Professional Trading through Option Theory

Lesson -> A study to Volatility and Normal Distribution

17.1 -Outlook

We discussed the potential trading range of Nifty, given its annualized volatility in the previous chapter. We came up with a lower and upper end ranges for Nifty, and concluded that Nifty will likely trade within this range.

It's fair enough. But how certain are we? Is it possible that Nifty might trade outside of this range? If so, what's the likelihood that Nifty would trade outside of the range and how likely is it to trade within the range? What are the values of an outside range?

These questions need to be answered for many reasons. It will provide a foundation for a quantitative approach towards markets. This is a different process from traditional technical and fundamental analysis.

Let's dig deeper to find our answers.

17.2 - A Random Walk

We are about to start a discussion that is very important, relevant and interesting.

Take a look at this image (image 1)

The 'Galton Board,' as it is commonly known, is what you see. A Galton Board is nothing but a board filled with pins. These pins can be collected using bins.

Drop a small ball above the pins. The ball will encounter the pin first, and then it can turn left or right before it meets another pin. This process continues until the ball falls into one the bins below.

You cannot control the trajectory of the ball once it is dropped from the top. The ball's natural path is unpredicted and cannot be controlled. The path the ball follows is known as the Random Walking.

Can you now imagine what it would look like if you dropped several of these balls at once? Each ball will naturally take a random path before falling into one of these bins. But what about the distribution of these balls within the bins?

  • They will all end up in the same place. Or
  • They will all be distributed equally in the bins. Or
  • They will likely fall randomly on the different bins.

People unfamiliar with the experiment might be tempted to believe that the balls would randomly fall across different bins, and not follow any specific pattern. This is not the case. There seems to be an order.

Take a look at this image -

(image 2

The Galton Board seems to work in this way: each ball takes a random walk and is dropped on the board.

  • The central bin is where most of the balls fall.
  • You will find fewer balls as you move away from the central bin (either left or right).
  • Extreme ends of the bins have very few balls

One can call this distribution" Normal Delivery''. The bell curve, which you may have known from school days is the normal distribution. The best part is that no matter how many times you do this experiment, the balls always form a normal distribution.

This popular experiment is known as the Galton Board experiment.

You might be asking why we are talking about the Galton Board experiment or the Normal Distribution.

This natural order is followed by many things in real-life. Take for example:

  • Take a group of adults and weigh them. Then, divide the weights into bins (or weight bins). You will get a normal distribution if you count the people in each bin.
  • You will get a normal distribution if you do the same experiment with different heights.
  • A Normal Distribution will be given to people based on their shoe size
  • Weight of fruits and vegetables
  • Time taken to commute on a route
  • Battery life expectancy

Although this list could continue, I want to focus your attention on another variable that follows the normal distribution: the daily return for a stock.

It is impossible to predict the daily returns of an index or stock. This means that if I were to ask you what TCS's return will be tomorrow, I would not be able tell you. Instead, it will more closely resemble the random walk that the ball takes. If I look at the daily returns for the stock over a period of time and the distribution of those returns, I can see a normal distribution. This is also known as the bell curve.

This chart shows the distribution of daily returns for the following stocks/indices.

  • Nifty (index).
  • Bank Nifty (index)
  • TCS (large cap).
  • Cipla (large cap).
  • Kitex Garments (small caps)
  • Astral Poly (small caps)
    (image 3)
  • As you can see, the daily returns of stocks and indices follow a normal distribution.

    You're right, it's fair enough. But you might be curious as to why this is important and how it relates to Volatility. You will soon understand why I am talking about this.

    17.3 -Understanding the Normal Distribution

    The following discussion might be too overwhelming for someone who is just starting to explore the idea of normal distribution. Here's what I will do: I will explain normal distribution, link it to the Galton board experiment and then extrapolate it onto the stock market. This should help you get the gist of it.

    There are many distributions that data can be distributed, in addition to the Normal Distribution. Different data sets can be distributed in different statistical ways. Other data distribution patterns include uniform distribution, binomial distribution and poisson distribution. The normal distribution pattern is the most studied and well-understood of all the distributions.

    The normal distribution includes a number of characteristics that help us gain insight into the data set. Two numbers can describe the normal distribution curve: the distribution's average (or mean) and standard deviation.

