# Mastering Pair Trading: Exploring Ratio Variable and Density Curve

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A quick recap
• If two companies have a comparable background in business, then they are probably quite similar.
• These range from the marketplace, the customer base and demand, technology, competition and costs.
• It can be assumed that if two firms have similar backgrounds, then their share prices tend to fluctuate similarly on a daily basis.
• The daily stock prices of two comparable companies usually remain in step with each other and therefore their daily returns often show a strong correlation.
• Occasionally, a local occurrence may substantially affect the direction of one business’ stock value, offering investors a chance to engage in pair trading.
• The stock prices of the two companies can be estimated using any of three variables – spread, differential, or ratio.
• We can expect these variables to be normally distributed, so we will compute the standard deviation and the basic descriptive statistics, including the mean, median, and mode.
• This ready reckoner is a standard deviation (SD) table, which covers 3 SDs in either direction.
• In conclusion, we are examining the two variants of pair trading. We began with Paul Whistler’s variation and will now be exploring an even more complex approach.
This brings us to our current position. In this chapter, we will explore the density curve and what instigates pair trading.   – Selecting the variable   We have reached a point where we must focus on one of the choices out of Spread, Differential, and Ratio. You may be wondering why simply one and not all three?   This is to make sure that we adhere to a plan and not be misled by mixed signals. I have presented three distinct possibilities for you, as an investor, to make the decision of which variable you are most comfortable with. I personally favour the ratio compared to the differential or spread because it comprises the current share cost and can give us a quick understanding of how many stocks of 1 we should buy or sell regarding stock 2.   For instance, if Stock 1 is priced at 190 whilst Stock 2 is 80, then the ratio between the two would be –   190/80   = 2.375   This means that for every one unit of Stock 1, 2.375 units of Stock 2 must be exchanged. We will explain the specifics later on, but we expect you get the idea for now.   One has the option to choose any of the variables – spread, differential, or ratio – for this discussion I will focus on the ratio.   – The trade trigger   The name implies a duo of stocks. We will explain the methods for purchasing and selling this pair later in the chapter, however, treat it as if you were merely trading with one stock.   Once you have identified the variable to track, which in this case is Ratio, the decision to purchase or sell rests on an analysis of that variable. The Ratio would be what determines if one will buy or sell a pair.   Contemplate it this way – stock prices fluctuate daily, resulting in a dynamic ratio for the pair. On many occasions, the alteration in the ratio is standard. However, some days could see a distortion from what’s anticipated—that’s when a pair trading opportunity surfaces.   A casual glance reveals two key facts –   The ratio chart generally sits between 1.8 and 2 – likely centering around this figure. I indicated it with a green line. Checking the mean value of the ratio we hand-calculated in previous sections is recommended.   The ratio is usually deviating from the usual value, either higher or lower.   Take a moment and consider the things we have gone over. Once you understand what has been discussed thus far, the remaining topics will be easy to understand.   The ratio, calculated by dividing stock 1 by stock 2, is variable and changes each day due to stock prices. If you graph the daily fluctuation of the ratio, you can observe a mean and that it alternately trades above and below this average. Despite its current position relative to the mean, there is a strong likelihood that it will revert back within the forthcoming days; a probability which we should be able to precisely quantify.   Mean reversion, or reversion to the mean, is an economic phenomenon.   On the chart, I’ve circled two points in red – one where the ratio is above average and another where it is below. In both instances, in time, the ratio returned to its mean. The 2nd circle from the left indicates a point where the ratio has deviated below the mean value. In both these cases, eventually, the ratio reverted to mean.   We can assume that the ratio will move in a predictable manner; at the first circle, it seems likely that it will retrace back to the mean, while at the second one, we anticipate that it will revert towards the average. Thus, providing an opportunity to short or buy accordingly. Likewise, the second circle points to an opportunity where one can buy the ratio, with an expectation that the ratio will move back to the average value.   The ratio can be treated like a stock or futures. Its direction can be forecasted, so we can wager on its directional progress.   I trust this is making sense.   The value of the ratio relative to the average serves as a significant prompt to commence trading. If the ratio is   The expectation is that the ratio will return to its average, so it’s a good idea to take a short position in the ratio.   The expectation is that the ratio will return to its average, meaning that it is advised to go long on the ratio.   So far, everything is going well. However, I have a few queries.   The ratio is either above or below the average; does this suggest that trading opportunities always exist?   At various points, the ratio appears to have hit a low or high; how do we pinpoint when to execute the trade? The responses to these puzzles are in the ‘Density Curve’. Let’s try tounderstand it. – The Density Curve I have identified four points on the graph which all show a ratio higher than the mean. Could this be a signal to initiate a trade? This same consideration could be made each time the ratio crosses above (or below) the mean. Agreeing that this is a great concept, we ought to keep a close eye on the ratio and activate a trade only when there’s an astounding probability that the mean reversion will occur. To put it differently, we should carry out the trade if we can be certain that the ratio will in no time return to its average value. It is much like a tiger lying in wait to hunt down its prey; the predator will not rush into action unless it is certain that its effort will result in a successful catch. To remain in ambuscade and anticipate a suitable opportunity to attack, what should we do? We take advantage of the Normal distribution and its properties. Hopefully you are already aware of this, but here is a brief summary anyway. To gain a full understanding I recommend reading up on the theory in more detail. Within the 1st standard deviation (SD) one can observe 68% of the data Within the 2nd standard deviation one can observe 95% of the data Within the 3rd standard deviation one can observe 99.7% of the data This means that the ratio is important. No matter its distance from the mean, the ratio has a standard deviation. For instance, it could only be a few points away from the mean, which would result in a 0.5 standard deviation from the mean. If the ratio shifts to two standard deviations away, there is only a 5% chance of further deviation. Generally speaking, there is about a 95% probability that it will return to normal. In the event that the ratio strays to the 3rd standard deviation, it has only a 0.3% probability of rising further, or more precisely, 99.7% chance of returning to the average. At every SD, we can assess the opportunity of the ratio reverting to its average. This filters out any potential trades and allows us to initiate only those with a high chance of success. This leads to an interesting insight – not just the ratio should be considered when initiating a trade, but also its standard deviation. Therefore, tracking the daily standard deviation of the ratio, rather than focusing on the ratio itself, is likely to be more beneficial. You can gain an understanding of the ratio with the help of a ‘Density Curve’, which is a value between 0 and 1. To get further information on this concept, I recommend viewing the Khan Academy video associated with it. To calculate the density curve, simply follow the image for guidance. Excel makes it relatively straightforward. You can take advantage of the built-in excel function, Norm.dist, which requires four inputs to use. X – this is the daily ratio value Mean – this is the mean or average value of the ratio Standard Deviation – this is the standard deviation of the ratio Cumulative – You have to select true or false, select the default value as true. I’ve calculated the density curve value for all variables, here is how the table looks –