# The Probability of Normality

When selling implied volatility you want the market to stay within the  expected range. But what is the historic probability that markets behave as expected? And what other analysis could be done to enhance your chances and find the periods when it is wise to sell an at the money straddle? This article will try to give some answers to this question.

## The normal distribution cone

The chart above shows a one- and two- standard deviation cone of implied volatility. S&P on top, the 30 day at-the-money implied volatility at the bottom. If you would sell an 30 day at the money straddle and hold it until expiry, you would win if the market is within the one standard deviation cone at expiry. But as you can see, this is not always the case. Sometimes future market moves are under estimated by the current implied volatility, and this would result in a losing trade.

## Statistics of normality

Normal distribution is only a mediocre assumption for market behaviour, and as Nicholas Taleb has shown in his books, the next black swan is just around the corner and you must not expect that things are normal distributed all the time.

Notice the fat tails on the left and the right shifted bell curve, the actual 25 day returns of S&P500 (since 1980) look quite different than you would expect from a normal distribution assumption.

And these differences to normal distribution are the cause why you will not get a 68% winning rate if you do a backtest on the one standard deviation cone shown at the beginning. Have a look at the table below:

The probability to be within the price range prognosticated by the 30 day ATM options volatility is by no means near the values you would have expected. Using the last 3000 daily bars to run this analysis, you can clearly see that not a single stock of the current DOW 30 stocks shows the expected value of 68%.  That’s no good news to options sellers. You have got a higher than expected probability that things go wrong. The 30 day at the money implied volatility seems to under estimate future volatility.

## An edge when selling implied volatility

One way to add an edge and bring the winning rate up might be to observe the relationship between the historic volatility of the market and the current implied volatility.

The chart above shows the 30 day at the money  implied volatility (black) and the historical volatility (red), based on the previous 30 daily bars. If you would sell an ATM straddle only if the implied volatility is above the historic volatility, meaning that the market seems to exaggerate volatility, it unfortunately would not have the expected result.

Compare the results on the left and right table: The left shows the probability to be within the range prognosticated by 30 day implied volatility after 25 trading days. The right side shows the results if you would have done this bet only if the current implied volatility is above the 30 day historic volatility. I can’t see a big difference, there obviously is no edge in following this approach. Nevertheless it can be seen in many books on options trading!

## Adding edge, attempt number 2

The basic idea of the attempt above, although it does not deliver the expected edge, is not too bad. As markets tend to calm down after a high volatility, the attempt to define areas of high volatility could be fruitful. Only selling options when implied volatility is high and thus placing a bet that things will be fine in the future is something I would also do in real trading.

To get better results, one can do the following: Compare the current implied volatility to the lowest levels during history and then only sell a straddle if current volatility is way higher. On the chart below you see how I do it in everyday trading. I am waiting until implied volatility has at least reached levels twice as high as the lowest levels on the 30 days before.

These areas give you a way better chance to stay within the prognosticated area than an average day would give you. Beside that, you get the highest premium for your short options and a high probability that volatility will contract, thus making the trade even more lucrative. See the statistics below and judge for yourself:

The right table gives you the probability to stay within the area prognosticated by the at the money implied volatility. The right table shows the probability if you only place this bet when the current implied volatility is at least twice as high as its lowest levels on the 30 days before. This does not happen every day, see the number of occurrences, but it improves the chance for a winning trade vastly. Your probability now is higher than the 68% a one standard deviation cone would suggest.

Have a look at the article on the statistics of VIX It shows why it is important to sell volatility only when it is high.

IV percentile will also be of interest, it shows the mean reverting properties and some statistics on VIX

Thanks to tradesignal for the software to run the tests and Refinitiv for the implied volatility data.

## 2 thoughts on “The Probability of Normality”

1. Feras

Thanks for the interesting post. I’m a fellow options trader and I’ve been engaged in short vol trading for quite a while now. I was wondering how did you calculate the % in move, as it seems to contradict a lot of papers and studies on the fact that IV generally overstates HV. I’ve attached a couple but there are many more to link to and it seems to be consistent with the options tick data I have over the last 5 years. What pricing models and methodology you use to determine your atm straddle? Happy to engage further in that discussion via email as the premise of this post is a big area of interest and ongoing quantitative research for me.

2. Kahler Philipp