A quick summary is a good idea at this point. This is because it is important to make sure we are all on the right page. To ensure that we stay on the right track, I strongly suggest you go through the recap. To make sure we don't wander off-track, I will keep this brief recap as a pointwise summary.
This brings us to the current stage. We will now discuss the density curve, and the trigger for pair trade in this chapter.
We are at a point where we must stick to one variable amongst Ratio, Differential and Spread. You may be asking why only one or not all of these variables are important.
This is so that we don't get confused by conflicting signals and stick to a system. To show that there are three options, I have included all three variables. You, the trader, can choose which variable you feel most comfortable with. Personally, I prefer the ratio to the spread or differential. The ratio captures the stock market valuation since it takes into account the current stock price. The ratio gives us an instant indication of the stock's market value.
If Stock 1 has a price of 190, and Stock 2 is 80 then the ratio of stock 1 to 2 is -.
This means that for every 1 share in Stock 1, 2.375 shares in Stock 2 must be traded. We'll get into the details later. For now, we hope you get the gist.
Of course, you can choose any variable, spread, differential or ratio. For the sake of discussion, I will use the ratio.
The name of the pair suggests that it consists of two stocks. We have not yet defined how to purchase or sell a pair. Later we will do that in the chapter. Assume for now that you can purchase or sell a pair the same way as you can sell one stock.
As you might have guessed, buying or selling a pair of pairs is dependent upon the variable you track. This variable could be the spread or differential or the ratio. We will be using the Ratio for this discussion.
It's simple: Stock prices fluctuate every day. Therefore, the ratio of the two changes each day. Most days the daily change of the ratio is within the range. There could be days where the daily change exceeds the expected range. These are the days that a pair trading opportunity is available.
Take a look at this chart (IMAGE 1)
Two things are obvious from casual eyeballing:
This is what I want to ask you to do. This is the tipping point for Pair trading. If you understand what we have talked up, the rest of Pair trading will be easy.
The ratio is itself a variable that is calculated by dividing stock 1 and stock 2 Because stock prices fluctuate every day, the ratio changes daily. The chart showing the daily changes in the ratio will show that it has an average (mean), and that the ratio trades between the mean and extremes of this value. There is a good chance that the ratio will return to its mean in the coming days, regardless of what it is right now (i.e. above or below the average). Notice that I used the word "great chance" here. This is a way to say that it should be possible to calculate the likelihood of the ratio returning to the mean.
This phenomenon is also known as "Mean reversion" or reversion toward mean.
In red, I have highlighted two points on the chart where the ratio is off the mean. The first circle to the left signifies a point at which the ratio is higher than the average value. The 2 and 3 circles to the left indicate a point at which the ratio is lower than the mean. In both cases, the ratio eventually returned to its mean.
If you think about it another way, we can now form an opinion on the direction that the ratio will likely move. The ratio is most likely to return to its mean if it moves above the average circle. You can also shorten the ratio at its highest point and then buy it back to the mean. The second circle indicates an opportunity to buy the ratio with the expectation that it will return to its average value.
The ratio can be thought of as a stock, or futures. We can place bets on its directional movement because it is predictable.
I trust you get the point.
Trades can be initiated by the ratio's value relative to the mean. If the ratio is 0,
Alright - so far so good. But, here are a few questions.
These questions can be answered by something called the "Density Curve". Let's find out how it works.
Take a look at this chart -
The chart shows 4 points. At all four points, the ratio traded above the mean. Let's suppose you were looking at the chart around the mark of the first circle. Would you trade if the ratio is higher than the mean? The same question can be asked each time the ratio trades above or below the mean.
This would be a fantastic idea, I'm sure. It is important to carefully monitor the ratio and only initiate trades when there is a high chance of mean reversion. We need to make sure that the ratio is moving down to the mean value as soon as possible.
This is a lot like a tiger hunting down prey in ambush. The tiger won't jump on the prey just because it is open to attack. Only if it is certain that it will be killed, will it attack.
How can we wait to get our chance at the kill and stay in the ambush?
We can find refuge in the old Normal distribution and its many properties. I hope you're familiar with normal distribution and its properties. This is a brief summary. I recommend you to read the entire theory. I have discussed this in Varsity throughout various chapters.
This is how it looks in relation to the ratio.
We can calculate the probability of the ratio returning to the mean for each SD. This allows us to filter out trade opportunities and only initiate trades at points that have high chances of success.
The interesting thing about this is that the trigger to initiate trades is not only based on the ratio's current value, but also on its standard deviation. It makes sense to track the daily standard deviation rather than the actual ratio.
You can track the 'Density Curve’ of the ratio to achieve this. The density curve is a value that lies between 0 to 1. The density curve is a non-negative value that lies anywhere between 0 and 1.
Excel makes it easy to calculate the density curve. This is how it works. Take a look at this image.
This can be done using the built-in excel function Norm.dist. This function needs 4 inputs
Here is the table I created after I calculated the density curve value of all variables.
We could probably end this chapter here. We will be discussing in the next chapter how the density curve can be used to trigger both long and short pair trading.