Pairs Trading - Copula Method vs Cointegration

Hi Jing, this is really great. Thanks for sharing your work. I am looking forward to exploring the math further. Your explanation seemed to really stream-line the difficult/important concepts into a more understandable framework, which I appreciated. Is there anything you were able to identify for why the strategy's activity slowed down from October 2014 to December 2015?

Hi FX trader,  For this pairs trading method, the chosen pair is “GLD” & “DGL”. In comparison to the profitable period from October 2012 to Feb 2014, if we take a close look at the historical log returns from October 2014 to December 2015,  the returns are less volatile and the 2 curves almost overlap with each other. Not much price divergence for pairs traing profits. Although we have trades during this period, the profits offset from longing and shorting this pair simultaneously 

It would be a good project to try copula pairs trading on not just ETF but stock pairs because of the volatility of stock market. It is promising to try high-frequency pairs trading with copula method. See Xiaowei's great example for High Frenquency Pairs Trading for cointegration Method Based on Stock Market 

 -- sorry for my late reply. Thanks for taking the time to look in to this and get back to me. This makes me interested in having a sort of dormant tag, that attempts to measure and perceive how similar current market conditions are with historical conditions that have been identified to place ideal trades. More for a reference framework. And thanks for sharing this strategy too

This is the new version of pairs trading algorithm with copula method. I fixed the bug of the unsupported decimal and float multiplication.

Hi Manoj Gupta, try to clone the new version algorithm in my last comment or check tutorial page here https://www.quantconnect.com/tutorials/pairs-trading-strategy/ for the updated algorithm

Hi Hugh Donnelly, it can be amended to trade multiple pairs at the same time, but the performance based on the candidate pairs you choose. It's kind of difficult because the copula parameter estimation process has to be adjusted for multiple pairs in Algorithm initialization and then the Ondata trading part accepts different input and trigger the different signals. Almost all the parts have to be amended. It might be more efficient to just change the candidate pairs in algorithm if you want to see the performance of copula method for other pairs.

Jing Wu -- Thanks, I will take a look.

I am very new here, and not sure yet how to do many even simple tasks, so apologize for not doing the work!  In this instance, visually speaking, it appears the algo is correlated with SPY, but I am not sure how to numerically prove this claim: i see beta=0.57, and given that volatility of your portfolio is most certainly lower than that of SPY itself, it means the correlation with SPY is definitely above that number. That should not be the case for a pair trade between SPY and DIA - but it seems there is a long market bias in the strategy right now.  Maybe there is something in portfolio construction that needs to be addressed?

Hi Satyapravin, dollar neutral is one of the methods to decide the hedging ratio in pairs trading. Here I use the beta neutral hedge ratio Price(Stock A) - beta * Price(Stock B) (beta is the coefficient in simple linear regression) not the dollar neutral ratio. Then I purchase x shares of stock A and short x*beta shares of stock B.

Hi Karen Chaltikian Thank you for your question! It's a very interesting perspective.

I understand what you are saying, and I test the market volatility within the same period, which is 0.14, and the algorithm's volatility is 0.104. In this case, the correlation coefficient between the market and the algorithm must be a number higher than 0.57. 

I will try to look into the portfolio construction! Hopefully we can figure it out.

Hi Jingw, I have some questions:

1). Is this supposed to pick only one pair out of 8 pairs?  2). Can I put 100 pairs in your code 'def _pair_selection()'?  3). I cloned the fixed version but it showed different result from the one you posted. What am I missing here? (See attached backtest.)  4). How does this algo control leverage? In the code, the quantity of leg1 is set up always 0.4. I am curious about total maximum leverage.  5). In your tutorial, you said "...We use the first 3 years of data to choose the best fitting copula and asset pair ("training formation period"). Next, we use a period of 5 years from 2011 to 2017 ("the trading period"), to execute the strategy...." What does this mean? Is this sort of Machine Learning algo? What does exactly mean by 'training formation period' and 'trading period'? 

