# How to use the Standard Deviation for Options Trading

When we trade options, we use mathematical and statistical parameters. The core of the option writer business is to have probability on our side. Great attention is paid to the price markers of the standard deviation.

From the school time the bell curve of Carl Friedrich Gauss may come to mind at this point. In this article I would like to simplify the connection to option trading.

## Basics from statistics

Based on statistical mathematics, certain facts are found in a number of different numbers or characteristics. Predominantly the body size of humans is used as an example. In this way, an average can be determined, which is called the expected value.

The normal distribution then describes the deviation from the average in scope and characteristics. This dispersion follows a recurring pattern in a wide variety of topics and areas of consideration.

The standard deviation can be calculated from the variance of the collected data. Within one standard deviation from the mean value upwards and one downwards there are then 68.27 % of the data sets. Often the data sets are also called case numbers, procedures, etc.

In the next area of the bell curve, another standard deviation is added up and down. This section then contains 95.45 % of the cases.

Consequently, adding the last standard deviation results in a further narrowing of the cases. Thus, within three standard deviations plus and minus of the expected value 99.73 % of all cases can be found.

### Graphical representation of the normal distribution

This can be illustrated by the following diagram:

This almost one hundred percent probability can almost put the options trader into euphoria. In order not to let hope grow immeasurably in the first place, I have to put this into perspective at this point:

The theoretical probability can be deceptive.

The remaining 0.27% probability also contains enough potential for a so-called black swan event. The values found in this area can exceed all expectations.

Moreover, attractive premiums for option writer transactions rarely arise outside the first standard deviation.

As is so often the case on the stock exchange, we can build the most beautiful scenarios from past data. And it is precisely in this past that we find indications that sometimes things can turn out exactly differently.

I do not want to go into the mathematical background further at this point. The formulas and derivation of the topic is explained very clearly under these links:

### The application in option trading

The price movements of an underlying asset can also be recorded statistically. It would be useful, for example, to document the trading range of a day. Similarly, the difference between the daily, weekly, or monthly closing prices could produce a validatable data set.

Since the strength of price fluctuations affects the prices of options, we first examine the historical volatility. Exactly this is determined on the basis of the closing prices for the last 30 days and expressed as a percentage. It is irrelevant here whether the price has risen or fallen compared to the previous day.

So if a share price falls by 1.5 percent every day, the historical volatility (abbreviated as “HV”) is the same as if it rose by 1.5 percent one day and fell by 1.5 percent the next. In the same way, a share with an HV of 0.4 percent can suddenly fluctuate by 10 percent on one day.

For the following example, I have deliberately selected two stocks that at first glance have comparable parameters. The price in each case is close to 320 US dollars. In addition, the historical volatility is around 1.8 percent. The AGM is also almost the same as the closing price of the previous day.

On this basis, it can be assumed that the price trend from the past is also similar. For a comparison I have placed the charts one below the other:

In my opinion, the difference of only 0.025% in historical volatility is expressed in the chart more strongly than expected. The gaps and individual long candles are larger for Cooper than for Apple. Because the gaps in the closing prices are relevant, the impact on the calculation of the HV remains small.

### A clear difference

But now one strong one special feature catches the eye. We find a kind of bell curve again. And these differ clearly in their expression. While Apple’s range at the end of the chart is around 70 dollars, Cooper’s is already 130.

So where does this difference come from, when the other indicators are almost identical? The secret lies in the implied volatility. In the screenshot from the TWS, I have outlined this in red.

Here, 1.5 % from Apple compares to 2.85 % from Coopers. That is almost double or, to be more precise, 1.9 times. Now we put the roughly estimated dollar spreads of the bell curves from the charts into relation and arrive at 1.86. This is no coincidence either. The curve reflects an assumed standard deviation for the future.

This means that the price of the underlying asset will move within the range of the graphically displayed bell with a probability of 68.27 %. In return, at 31.73 it can of course also find itself outside this range.

### The bell as a glass ball?

The implied volatility results from the prices of the options for the underlying. The prices are determined in the market by supply and demand. Demand arises from the need of market participants to profit from price movements and/or to hedge existing positions.

Accordingly, it can be argued that volatility is a measure of uncertainty. This can be caused by news about the company. Similarly, uncertainty in the market increases when events with an uncertain outcome are imminent.

In the case of shares, the date for the quarterly report is a good example. Cooper’s screenshot was taken before the quarterly figures were released. Looking at the chart the day after, it is easy to see how the implicit volatility has also decreased as the uncertainty about the outcome has disappeared:

### The past is no guarantee for the future

This ensures that the forecast of future share price developments always represents a snapshot of the current situation. On the stock exchange, expectations are traded predominantly. If the facts cannot follow the expectations, disappointments will result. At least disillusionment in the form of consolidation.

### Representation of the standard deviation in the option chain

In the option Chain of the broker software the standard deviations are also displayed. In addition to the marks from the chart, it is then easy to find a suitable strike for the current trading idea.

### Conclusion on standard deviation

The standard deviation comprises price ranges to which market participants can orient themselves on the basis of statistical probabilities. In options trading, it helps us to find the strikes for corresponding trading ideas. In addition, the strikes can be set in relation to the expected price movement.