Volatility And Normal Distribution

We have learnt how to calculate the upper and lower range of a Nifty or a stock by calculating volatility for the same. But how sure are we about this range? There can be chances that Nifty trades beyond this calculated range. Let’s now figure out these chances and also the new range that Nifty trades in.

What we see here is a Galton board, where if we drop small balls from the top, it randomly makes way through the pins and gets accumulated in the panels below. We can notice:

• Most of the balls tend to fall in the central bin.
• As we move further away from the central bin (either left or right) there are fewer balls.
• The bins at extreme ends have very few balls

This kind of distribution is called ‘Normal Distribution’ (the bell curve). The normal distribution curve can be fully described by two numbers- the average and standard deviation.

The mean is the central value where maximum values are concentrated.

Keeping the average in mind, the data is spread out on either side of this average value. The way the data is spread out is expressed as standard deviation. 1 SD will consider the area shaded in green.

2 SD will consider the area shaded brown and green together on either side.

3 SD will further include the blue area from both the sides.

Now,

• Within the 1 SD we can see 68% of the data.
• Within the 2 SD we can see 95% of the data
• Within the 3 SD we can find 99.7% of the data

If I drop a ball, can you guess which bin it will fall into? I guess not because the ball will take a random walk. However, you can predict the range in which it may fall.

So what will be the range? It can be within 1st SD or the green region in the above chart.

How sure are you about it falling within that range? The answer is 68%

Can we estimate a better accuracy?

Yes. We can increase the range to 2 SD (both the green and brown area) and we can be 95% sure about it.

For even better accuracy of 99.5%, I can say that the ball will fall anywhere within the two blue areas.

So is there no chance for the ball falling in the red region on either side? There is, as low as spotting Black Swan in a river, the probability of the same being as low as 0.5%.

Amazing Fact: At extreme temperatures, Celcius and Fahrenheit tend to be equal. They are the same at -40 C and -40 F. Similarly, at extreme volatility, everything becomes Delta 1.