Return Differential Arbitrage

Return Differential Arbitrage

I have my hands in many different projects and areas of study these days, one of those being arbitrage. I see opportunities many times a day between spot and futures prices where pure arbitrage opportunities exist that are big enough to be exploited. Sometimes the spread widens up to around 10 pips long enough to execute the trades manually. I notice this on Oanda BTW.

I bring the above example up to demonstrate a real arb setup where you have identical instruments trading at different prices. Another type of arbitrage exists that uses statistical and quantitative measures for finding spreads in the values of correlated instruments.

The idea is that similar instruments should have similar values. This has to do with what economists call the ‘law of one price.’ That law states: "In an efficient market all identical goods must have only one price." (Wikipedia) If were are using similar instruments the idea becomes tweaked to some degree, but still holds true in many cases.

I’ve been scouring the net for good examples of stat arb or pairs trading only to discover that nothing really exists that is of much value. Maybe it’s out there somewhere, but I could not find it. All the quantitative forums are pretty conservative when it comes to actual strategy discussions. I’m not a huge math buff, so lot’s of what they do goes over my head, but I have begun working on some ideas of my own.

One of the biggest questions throughout my studies was “How do I measure for differences in value between two correlated instruments? “ I have long looked at the correlation of currency pairs over at dailyfx.com using the overlay feature and noticed they use a simple percentage rule. Basically, to measure the correlation between two different pairs, a calculation is started at the same point in time in each pair and the percent change for each pair is tracked as time progresses. There is a fatal flaw though in the dailyfx charts. Their charts begin the correlation calculation at the first bar or candle on the chart, BUT, every new bar on the chart removes the original bar the calculation started on. This removal of one bar each time a new one pops up accelerates the convergence of values in a false way. Usually the pairs are not actually converging, but the calculation keeps getting moved forward, thus giving one the perception that the values are always spreading and converging, spreading and converging ect. The truth is, this relationship can only be seen in hindsight, making the apparent setups untradeable.

So what’s the solution? Simple, the initial point the calculation is started must remain the same. All I’ve been doing is using a modified fib tool and making a percentage change grid on the pairs I’m following. I simply put in whole numbers in my fib (0,1,2,3…), this gives me an equidistant lattice that I can use to track % change instead of price. I usually use the weekly opening prices for each pair because time is the most important factor in these studies. If you start your measurements at different times in each pair, the running relationship between the pairs will be screwed up.

I’ll analyze EU and NZU to serve as an example for the method I’m describing. I’ve taken the opening price from 4/12/2010 in each pair to begin measuring % change in both pairs. When you first start these measurements the pairs are 100% correlated because price has not had a chance to move yet. As time moves on , the pairs will tend to move apart and then back together. Often times the pairs will diverge and then converge a few times within a certain window of time to eventually diverge to larger degrees. Some of these trades can become longer term opportunities. To keep these trades more current, I’d suggest keeping the starting measurements not more than a couple weeks back.

The risk with these trades is called drift. This is where you sell the higher valued instrument and buy the cheaper one only to have the spread in value widen leaving you with two bad positions. This is where keeping your measurements fairly current is helpful. The most efficient way to do these trades would be to make a program that could run through all time points to identify the places with the largest spreads in value. If you know where the largest spreads in value exist, you can pretty safely do a longer term arb trade. For instance, there is a ~ 27% spread in value between EU and EJ, based on a price/time back in 1991. The value spread has opened and closed several times throughout the life of the measurement point, but it was one of the widest points I found in those two pairs. That setup is a sell EU buy EJ trade. If one of you programming gurus out there can come up with an easy way to find these spreads I’d be much appreciative.

I chose to analyze EU and NZU because of their high correlation. I am using Oanda correlation charts to see the correlation.

At this point in time, measuring from 4/12, a .07% spread exists in the pairs. This is a possible buy EU Sell NZU trade. When I plot the grid, I take the starting price and I multiply it by .001 so I know what .01% change in price is per pips. I then plot the fib so that the 0-1 range is that amount of pips. So my grid is setup to show me .01% change increments. I have to do it this way because I have not been able to find an indicator that will let me pick where I want the calculation to start.

One of the problems with these trades is that one pair is usually more volatile than the other, so you have to lessen the position size in the more volatile pair. This can be tricky in itself, but I have had positive results by lessening the position size in the over performer by the factor it has over performer. So if it’s 15% more volatile, I’ll decrease the position size by that amount or increase the position size by 15% in the underperformer.

The first attached chart show the current view of the EU NZ trade. The spread at this moment is ~.8% I’ll attach a couple other charts of trades that have happened so far within the same grid.

