Volatility and Normal Distribution

From Volatility to Price Range

In the previous chapters, we learned that volatility measures the amount of variation in an asset's price.

But knowing the volatility percentage is only the beginning.

A more useful question is:

"Given this volatility, what range could the asset trade within?"

For example, if an index has an annualised volatility of 16%, we can use that information to estimate a possible range for its future price.

However, this range is not a prediction of the exact price.

It represents a range of possible outcomes with a certain level of probability.

Understanding Random Walk

Stock prices do not move in a perfectly predictable pattern.

One way to understand this is through the idea of a Random Walk.

Imagine dropping a ball through a board filled with pins.

At every pin, the ball can move either left or right.

You cannot know the exact path the ball will take.

If you drop many balls, however, a pattern begins to appear in the way they are distributed.

Financial prices behave in a somewhat similar way.

Individual price movements are difficult to predict precisely, but a large number of observations can reveal statistical patterns.

This is the foundation for using probability and statistics to understand market movements.

What is Normal Distribution?

When a large number of observations are plotted, they can often form a pattern resembling a bell-shaped curve.

This is called a Normal Distribution.

In a normal distribution:

• Most observations are concentrated around the average.
• Fewer observations occur as we move away from the average.
• Very extreme observations occur less frequently.

For market returns, the distribution is often treated as approximately normal for basic statistical analysis, although actual markets can behave differently and may experience extreme movements more often than a perfect normal distribution suggests.

Mean and Standard Deviation

Two important concepts help us understand a normal distribution:

Mean

The Mean represents the average of the observations.

It gives us the centre of the distribution.

Standard Deviation

Standard Deviation (SD) measures how widely the observations are spread around the mean.

A higher SD means greater dispersion.

A lower SD means the observations are more closely grouped around the average.

In market terms:

Higher SD → Higher Volatility → Greater Price Range

Lower SD → Lower Volatility → Smaller Price Range

The 68–95–99.7 Rule

A normal distribution gives us a useful rule of thumb.

Within 1 Standard Deviation

Around 68% of observations fall within one standard deviation of the mean.

Within 2 Standard Deviations

Around 95% of observations fall within two standard deviations.

Within 3 Standard Deviations

Around 99.7% of observations fall within three standard deviations.

This helps us estimate the probability of an asset remaining within a particular range.

What Happens Beyond 3 Standard Deviations?

An event occurring beyond the 3 Standard Deviation range is statistically unusual.

Such extreme events are often referred to as Black Swan events in market discussions.

However, this does not mean that such events are impossible.

Markets can and do experience movements much larger than normal statistical expectations.

Therefore, a calculated range should never be treated as a guaranteed upper or lower limit.

Using Volatility to Estimate a Price Range

Suppose:

• Current price = ₹1,000
• Annual volatility = 20%

A simplified one-year range based on one standard deviation would be:

Lower Range

₹1,000 − 20% = ₹800

Upper Range

₹1,000 + 20% = ₹1,200

This suggests that, under the assumptions of the model, the price could fall within approximately ₹800–₹1,200 over the period represented by the volatility.

The important point is that this is a probability-based range, not a guaranteed forecast.

Confidence Levels Matter

The range changes depending on how much confidence we want.

1 Standard Deviation

Approximately 68% confidence

The range is relatively narrow.

2 Standard Deviations

Approximately 95% confidence

The range becomes wider.

3 Standard Deviations

Approximately 99.7% confidence

The range becomes much wider.

Therefore:

Higher Confidence → Wider Expected Range

This is an important concept when using volatility for trading decisions.

Estimating Range for Multiple Days

Annual volatility can also be converted into volatility for a shorter period.

The general relationship is:

Period Volatility = Annual Volatility × √(Number of Days / 365)

For example, if annual volatility is 20%, the expected volatility for a shorter period will be lower because the time period is shorter.

This allows traders to estimate possible price ranges for:

• A few days
• A week
• A month
• Other specific time periods

The source material uses this approach to estimate possible ranges over 30 days.

Why This Matters in Options Trading

This concept becomes particularly useful when selecting option Strike Prices.

Suppose statistical analysis suggests that an underlying is likely to remain within a particular range with approximately 68% probability.

A trader can compare that range with available Strike Prices.

This can help answer questions such as:

• Which strikes are relatively far from the current price?
• Which options may have a lower probability of finishing ITM?
• How much premium is available at different Strike Prices?
• What level of risk is being taken for that premium?

The next step is to apply this information practically while selecting option strikes.

Practical Insight

Volatility does not tell you exactly where the market will go.

Instead, it helps you estimate how widely the market could move.

This distinction is important:

Direction = Where the market may go

Volatility = How much the market may move

Common Beginner Mistake

A common mistake is treating a volatility-based range as a guaranteed prediction.

For example, if a calculation gives a 1-standard-deviation range, it does not mean the asset cannot move outside that range.

It only indicates that the range covers approximately 68% of outcomes under the assumptions being used.

Extreme market movements can occur.

Key Insight

Volatility tells us about the size of possible price movements, while normal distribution helps us assign probabilities to different ranges.

Together, they provide a basic quantitative framework for thinking about market risk and option Strike Prices.

Key Takeaways

• Price movements are difficult to predict precisely and can be viewed as a form of Random Walk.
• Normal Distribution provides a statistical framework for studying the distribution of returns.
• Mean represents the centre or average of the observations.
• Standard Deviation measures the dispersion around the mean.
• Higher Standard Deviation generally means higher volatility and a wider expected range.
• Approximately 68% of observations fall within 1 SD.
• Approximately 95% fall within 2 SD.
• Approximately 99.7% fall within 3 SD.
• Events beyond 3 SD are statistically unusual and are often called Black Swan events.
• Volatility can be used to estimate possible price ranges over different time periods.
• A higher confidence level results in a wider estimated range.
• Volatility-based ranges are probability estimates, not guaranteed price predictions.