Earlier this week we highlighted the **strong correlation between the Yen and US Treasury bonds **due to the carry trade**.** We suggested the existence of stat arb strategies which could take advantage of the interrelationship. Today we will introduce one possible strategy: a **pairs trade**. This is one of the simplest forms of stat arb and consists of:

- Identifying a pair of financial instruments which are highly correlated (or even cointegrated)
- Creating a tradable linear combination (a
**spread**) of the two instruments which is mean reverting **Buying the spread when it goes down, selling the spread when it goes up**(according to some algorithm), ad infinitum or until the relationship falls apart

To **create a spread between futures**, we have to take several things into account. First, we need to determine how much each *minimum price change*, or tick, is worth in dollars. In the case of Yen futures (**6J**), the minimum tick is 0.000001 worth $12.5. On the other hand, the 30 Year (**ZB**) has a minimum tick of 1/32 (0.03125) worth $31.25.

Next we need to construct a linear combination which will serve as our trading model. A simple way to do this would be to look at the price changes in each contract and construct a linear regression model. In our case this is not very useful: we are** trading two futures with prices that differ by several orders of magnitude**. Moreover the contracts have different minimum tick values. How do we reconcile the two contract specifications and create a useful trading ratio?

One way to normalize the two series would be to **convert each price change into dollars**. Suppose we are looking at the 5 minute raw price changes. Multiply ZB’s raw price changes by 32 and they become the number of ticks. Likewise, if you multiply 6J’s changes by 1,000,000 they transform into ticks. Now we can multiply each series by the contract’s minimum tick value in dollars and have two series which are directly comparable.

For instance, looking at data from mid-December until now, we see that the **5-minute dollarized volatilities** are:

- ZB 5-minute dollarized vol: $48.64
- 6J 5-minute dollarized vol: $38.72

We can use the realized vol’s to **create a hedge ratio**. Suppose we want 6J to be our initiating contract and ZB our hedging contract, a trade ratio from Yen to bond could be 38.72 / 48.64 = ~ 4/5. Thus we could **price our spread**:

**Spread_vol = 5 * 6J – 4 * ZB**

This type of spread ignores the correlation coefficient between the contracts (0.41 for price changes and 0.76 for price levels). Another method of spread construction multiplies the correlation by the volatility-derived trade ratio, e.g. 0.41 * 0.8 = ~ 1/3, changing our spread forumula:

**Spread_cor = 3 * 6J – 1 * ZB**

Which spread performs better?. **Market conditions are the biggest determinant of which spread to chose**. In our experience, the volatility derived spread is better for short term intraday trading, especially when each contract is active and moving around. The correlation derived spread is better for lower volatility periods where you are essentially just scalping the initiating contract and using the hedge as insurance in case of a level change from the market. If we had to chose one we would unambiguously pick the volatility-derived spread but then we have a **personal preference** towards short term trades and we don’t mind the potential of being overhedged, especially in the recent volatility regime.

**Before diving right in** to trading the spread, there are several considerations:

- How stable is the estimated relationship over time?
- What is the best algorithm for determining when to buy or sell the spread?

Parts II and III will address these concerns in further detail. Specifically, Part II will examine the stability of the relationship between 6J and ZB and show how to create a **hedge ratio which is dynamic through time**. This ratio will update as new information becomes available using a Bayes filter. The filter must strike a balance between how quickly it adapts to new information. Adapt too quickly and your hedge ratio will always be perfect for yesterday at the expense of today; adapt too slow and risk being blind-sided by a shift in volatility regimes.

Part III will examine several competing algorithms for determining buy and sell signals. Again we must strike a balance between risk and reward. **Fade the spread too quickly and you risk getting run over; fade too conservatively and you risk missing out on profitable trading opportunities**.

You can visit the reading room to find sources for more information on statistical arbitrage strategies and Bayes filters.

Categories: Quantitative Trading

Have you come up with Part II and III yet ?

I searched a while, but couldn’t find them 😦

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sorry they don’t exist yet, when we published this it didnt garner the most exciting reaction, that being said, comments like this upgrade it in our priorities list for sure!

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Please let us know when pt2 or even pt3 are out.

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Part II is launched today with name “The Kalman Filter and Pairs Trading”, and is very interesting.

Still waiting eagerly for part III. When will it be available?

I am not able to understand what will be trading signal when using kalman filter. I do not want to use trading window, because that can give false signal in case of large market move.

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just a heads up part III is available here Pairs Trading Strategies

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