The second chart I’ve attached shows a spread of about 1% on 4/14 @ 01:00 with NZ being a buy and EU being a sell. A couple days later on 4/16 @ 12:00 the spread had completely closed and the pairs actually crossed momentarily. The idea would have been to close the arb trade when the pairs converged to a zero spread condition.

This type of trading is fairly time intensive and I don’t trade it seriously at this point, but I thought that If I could round up some of you to help me flesh it out a little more that it may become more viable. There is a lot of work that needs to be done, mainly statistical work which is not my cup of tea. We need to identify things like maximum spreads in value per time window, and try to come up with some better programming to make the method easier and more efficient to carry out.

Thanks,

Scotty B

I don't want to set things off on a negative footing, but I've already tried this stuff with forex and it doesn't really work. There are a number of problems...

1. By taking the spread of NZD/USD and EUR/USD all you are really doing is putting directional bets on EUR/NZD, pairs trading is supposed to dispense with this market directionality.
2. Proper neutral position sizing is quite hard (but not impossible) to calculate because of the different ways of calculating 'pip value' depending on the relationship of any given forex market with USD.
3. You're only really limited to a couple of viable pairs due to nearly all forex markets correlation with the USD.
4. Correlation is a weaker relationship than co-integration, co-integrated forex markets are...elusive.

Anyway, I know nobody wants to hear any of this.

You were saying the there was nothing around on pairs trading, I beg to differ, here is some stuff...

The classic starting point.
www.yats.com/doc/cointegration-en.pdf
Books.
http://www.amazon.com/Pairs-Trading-...1818536&sr=8-1
http://www.amazon.com/Trading-Pairs-...1818536&sr=8-2
http://www.amazon.com/Quantitative-T...ref=pd_sim_b_3
Also, take a look at that blog BlackMage linked to, it's full of papers on this type of thing.

Good luck with this stuff, it will at least get you thinking about the right things.

Thanks Craig, I appriciate your thoughts. I agree, it's a very difficult approach to carry out, but I'm not quite ready to say it's impossible. I am having a hard time grasping how taking the spread becomes a directional bet in ENZD. Would doing this in the real market place cause ENZD to consolodate? This is assuming that traders were intentionally trading this way. Could you explain it to me?

Apart from the percieved arb type setups, I see value in how the pairs react when they are at relative % parity. It's a sort of S/R that sets up. Also, many times the lead pair that visits a % level or zone first will tip you off on how to trade the other pair when it gets to that level. This is especially true if you use the weekly open price.

I was half casting out my line by starting this thread, hoping someone who knew more would come along and offer some hints about this avenue of trading. Thanks for doing just that C++ Money!

That reminds me, I still need to write you a PM I've been meaning to get to.

Actually now I think about it I can't justify it to myself, I remember reading something where somebody was trying a hedging approach with EUR/USD & GBP/USD and somebody made a argument which seemed valid at the time that this was the same as taking a single position in EUR/GBP. I seem to remember that the argument was if you buy EUR/USD and sell GBP/USD that you actually had (EUR/USD) x (USD/GBP) which is EUR/GBP. But this is not how the maths works in equity examples, so now I'm a little confused.

This is the other reason I stopped trying to do this in forex, because it's bloody confusing!

Here is an interesting read (http://www.kreslik.com/forums/viewtopic.php?t=307), not quite the same, but there is a more lucid explanation of the above idea.

No no..It's all good man. The reason I came to the conclusion I did was due to the pair overlay script or indy I have for MT4. I use to overlay GBPUSD and USDJPY over a GBPJPY chart just to learn what happens as the GU and UJ move about. GJ moves the hardest and fastest when both GU and UJ move togeher in the same direction. When GU and UJ move in opposite directions GJ just sits there range bound. If just one of the two pairs moves strongly, GJ will move, but only modestly. Anyways..

I looked into this kind of thing a while back. There is math involved that helps to clarify exactly what positions or risk you have on. Presumably you're using equal sized positions on the E_U / N_U trade.

If you go long NZD/USD 10K and short EUR/USD 10K, you really have the following currency positions:

Price: NZD_USD = (bid/ask) 0.7116 0.7118, EUR_USD = 1.3395 1.3397

long N/U = 10K NZD long, short 7,118 USD
short E/U = 10K EUR short, long 13,395 USD

You should see something similar in the Oanda position exposure tab if you use that. So in reality this "pairs trade" has three different exposures and won't trade exactly like an EUR/NZD short due to the excess USD long position (+6,277 if my math is correct).

The interesting thing about positions like these is that it is possible for all three core currencies to show a profit, or for all three core currencies to show a loss. The latter case is when the trade falls apart and typically occurs when everything becomes very serially correlated and volatility goes through the roof - like a panic. Your short EUR position should be a partial hedge in such an occurrence but as stated before it's possible for all 3 positions to move either for profit or for loss.

