting the price series to be mean-reverting? Most price series are not mean-reverting -- only spreads are.

Ernie

is there a difference between terms "quantitative trading" and "algorithmic trading" ??

and, what are best materials to learn quantitative trading and investment strategies ??

Hi Sunil,

Algorithmic and quantitative trading are basically the same.

One of the better places to learn about the subject is my book Quantitative Trading!

Ernie

Dear Ernie,

If I run a OLS regression on the dz vs. prevz-mean(prevz) on pg141 of your book to estimate the theta for the Half Life. This regression has very low Rsquare in general. And the t-statistic for theta is usually quite negative which we will reject the null hypothesis. So does this mean this estimate of half life is not very accurate in general as opposed to what you suggest in your text which is "Since we make use of the entire time series to find the best estimate of Theta, and not just on the days where a trade was triggered, the estimate for the half life is much more robust than can be obtained directly from a trading model."

Suny,

Half-life estimates do not need to be very accurate to be useful. If you calculate the t-statistic of the mean return of your trading strategy, you will likely find a much worse result.

Ernie

Whenever I short one of the positions in the hedge it seems thst a few weeks later I get a buy back notice from the brokerage firm so that I have to close out the position. Any suggestions ?

Thanks

Anon,

I suggest you find a better broker.

Or short instruments that are in better supply.

Ernie

Please Dr Chan, specify the input data (which stock, which dates) you use when you calculate something in your book. Confusion arises on several blogs because of this:

http://pcweicfa.blogspot.se/2010/08/r-implementation-of-ornstein-uhlenbeck.html

He gets halfday of 8.8 days. The reason? Because you use more data points than he does. This took me several hours to figure out, which code I should trust: his or yours? The discrepancy is because he did not know which input you used.

Now I am trying to find out why your code can not find the half life lambda (which is 3) here

http://www.sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model/

His ansatz gives 3.12 which is close to the correct answer. Your code gives an answer of -20.787, which is totally off. Could you please investigate this further? I am using your code in R version (see the first blog). Why can your ansatz not find the correct halflife, which is 3?

Hi Anon,

Sorry, but I am mystified by why you find the input confusing. The entire set of input data in example 7.2 is available for download (from epchan.com/book) as an Excel file, and there are just 2 ETFs in questions: GLD and GDX. I don't see how anyone using this same data could be using different number of data points.

Ernie

Dr Chan,

I want to calculate the expected time for the current price to revert to the mean. How can I use the OU half life for this?

My reason is because the de-trended (stationary) time series has significant drift. For example 30% per annum. The expected price is the current mean price + time to revert x drift.

Is the z-score also a factor?

Imagine one time series with constant OU half life. At different samples, the price is 0, 1 and 2 stdevs from the mean. Is the expected time to revert constant or sensitive to the z-scores 0, 1, 2?

Ken,

By definition, a detrended price series should have zero returns. How can it be 30%?

The OU halflife is the expected time to revert about half of the current deviation from the mean. It is independent of what the current zScore is. If the zScore is 2, it is the time to get to 1. If it is 1, then it is the time to get to 0.5.

Ernie

Dear Ernie,

In your post about calculating the half life of the spread you talk about linear regression of the daily change of the spread versus the spread itself.

so if yt is my daily spread at time t,am i correct in doing the following regression.Where C is the regression symbols?

yt - y(t-1) c y(t-1)

Hi JPS,

Yes, your regression is correct.

Ernie

Dear Arnie,

I have two stocks which are cointegrated as per the Johansen co integration test.An on running the VECM i check the residuals of the regression.My observations are as follows.

1. There is no serial correlation among the residuals

2. There is HETEROSCEDACITY in the residuals

3.The residuals are not normally distributed.

Is this model acceptable?

is there any way of removing heteroscedacity in VECM?

Hi JPS,

To avoid non-constant variance, try log prices instead of prices. But I don't see why heteroscedacity is a problem for creating a profitable trading model. We also don't particularly care if the residues are normal.

Ernie

HI Ernie,

Thanks a lot for the clarifications.I was reading your blog about creating more efficient pairs using combination of more stocks.My query regarding creating combo of such type (for example a combination of stocks taken from index to be combined with the index itself) is as follows.

1.Do these stocks individually need to be cointrgrated with the index

2.Do these stocks individually need to br Cointegrated with each other also

in other words what I am conjecturimg can we have two stocks which are not cointegrated on one to one basis become conitegrsted when combined with a third stock in a combination of three stocks.

Hi JPS,

For the index arbitrage strategy between stocks and an index instrument, the stocks individually should cointegrate with the index, but not necessarily with each other.

Ernie

Dear Ernie,

While going through some literature for Unit Root testing( which I implement on the stock price time series ,I came across PPURootTest ( The phillip Perron Unit Root Test) that also checks for the Structural breaks in the data.The NUll of the test is that the Series has the unit root ( at Level) with a structural break at a particular date which I interpret as that accepting the Null Hypothesis means the Series is Non Stationary at Level and the data has a structural break at a particular date at a specified date suggested by the test.But What to do with the information abut the structural break ?

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