The mean-reversion of this spread is even more obvious than my plot in the earlier article. Also, with the longer history, we get a much better feel for the range of fluctuations. While the value of the spread is about -$213 as of the close of Nov 9, it can certainly go much lower before reverting, based on the highs and lows of the last 3 years.

FOOTNOTE

A reader of my earlier article made an interesting comment about shorting ETF’s such as GDX and GLD. He argued that since ETF shares can be constantly created, it should not require existing shares to be borrowed for shorting. I asked Mr. Phillips of Van Eck Global about this, and he confirmed to me that a newer ETF like GDX can in fact be hard to borrow. He went on to say that the borrowing of ETF’s has nothing to do with the issuer. The issuer can indeed create an unlimited supply of the shares, but the trader still need to borrow them from his or her broker for shorting. He also told me he is currently working hard to eliminate any borrowing problems in GDX that may have existed.

## Sunday, November 12, 2006

### An updated analysis of the arbitrage between gold and gold-miners

In my article about the arbitrage opportunity between gold and gold-miners, I cautioned that we should take the analysis with a grain of salt because of the short history of GDX (a gold-miners ETF). Adam Phillips of Van Eck Global, the firm which created GDX, has kindly pointed out to me that GDX is designed to track the Amex Gold Miners Index, GDM, which has a much longer history. Hence I repeated the analysis with gold spot prices vs. GDM for the last 3 years. The results confirm my earlier analysis with much higher statistical significance: GDM cointegrates with gold prices with over 99% probability. Here I plot the difference between the spot prices of 6.1 troy ounce of gold and the GDM index multiplied by 3.68 (to compare with my earlier plot, I normalize the gold prices and the GDM index so that the Gold-GDM spread yields roughly the same dollar value as the GLD-GDX spread at any time):

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## 32 comments:

Ernest,

Excellent blog! I just found this blog through your epchan.com website.

On the subject of cointegration, have you found any interesting opportunities in the currency markets? Namely between the SPOT contracts and the futures contracts? (EURUSD and EC traded on the CME)

Or in general, any interesting cointegration opportunities in the currency market themselves?

Many thanks in advance!

In trying to correlate/cointegrate two entities like Spot prices of Gold and the Gold ETF GDX, is there a fundamental problem? Here's my doubt.

Being long GDX is owning the underlying companies. Companies can turn a healthy profit EVEN IF prices go down-witness the computer industry. If you plot memory prices vs. Samsung Stock you will see that the theory of commodity prices having much to do with the actual price of the companies doesn't hold water-memory prices have been going down forever but samsung keeps making more and more money-and the stock keeps going up. Same can be said about prices of computers-prices of computers have been falling forever but as a group hardware makers like Dell and HP are still making tons of money.

Comments?

Dan: no, I haven't looked at currencies so far. But it will not be hard. If you want to discuss this in more details, feel free to email me at ernest AT epchan DOT com

-Ernie

Sanjay: I would argue that chip making and computer companies cannot be valued the same way as commodities company. I think that even though the memory chips price goes down, the cost of making them are going down even farther, therefore the profits can still grow. On the other hand, the cost of extracting the commodities like gold and oil are not decreasing much. In fact, as the "low hanging fruits" are already extracted, we have to turn to more challenging places to extract these commodities (e.g. deep sea drilling, Alberta tar sand, etc.) So in this case the costs are getting higher even as the commodities price may be getting lower.

Like I said, I am not much of a fundamental analyst, so all this are just a layman's thoughts. I can only present historical statistical evidence to show that commodities price cointegrate with commodities producing companies. Your comments are interesting in that we can certainly test whether memory chip prices cointegrate with memory chip producer statistically-- though since we cannot easily buy or short "memory chip futures", this exercise may be academic only.

-Ernie

which function do you use for this? I am unable to replicate your results using the cadf function on the GDX-GLD spread.

BTW where does that multiplier of 60/100 come from?

Wayne: I used the same cadf function as you did for GDX-GLD. The t-stat is -3.19, while the 10% critical value is -3.09. The t-stat using GDM-GOLD is much higher (in absolute value) because of the longer history.

The ratio 60:100 comes from a regression between GDX and GLD.

If you want me to compare our 2 programs, please feel free to email me yours and I will take a look.

-Ernie

Hi Ernie,

Good to see you being "pure quant". Commodity production costs-are they really going up? In general, if you buy the productivity theory of humans, the cost of producing everything is going down. That's why we have better standards of living in general. Anywhere where costs go up is 1) short term, and 2) artificial-management, government problems, wars, natural distasters, etc.

But a correlation or cointegration model is cool-I would rather be wrong (fundamentally) and make money that be right and lose money.