So as far as position sizing if you want to eliminate the USD portion, you will have to trade more N/U long or less E/U short. I can help with the math on that if you're interested. But even then, if you properly hedge your position you will simply have a synthetic EUR/NZD short and at that point you should be able to see the P&L fluctuations mirror that pair's movement. The kreslik site got me going on the math and is a good reference. The true riskless arb opportunities are hard to achieve because the time is limited & fill critical, and if you do it consistently, your broker will change the game on you. Just my experience.

I am not sure I fully understand what you're trying to say in the OP.
In academic use, arbitrage is the opportunity for a "risk free" profit. I'd like start off by emphasizing that statistical arbitrage is in no way risk free. Differential arbitrage, however, is (or under market circumstances should be!)

The process you're explaining in the quote is of course the consequence of the triangular nature of cross rates.

If EURUSD ≠ EURGBP * GBPUSD, a triangular arbitrage opportunity exists.

For example if
EURUSD = 1.3300
EURGBP*GBPUSD=1.3320
Then you sell EURGBP and GBPUSD. This leaves you with a net short position in EURUSD, which has to be offset by buying EURUSD. In case of no transaction costs, your risk free profit is 20 pips. The position sizing calculations are fairly easy.

Pairs trading is a method of statistical arbitrage, attempting to exploit drift between correlated assets (ie not the same instruments) - hoping that the correlation will hold in the future. Currencies are not that suitable for pairs trading in the sense that an uncovered currency pairs trade is, like Craig said, a directional bet on the two underlying currencies! Arguing that neutralizing the directional bet by entering a third position in turn removes the essence of the pairs trade.

A possible pairs trade that involves currencies could for example be CADJPY/Oil.

I apologize if I totally misunderstood your point. In that case, please clarify yourself, as I'm quite interested in these topics.

There is pip variability until you lock the triangular arbitrage. You cannot game the synthetic using a single dealer... they use triangular arb to maintain the synthetic pairs.

All dealers use some sort of natural/synthetic algo to maintain a 0-arb condition. There can be instances of option arbitrage (I closed the Euro/American arb at Oanda), but not in spot.

True

I had a hard time when coming up with the name for this thread. When I say 'return differential,' I mean a differance in the % change performance of the two markets. When looking at EU and NZDU I am thinking about the value of the dollar that and the relationship that exists between the currencies. I look at there always being a situation where you have cheap dollars and expensive dollars (relatively of course), or cheap euros and expensive one's ect.

What I'm doing would be considered a sort of quanitative analysis arbitrage. I am going off the observation that the huge % differance gaps open and close back to 0 all the time. I don't fully understand why this happens, it just does. Right now there is a 2%+ spread in EU and NZD based on the time/price I started meassuring. There is no guarantee the gap will close without widening further (risk), but this is why I need a program to examine data from every price/time point available so I know where all the % gaps are from smallest to largest. Arbing the largest gaps would in theory take much of the risk of drift out of these trades.

I'll post some more charts up later with example of these gaps and how they open and close.

I have not found another example of anyone doing exactly what I'm doing with this..I even bought a $20 system off Ebay because the description the guy gave sounded similar to this. I was just curious. He was not doing this BTW.

It's all about the % change grid, setup from many differant points in time on correlated pairs. I beleive there is something to be exploited in it, all based on my humble observations.

Hi Scotty- Not sure this is what you are doing but a while back i tried overlaying the eur/usd and usd/chf on the same chart (the usd/chf had to be inverted as well) When ever they diverged from each other by a certain amount i would take opposite directions on each currency so that in theory whichever direction they went as long as they converged i would make money. I also had to trade more lots on the usd/chf to account for the fact that it moved less. Sometimes they converged -sometimes not so i gave up but there maybe something in it. I have some mt4 indicators that overlay the currencies if you need them?
Anyway good luck with it

Hi Arbitton,

Thanks for stopping by. You have the gist of this (as most of us do because it's simple), but I'd stay away from most of the overlay indicators. What I'm doing that is differant from all the overlay programs/indys out there is that I'm meassuring from fixed points in time. When you are relying on the overlays indies the meassuring point is always changing or you are stuck with the point the program picks. I hope this makes sense. So my advice would be to do the calculations manually for now.

Alright guys, I don't have a lot of time tonite and I'm away from home so I have to rely on dailyfx charts. But I want to post a chart I just put together to show some of the cool things you can do with this type of meassuring system, both for arb opportunites and for S/R forecasting where one pair leads the other.

I am analyzing EU and EJ here using a monthly chart and the closing prices of Oct 1 2000. The starting point is not really important as long as it's the same in each pair. As I've said, to do this right you really need to run calculations on each time point in these studies, but it would take forever doing this manually. We really need a program that can do this for us.