    The central value that contains the maximum values is called the mean. This is the average distribution value. In the Galton board experiment, the mean bin is the one with the most balls.
    (image 4)

    If I were to count the bins starting from the left as 1, 2, 3,... all the way up to 9, (rightmost), then the 5 th bin marked by a red Arrow is the 'average bin. The average bin is used as a reference. Data is then spread on both sides of the average reference value. The standard deviation is the measure of how the data is distributed (or dispersion, as it is known). This is also the volatility in stock market context.

    This is important information. When someone refers to 'Standard Deviation' (SD), they mean the 1 SD. There are also 2 nd standards deviation (2SD), 3 rdstandard deviation (SD), etc. When I refer to SD, I mean the standard deviation value. 2SD refers to twice the SD value and 3SD refers to three times the SD Value.

    As an example, let's say that the SD for the Galton Board experiment is 1 and the average is 5. This is the result:

    • One SD would include bins between 4 th bin (5-1 ) and 6 th bins (5+1 ). This means that there is one bin to the left and one bin to its right.
    • 2 SD would include bins between 3 rdbin (5 - 2*1) or 7 thbin (5 – 2*1)
    • 3 SD would include bins between 2 nd bin (5- 3*1) or 8 th bin (5 + 3*1)

    Keep the above in mind. Here is the general theory of the normal distribution that you should be able to understand -

    • The 1 st default deviation can be used to observe 68% data
    • The 2 and standard deviations can be used to observe 95% of data
    • 99.7% can be observed within the 3 rd normal deviation

    This image will help you to visualize the above.
    (image 5)

    This is how you apply it to the Galton board experiment

    • We can see that 68% of the balls were collected within the 1 -st standard deviation, which is between 4 _ th to 6 _ th bin.
    • We can see that 95% of the balls collected within the 2 nd standard deviation, i.e. between 3 _rd to 7 th bin.
    • We can see that 99.7% of the balls collected within the 3 rd standard deviation, which is between 2 nd to 8 th bin.

    Let's say you are about drop a ball on Galton Board. Before doing so, we have a conversation.

    You I'm about drop a ball. Can you guess the bin it will end up in?

    Me -- No, it's not possible. Each ball is a random walk. But I can predict which bins it will fall into.

    You Can you predict the range of numbers?

    Me- The ball will most likely fall between the 4 th, and 6 _ thbin

    You. - How sure are you?

    Me I'm 68% certain that it would fall somewhere between the 4 th and 6 th bins

    You -- 68% accuracy is a little low, but can you estimate the range more accurately?

    Me I'm sure. The ball will likely fall between the 3 rdand 7 th bins, and I'm 95% certain about this. For a higher level of accuracy, I would say the ball will likely fall between the 3sup>rd and 7sup>th bins. I'm 99.5% certain about this.

    You -- Does that mean the ball is unlikely to land in either the 1 first or 10 second bins?

    Me - There is a possibility that the ball will fall in any of the bins other than the 3 RD SD bins, but it is very unlikely.

    You How low?

    Me - This is the chance of spotting a Black Swan'in a river. Probability-wise, it is less than 0.5%

    Tell me more about Black Swan

    Me Black Swan events, or as they are known, events that are low in probability (such as the ball hitting one of the 10 th bins) are called. Black swan events are not impossible to happen, and they have a non-zero probability. It is difficult to predict when and how often it will occur. The picture below shows the possibility of a black Swan event.
    (image 5)

    The above image shows many balls being dropped. However, only a few of them are collected at the extremities.

    17.4-View on Normal Distribution and stock returns

    The normal distribution should be familiar to you from the discussion that has been just begun. Because the daily returns of stock/indices form a bell curve, or normal distribution, the reason we're talking about normal distribution. Knowing the stock's mean and standard deviation will allow us to gain a better understanding of the stock returns and its dispersion. Let's take Nifty as an example and analyze it.

    Here is Nifty's daily return distribution: -
    (image 6)

    We can see that the daily returns are distributed in a normal way. This distribution has been calculated using the standard deviation and average (if you're unsure how to calculate it, please refer to the previous chapter). These values are necessary to calculate the log daily return.

    • Daily Average/Mean = 0.04%
    • Daily Standard Deviation/Volatility = 1.046%
    • Current market price for Nifty = 8337

    Note that an average of 0.04 percent indicates that daily returns for nifty are at 0.04%. Let's now put this information into perspective and calculate the following:

    • Nifty's likely trading range in the next year
    • Nifty's trading range for the next 30 Days.

    We will use 1 and 2, respectively, to calculate the above calculations with 68% confidence and 95% confidence.