:-)

Sanjay

Dan: I took a quick look at some currency pairs to look for cointegration. The most likely candidates are EUR/USD vs GBP/USD, or

AUD/USD vs NZD/USD. It all depends on the timescale. For e.g. on a 1-year timescale, the EUR/USD vs GBP/USD pair does appear cointegrated, whereas it doesn't on a longer timescale. This often happens with cointegration: over a long period of time, economic fundamentals do change, and cointegration will fail when that happens.

Hi, Ernie:

Thanks for your blog. Very informative and practical. A beginner question. How do you determine when to put on a trade? In other words, how to determine whether the spread is positive or negative enough? As Let's say the gold price now is 600. One year later, it may go up to 1000. Therefore, the swings of the spread will be wider. How do you determine when it's enough to trigger a trade?

Thanks!.

Yue

Yue,

Typically people enter a trade when the spread reaches several standard deviations from its historical mean. As to how many standard deviations, it is a matter of personal risk/reward preference. A lower number generates more trades, and therefore potentially more annual income, but also incurs greater risk.

Ernie

Ernie,

Thanks for your reply. That's exactly what I am asking. If you use historical mean of the spread to determine, say, a trade should be put on when it exceeds 2 standard deviation, it may not be appropriate. Let's say the historical gold price is 600. If gold goes up to 1000, the spread between GLD and GDM will be on average much bigger than its historical mean, simply because both GLD and GDM are much higher than their historical averages. Therefore, if you use the historical mean and standard deviation of the spread to determine trade signals when gold is trading at 60% higher than its historical average, it may trigger false signals. Am I correct?

Thanks.

Yue

Yue:

If a cointegration analysis based on the share prices of the 2 different assets indicates that these 2 assets do in fact cointegrate historically, it means that their spread was bounded, or mean-reverting, and did not drift to infinity as in a random walk. In such a case, no matter how their individual share prices go (either up or down), their spread will have a well-defined standard deviation that remains bounded, if not fairly constant, in time. The spread will not increase monotonically even as the share prices go up, as long as the 2 assets remain cointegrated going forward.

Ernie

Thanks, Ernie. That makes sense. However, I still have confusions about this. :-) You got the ratio 60/100 by regressing GLD on GDX, correct? If I am correct, it means you are using in-sample regression to generate in-sample trade signals from 05/2006 to 12/2006. Of course the graph will look very bounded with trading opportunities. Shouldn't you use out of sample data regression to determine the ratio? The reason why I am confused is that I regressed GLD on GDM from 11/2004 to 05/2006 before the introduction of GDX. Then I mulitplied the regression parameter above by the ratio of GDM/GDX which is around 27.18. The result turned out to be approximately 1.08 GDX/GLD. It's far from the 0.6 you will get by regressing GDX and GLD directly. Did I do anything wrong?

Thanks.

Yue

Yue,

Several points:

Firstly, out-of-sample testing is only for the confirmation of the validity of cointegration. To determine the best trading parameters such as hedge ratio or standard deviation, one should use all the data you have got (or at least the more recent portion) to give the best estimate.

Secondly, I performed the regression test on GDM vs GDX with same adjustment factor as you stated from 20030924 to 20061103 and obtained a hedge ratio of 0.61. If you use this period, hopefully you will get the same number.

Thirdly, you will find that this 0.61 ratio works well out-of-sample. For several recent months, this spread seldom moved beyond half a standard deviation from its historical mean!

Ernie

Ernie:

Thanks for the suggestions. Sorry for being a pain. But could you please elaborate how you got the hedge ratio 0.61 after you regressed GDM on GDX? I can only get 0.61 if I regress GDX on GLD directly. If I regress GDM on GLD, then multiply the regression coefficient by the adjustment factor of 27.18, I get a number close to 1.

Thanks.

Yue

Yue,

It seems that the only way to resolve the difference between your regression results and mine is for you to email me your data so that I can do a comparison. Often when my readers told me they find their results differ from mine it is because of data discrepancies. You can find my email in my profile.

Ernie

Hi, I only a measily programmer, so I might be wrong, but when I looked at those "mean reverting" spreads of the correlated instruments I noticed that relative strength and trend possibly do have some influence.

If one's instruments are correlated, than one should go long with relatively stronger instrument and short with relatively weaker instrument. As well, there should be a trend up in stronger one.

If one's instruments are anti-correlated, than there should be trend up in the long leg and trend down in the short leg, before the entry.

There seems to be more profit in trading anti-correlated instruments versus correlated (I might be wrong here), since one can "let profits run". One enters when spread is zero, and hopefully rides trend "outwards".

Is there a way to test this idea?

Dear insight2:

1) Regarding correlated pairs, it is possible that there are trending characteristics of one side or the other, either prior to entry, or after. However, I have long given up trying to disentangle such subtleties after much research. I have found that one can be quite profitable just trading the basic mean reversion.