Before I explain more, I have to say that the grids you see on my chart represent a 3% appriciation per line. I took a fib tool and programed it 0-25 and each line represents 3% of movement to accomodate the huge timescale. You can set your scales however you want, just make sure they are the same on each pair.

Anyways..Some observations.

Arb opportunities:

On May 1 2003, EJ's closing price was right around the 17 level as seen on my chart which equates to a 51% appriciation rate. On the same date EU was only at the 12 level, a 36% appriciation. So 51-36= 15% spread in value. To do an arbitrage we would buy EU and sell EJ. Fast forward..On July 1 of 2008 EU was at the 27 level while EJ was at the26.5 level, leaving only a ~1.5% spread in value. I'm not saying I would take such a long term trade, but the spread almost dissapeared at this point and would have made money. I'm not sure what the interest cost's would have been carrying the trades such a long time, but the long EU would have been at + 3850 pips while the EJ short would have been @ -2696 pips leaving a differance of + 1154 pips on 7/1/08. I'd be curious to see if the cost of carrying the positions would have canceled each other. If so, this would have been an example of a possible profit opportunity.

Info opportunities:

If you notice on the attached chart. EJ consistantly led EU in appriciation and depriciation. In july of 2007 EJ was rejected off of the 27 level (81%), Almost a year later Eu makes it to the exact same level and only exceeds it by ~4%, being rejected in the same zone. EJ then falls sharply all the way down to the 7 level (21%) before making a bottom. It will be interesting to see if EU follows suit. This would put EU at almost parity with the dollar. While EJ fell down to 7, EU fell to the 15 level and found support during the month of October 2008. Notice that the 15 level is resistance for EJ when it returns to this level in April-Oct of 2009. I hope you guys can see the significance of this.

I could go on all night about other opportunities just on this chart. It's a time intensive method though, hopefully somebody will come along and make it a little easier on us...Programmers..Where are you?

Have you considered that the 1154 pips is exactly equivelent to the price change in U/J during the time period in question?

And putting aside the daily fx reports, is there any compelling reason why nzd/usd and eur/usd would be correlated?

Hey Darkstar,

No I had not considered UJ. One of the ideas behind this approach is that if you can enter the market in a place where the percentage spread is at it's widest or retracting from it's widest point, you could hedge yourself to some degree from directional market movement.

A directional trade in UJ would not offer the benefit of the hedge effect. Also, the reason you put on the arb type of trade is because there is really no way of knowing which pair will end up closing most of the gap. Even if one of the pairs is losing in the end, when the spread closes there will be a net profit. I admit this is all pretty tricky and I only half know what I'm talking about in this area of the market.

As far as EU and NZD being correlated..I'd say that with USD being a part of each pair that you'd have a case for correlation. While the exchange rate has to do with Euro and NZ strength or weakness it also has everything to do with dollar strength and weakness. If NZD is depreciating while EU is going up you have the dollar that is simultaneously going up and down in different markets. I don't know whether the law of one price applies in these cases, I'm just going off of my own observations of how the market tends to close these gaps eventually. It's almost as if the individual currencies are only elastic to a certain point, eventually the different markets move in such a way as to bring prices back to a kind of parity. It could be that it's all nonsense, but I think it at least deserves some thought and testing.

I couldn't skip not answering...

What in the world makes you think NZD/USD and EUR/USD are SIMILAR PRODUCTS ??

They are both hinged to the US Dollar.

I beg your pardon....

So - lets draw some parallels....

Lets say I am a schoolboy as you are. And we are HINGED to SCHOOL.

Does this mean my marks will be THE SAME AS YOURS?

Bake a bread and forget the markets, cause you are in the way of destruction my friend

huh?

You're right, there is some correlation between EURUSD and NZDUSD because of the fact that the USD is reflected in both pairs. However, I would argue that the correlation between these two currencies is pretty low, because both the euro and the NZD have their own 'agenda.' EURUSD and CHFUSD should be far more correlated, as the economies are more related. Same goes for AUDUSD and NZDUSD.


Any divergence (or spread, as you call it) in the (relative) prices of EURUSD and NZDUSD is reflected in a change of the EURNZD rate. As I see it, this divergence doesn't mean that the dollar is going up and down in different markets, but instead it implies that the euro either gained or lost strength vis-a-vis the NZD. The law of one price (in the non-Purchasing Power Parity sense) therefore shouldn't be violated.

You are right that in forex, there is some reversion to the mean. I would argue, however, that it is going to be really hard trying to find an edge here.

On the other hand, as Einstein said: "If at first, the idea doesn't sound absurd, then there is no hope for it."