Pair Trade Strategy: Understanding Density Curve for Profitable Trading Opportunities

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The Pair Trade

– Digging into Density curve

We should take note of the density curve in order to detect possible trading prospects. Make sure you focus on two particular aspects.

The density curve is determined by the time series data – specifically, the ‘ratio’ as discussed in the prior chapter. The main factors for calculating this curve include the ratio’s time series, mean and standard deviation.

The density curve has a value between 1 and 0 which provides us an insight into the probability of the ratio reverting back to the mean.

I understand that the 2nd statement may cause confusion for some readers, but I urge you to bear it in mind as we continue. You will see what I mean eventually.

Let us devote a moment to the normal distribution. We have already discussed this subject on several occasions, yet I ask that you indulge me one more time.

Time series data, such as the ratio, normally have an average or mean. In our earlier chapter we calculated that the ratio time series average was 1.87. Generally, the value of the ratio stays near this mean. When there is a shift away from it, the value usually comes back to where it started.

If the ratio reaches a high of 2.5, then it will eventually decrease to 1.87. Conversely, a considerable drop in the value of the ratio is also possible.

Can we measure the likelihood of the ratio returning to its average when it deviates from it?

We can predict that it will drop to an average of 1.87, but what are the odds this will happen? Can we estimate that the chance is 10%, 20% or 90%?

The density curve provides us with the ability to determine how much a ratio has deviated from its mean. This level of deviation is given in terms of standard deviation, which gives us a probability value that can aid in making trading decisions.

I’d like to illustrate this with an example.

Have a look at this data –

Latest ratio – 2.8

Ratio Mean – 1.87

Density curve – 0.92

This data suggests that the ratio of 2.87 has largely diverged from its expected value of 1.87, and this deviation can be calculated to the 2nd standard deviation. It is probable that it will revert back to its mean with a 95% probability.

This leads us to conclude that the ratio of 2.87 is close to the 2nd standard deviation. We can come to this conclusion by looking at the corresponding density curve value, which is 0.92.

The range of values between 0 and 1 denotes the standard deviation. For instance,

The density curve of 0.16 is indicative of a value that falls one standard deviation below the average.

The density curve value of 0.84 suggests that it is +1 standard deviation above the mean.

This density curve value of 0.997 indicates that it is 3 standard deviations above the average.

Once I know the standard deviation, I can figure out the probability.

I arrived at 0.16, 0.84, and 0.997 as they are standard deviation values. To avoid further discussion on this topic, I will give you a table that you can use to help you along your way.

Based on the density curve value, if it is around 0.19, then the ratio would be around -1st standard deviation. This would give a probability of 65% for the ratio to return to its mean. For a value of 0.999, the ratio would be near -3rd standard deviation and the likelihood of reaching back to mean would be 99.7%.

And so on.

– The first pair trade

We finally have our chance to demonstrate our Pair trading. There are a few reminders to keep in mind as we do so:

The calculation of the ratio is done by dividing Axis Bank by ICICI Bank, as an example. Thus, Ratio = Axis Bank / ICICI Bank.

The ratio of Axis Bank and ICICI Bank’s stock prices shift daily.

The ratio and density curve value need to be determined each day.

The trading philosophy is as below –

Similar businesses, such as Axis Bank and ICICI Bank, tend to have stock prices which mirror each other if they work in the same market environment.

Any alteration in the economic environment will have an effect on both companies’ stock prices.

A single occurrence can lead to the market value of one organisation diverging from that of another. On these days, the proportion consequently varies.

We search for these discrepancies to locate sound trading prospects.

Essentially, a pair trader monitors the ratio and its related density curve value. They establish a pair trade when the ratio (and corresponding density curve) noticeably differs from the normal value.

This brings us to the next logical question – what is a sufficient level of confidence? In other terms, at what point on the density chart would we start the transaction?

Here is a general guideline to set up a pair trade –

The aim is to enter into a trade (whether long or short) when the ratio falls between 2nd and 3rd standard deviation and close the position as it falls under the 2nd standard deviation. The greater the proximity to the mean, the higher your gain.

