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Volatility is the estimated standard deviation (σ) of a price change in a security over the number of trial periods and represents the probability of price outcomes.

This is the most important variable in the BSM model, as all of the other four inputs are easily known at any point in time, while volatility is not. σ serves as a measure of the variability in a dataset, and when used in an options-related context, represents the probability of price outcomes of underlying asset prices. As stated earlier, the BSM model has some flawed assumptions on volatility regarding log-normality - I might address this later on, but to make it simple, let’s just let it fly.

Normal Distribution

Stats review time.

AKA the Gaussian Distribution if you want to be a snob about it

If an underlying asset has a 10% volatility and a current price of $100, then there is a 68% chance (with the normal distribution assumption) that the asset price at EOY will be in the range of 90-110.

A normal distribution has moments, which are a set of mathematical properties that describe the shape, location, and spread of a normally distributed random variable.

1. Mean (μ) represents the center or average value of the distribution.

2. Variance (σ^2) gauges the dispersion or spread of data points. Smaller variance values indicate that data points cluster closely around the mean, while larger variances signify greater dispersion.

   • The standard deviation (σ) is the square root of the variance and is often used as a more interpretable measure of spread.

3. Skewness characterizes the asymmetry of the distribution.

   • A positive skewness indicates that the distribution is skewed to the right (tail on the right side), meaning there are more data points on the left side of the mean.

   • A negative skewness indicates that the distribution is skewed to the left (tail on the left side), with more data points on the right side of the mean.

   • A perfectly symmetric distribution (like the standard normal distribution) has skewness equal to 0.

     

4. Kurtosis measures the "tailedness" of the distribution, i.e., how heavy or light the tails of the distribution are compared to a normal distribution.

   • Positive kurtosis (leptokurtic) has heavier tails and is more peaked at the center compared to a normal distribution.
   • Negative kurtosis (platykurtic) has lighter tails and is flatter at the center compared to a normal distribution.

   

   Source: PinkMonkey.com

Standard Deviation

In the BSM model, volatility simply refers to the standard deviation of asset prices around their mean during a one-year per

$$ \sigma = \frac{\sqrt{\Sigma(x-\mu)^2}}{N} $$

$$ \\x = \text{Price}\\\mu= \text{Mean Price}\\N=\text{Number of Trials} $$