Hi 

I changed a little: trading everyday after 15 min. after market open, and added 'QQQ', 'XLK'. I got en error message as below around November 7~14, 2016:

Runtime Error: System.Exception: BacktestingRealTimeHandler.Run(): There was an error in a scheduled event XLK: EveryDay: XLK: 15 min after MarketOpen. The error was Python.Runtime.PythonException: KeyNotFoundException : 'QQQ' wasn't found in the Slice object, likely because there was no-data at this moment in time and it wasn't possible to fillforward historical data. Please check the data exists before accessing it with data.ContainsKey("QQQ")
at QuantConnect.Data.Slice.get_Item (QuantConnect.Symbol symbol) [0x0002d] in <8f9e699bbc3f46739adc6b359ebcd5b9>:0
at (wrapper managed-to-native) System.Reflection.MonoMethod:InternalInvoke (System.Reflection.MonoMethod,object,object[],System.Exception&)
at System.Reflection.MonoMethod.Invoke (System.Object obj, System.Reflection.BindingFlags invokeAttr, System.Reflection.Binder binder, System.Object[] parameters, System.Globalization.CultureInfo culture) [0x00038] in <dca3b561b8ad4f9fb10141d81b39ff45>:0
at Python.Runtime.Dispatcher.Dispatch (System.Collections.ArrayList args) [0x00018] in <387056c9810b431d9b668f2df5d6c027>:0
at __System_ActionDispatcher.Invoke () [0x00006] in <166e461c70884c28abc297d007aa4fef>:0
at QuantConnect.Scheduling.ScheduleManager+<>c__DisplayClass16_0.<On>b__0 (System.String name, System.DateTime time) [0x00000] in <8f9e699bbc3f46739adc6b359ebcd5b9>:0
at QuantConnect.Scheduling.ScheduledEvent.OnEventFired (System.DateTime triggerTime) [0x00036] in <8f9e699bbc3f46739adc6b359ebcd5b9>:0

To me, it's impossible not to find the symbol 'QQQ'. What's happening here? Should I change the trading schedue? Thanks :)

I am not sure why you are having a problem. I have modified the algorithm to only trade 'QQQ' and 'XLK', and set the schedule to 30 minutes after market open on the same month start days. Maybe you can see what is different between this version and yours to help you.

Hi Jingw, Karen Chaltikian, As I said above ("...trading everyday after 15 min. after market open..."), I changed the trading interval from once a month to a daily basis. I tried different schedule but in November 2016, the same error message showed up. I tried with Jingw's version and Karen's version, but both had same error messages.                                           

I don't know what happened then. I guess there's a bug we missed. I will look into this more in detail. Any suggestions and comments are welcome. Thanks :)

Jingw, I have more questions. Please see the attached backtest for the period from 2016,01,01 to 2017,9,20 and its trades logs as below. As you can see below, the quantity for both legs decreased dramatically from 2016-02-19. 

And I thought that it trades just once a month on the first day of the month but it does not. Is this supposed to do this? What am I missing? Thanks :)

Jingw, I attached a backtest for the year 2017. It looks very disappointing and has a lot different outlook from the one above. Do we need some sort of warm-up period? Can you explain about this? Thank you :)

Hi HanByul, yes this algorithm needs model selection and training period. For the two parameters here,

history data to find the best fit copula from Archimedean copulas class, and use 250 days to train the chosen copula function to find the best parameter estimation. For pairs trading, when to trade and how many trades each month really depend on if the criteria is satisfied and can't control manually. But for this reason, when there is a trading opportunity for buying the pair(buy first sell second), it could happen that the last trading opportunity is still buying signal, then since you already hold the long position, you have limited money to buy new ones. That's why the quantity decreased dramatically sometimes. But the decreasing trading quantity can be modified by changing the leverage or you can set your own trading rules when there is a pairs trading opportunity.