Let us agree to a trade on the basis of the given table; you can obtain an Excel sheet at the end of the preceding chapter to help with this.

On 25th Oct 2017, the density curve value was 0.05234 and the corresponding ratio figure came to 1.54, which falls just outside the parameters of a typical long trade (where we’re looking for values between 0.025 and 0.003). Despite not being ideal, this is still the best setup within our current time series.

The quotient of Stock A and Stock B is known as the ratio.

If you are looking to take a long trade, then you should buy Stock A and then sell Stock B.

If you’re looking to make a short trade, you should consider selling Stock A and buying Stock B.

We have established the ratio as Axis to ICIC; therefore, at the closing of the 25th one can expect to –

Buy Axis Bank @ Rs.473

Sell ICICI Bank @ 305.7

Axis has a lot size of 1200, resulting in a contract value of Rs. 567,600/-. ICICI Bank’s lot size is 2750 with a contract value of Rs. 840,675/-.

Ideally, we should aim for a Rupee-Neutral approach, which I’ll avoid discussing for now. When it comes to the next pair trading technique, we’ll look at this concept through a different lens.

Once the trade is established, we have to allow the exchange rate to drift towards the mean. It is most advantageous to open a trade close to the 3rd SD, yet it can take a long time for the ratio to arrive at this position and the day-to-day fluctuations can be difficult to bear. Without strong financial reserves, one must act quickly and close out these trades as soon as feasible.

On 31st Oct 2017, the ratio moved up to 1.743 and the corresponding density curve value reached 0.26103. This is nearly the desired density curve value, so one ought to look at closing the trade.

We Sell Axis Bank @ 523 and buy back ICIC at 300.1. The P&L and other details are as follows –

It’s evident that the substantial portion of profits originates from Axis Bank, hinting that it has shifted from the usual course of trading.

Pretty good, right?

Let’s examine a short position at this time.

The density curve on 9th August 2016 printed a value of 0.99063156, which was near enough for us to initiate a short trade; this involves selling Axis and buying ICICI.

If it’s confusing as to which stock to buy or sell, consider that the numerator usually is the dominating stock – so if a pair trade requires a long position, purchase the numerator. Conversely, if you’re shorting, do the same with the numerator and remember that the denominator’s trade will be inversely proportionate.

We offer Axis Bank and ICICI Bank, both of which can be purchased.

Trade details are as follows –

Short Axis @ 574.1

Buy ICICI @ 245.35

Ratio – 2.34

The Density Curve is equivalent to 0.99063156.

Once the transaction was commenced, it was available for closing on 8th Sept. Its particulars were as follows –

Buy Axis @ 571

Sell ICICI @ 276.33

Ratio – 2.27

Corresponding Density Curve value – 0.979182

Agreed, one may have pondered waiting a bit longer to let the density curve drop further, yet the pair trader has to consider the time and mark to market factors in balance.

The majority of the gain can be attributed to one stock – ICICI – suggesting that it has clearly diverged from its original trajectory.

I must admit, neither of these trades fit the criteria outlined in the table for entering and exiting a pair trade. However, I reiterate that you should use the table as a reference to help hone your skills.

I strongly suggest that you take the time to search for any other chances offered by Axis & ICICI Bank.

I hope that the P&L of pair trading has inspired you to learn more about it. That’s all for now, but let me leave you with some final points.

So far, we have only covered 25% of the material I plan to cover.

These first 7 chapter discusses a very basic pair trading technique, mainly to help lay a foundation

We haven’t followed traditional trading parameters like stop loss and targets. I’ve opted to keep my remarks quite broad.

One angle we have yet to consider is the importance of neutrality in both parties’ positions.

We still need to talk about the risks associated with Pair trading.

Pair trading requires a significant capital outlay, so it is essential to have sufficient funds to cover the investment. However, its potential return makes the expenditure worthwhile.

For a given pair of signals, you can anticipate no more than 2-3 in the course of a year. Therefore, it is necessary to observe multiple pairs to discover consistent opportunities available in the market.


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