    Solution 1 (Nifty's range of next 1 years)

    Average = 0.04%
    SD = 1.046%

    Let's convert this into annualized numbers.

    Average = 0.04*252 = 9,66%
    SD = 1.046% * Square (252) = 16.61%

    With 68% confidence, I can state that Nifty's value is likely to fall in the range of -68%.

    = Average + 1 SD (Upper Range), and Average – 1 SD (Lower Range).
    = 9.66% + 16.61% = 26.6%
    = 9.66% - 16.61% = = -6.95%

    These % are log percentages, as we calculated them on log daily returns. We need to convert them back to regular%. This will give us the range value (w.r.t. Nifty's CMP at 8337).

    Upper Range
    = 8337 *exponential (26.66%)

    For a lower range, -

    = 8337 * exponential (-6.95%)

    This calculation indicates that Nifty will trade between 7777 to 10841. This is how confident I feel about it. You know that I am 68% confident in this.

    Let's increase our confidence to 95%, or the 2 nd normal deviation, and see what results we get.

    Average + 2 SD (Upper Range), and Average -2 SD (Lower Range).
    = 9.66% + 2*16.61% = 42.7%
    = 9.66% - 2.161% = -23.56%

    The range is therefore -

    Upper Range
    = 8337 *exponential (42.87%)

    For a lower range, -

    = 8337 * exponential (-23.56%)

    With 95% confidence, Nifty will trade in the range 6587 to 12800 for the next year. As you can see, the range gets larger when we need higher accuracy.

    You might also consider doing the same exercise with 99.7% confidence, or with 3SD, and see what range numbers you get.

    Let's say you calculate the range of Nifty at 3SD Level and get the lower range value as Nifty 5000. (I'm just using this number as a placeholder number). Does this mean that Nifty can't go below 5000. It can, but it is unlikely to go below 5000. If it does drop below 5000, it could be called a black swan. The same argument can be extended to the higher end.

    Solution 2 (Nifty's range of next 30 days)

    We are familiar with the daily mean and SD.

    Average = 0.04%
    SD = 1.046%

    We are interested in the calculation of the range for the next 30 days. Therefore, we will need to convert it for the time period we desire.

    Average = 0.04% x 30 = 1.15%
    SD =
    1.046% * sqrt (30), = 5.73%

    With 68% confidence, I can tell you that Nifty's value over the next 30 day is likely to be around -

    = Average + 1 SD (Upper Range), and Average – 1 SD (Lower Range).
    = 1.15% + 5.73 = 6.88%
    = 1.15%-5.73% = - 4.58%

    These % are log percentages so we must convert them back into regular %. We can do this directly and get range value (w.r.t. to Nifty's CMP 8337).

    = 8337 *exponential (6.88%)

    For a lower range, -

    = 8337 * exponential (-4.58%)

    This calculation shows that Nifty will trade between 8930 to 7963 in the next 30 trading days with 68% confidence.

    Let's increase our confidence to 95%, or the 2 nd normal deviation, and see what results we get.

    Average + 2 SD (Upper Range), and Average -2 SD (Lower Range).
    = 1.15% + 2* 5.73% = 12.61%
    = 1.15% - 2* 5.73% = -10.31%

    The range is therefore -

    = 8337 *exponential (12.61%)
    9457 (Upper Range)

    For a lower range, -

    = 8337 * exponential (-10.31%)

    I trust you find the above calculations easy to understand.  

    You may be able to make a valid point at this stage. Normal distribution is fine. But how can I use the information to trade? As such, this chapter can accommodate many more concepts. We will therefore move the application section to the next chapter. The next chapter will focus on the application of standard deviation (volatility), and its relevance for trading. In the next chapter, we will be discussing two key topics: (1) How to choose strikes that can sell/write using normal distribution; (2) How to set stoploss using volatility.

    Remember, however, that eventually we will be discussing Vega and the effect it has on the options premium.


    1. Stock returns are unpredictable and highly unpredictable.
    2. The stock returns are distributed in a normal manner or very close to normal distribution
    3. Normal distributions have data centered around the mean. The standard deviation measures the dispersion.
    4. We can see 68% of the data within 1 SD
    5. We can see 95% of the data within 2 SD
    6. We can see 99.5% within 3 SD
    7. Black Swan events are events that occur outside of the 3 rd normal deviation.
    8. We can use the SD values to calculate the upper/lower value of stocks/indices