2) Regarding anti-correlated pairs, I have not done very much research on them. I think it is very possible that they exists under some circumstances, but generally in much smaller number than correlated pairs and they generally last only for a short time, so it is hard to study them. Furthermore, as I have commented elsewhere on this blog, trending behavior tend to be more unreliable, with the duration getting shorter as more traders discover the trend. You will generally find potential trending behaviors after major news release or changes in the fundamental valuation of the companies.

Ernie

I hope this would be useful to this thread: http://www.mrci.com/special/correl.htm . There is this amazing correlation between Soya Oil and Crude Oil future.

Would this pair be a good candidate for mean reversal approach?

regards, dejan

Dear Dejan,

I think you meant the correlation between crude oil (CL) and HEATING oil (HO) is high. This is in fact true, and can certainly be a candidate for a mean-reversal trade.

Ernie

Hi,

There is a their symbol list: http://www.mrci.com/client/symbols.php

I do realy mean Soyabean Oil (BOH) v. Crude Oil (CLG). Their table shows 95% correlation and that holds well for other NYMEX oil futures like RBOB Gasoline (RBG) and Heating Oil (HOG). My guess is that production of the Soyabean Oil is so heavily dependent on the price of petrol, that tight correlation is created. Advantage of this spread would possibly be the fact that it is relatively "undiscovered" and not many speculators are competing for it.

Only real snag is that CBOT's Soyabean trading session is closing earlier (1:15pm CET) than NYMEX Oils (2:30 EST), so EOD data on those two contracts do not reflect the same moment in time.

Otherwise, good anti-correlated pair would be German Bunds and Euro STOXX 50, both traded on EUREX.

regards, dejan

Dear Dejan,

That is indeed an interesting and surprising observation.

The same problem with different closing times occur with pairs of ETF's too (NYSE's ETFs closes at 4, AMEX's ETFs closes at 4:15)

I don't think it will be a huge problem though if you exit them at the same time.

Ernie

Hi Ernie,

My sole objective was to delegate some work from programmer to quant, in reverse from usual ;-). Just joking.

Thank you for showing how to match position sizes in the beginning of this thread. I'll get few months of intra day soya data and check this.

regards

With the spread between GLD-GDX being at an all time high, what would be the arbitrage scenario today?

When GLD-GDX is at a high, we would want to short GLD and long GDX.

Ernie

Hello Ernie,

Have you done any pair trading between futures and equities/ETF's?

Can you offer any insights or explanations regarding the need to align the futures' particular minimum tick increment and tick value attributes with the equity?

For example, the minimum increment for an S&P 500 ETF is 0.01, which is worth $0.01. Whereas the minimum tick increment on the E-Mini S&P 500 is 0.25, which is worth $12.50. In order to run a cointegration analysis between the two, we need to account for this, right?

Thank you. Any input is much appreciated as always. Your book, blog posts, and replies to our comments are extremely helpful.

Cheers!

Jacques,

If you are not concerned about dollar-neutrality, and merely wish to construct a cointegrating pair of ETF vs. futures, then there is no need to worry about the so-called futures multiplier.

When one computes the hedge ratio between the ETF price and the futures price, the regression will automatically take care of the relative volatility (or "beta") between the 2 instruments.

However, if dollar-neutrality is required, then you have to find out the futures multiplier so that the long and short side has the same market value.

(The minimum increment is not relevant to the strategy.)

Best,

Ernie

Hello Ernie,

Please pardon my ignorance in this matter... :)

Could you please elaborate upon any instances where a quantitative trader intent on engaging in pairs trading based upon cointegration would NOT be concerned with dollar neutrality?

Aren't we seeking market and dollar neutrality?

Any explanations that you can provide are greatly appreciated.

Cheers!

Jacques,

One cannot ensure both dollar-neutrality and market-neutrality simultaneously for the same pair.

It is up to the trader to decide which option to choose, and there are merits for either choice. However, in any case, if you calculate the hedge ratio based on regression, the pair will not be far away from either type of neutrality.

Ernie

Hi, Ernie:

Thanks for sharing your trading strategy and ideas. In you book and S&C’s interview, you mentioned you trade intraday, but all these pair trading strategies have half life more than 10 days, so I am kind of confused, are you applying these pair trading strategies to intraday time frame? Thank you.

steve

Steve,

Most of our strategies are indeed intraday. The only exeception is pair trading, where we hold multiple days.

Ernie

If you want to get really interesting, take a look at spreads like the call options on gold producers against the futures or against the GDX or GLD. Keeping it simple like pairing one gold stock against another gold stock works fine and it's what we do all day long. Just make sure the 2 gold stocks are similar in size and stock prices.

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