How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega

Chapter 2The Normal Probability Distribution

The “Bell” curve or Gaussian distribution, called the normal standard distribution, displays how data/observations will be distributed in a specific range with a certain probability. Think of the height of a population; let's assume a group of people where 95% of all the persons are between 1.10 m and 1.90 m, implying a mean of 1.50 m . Looking at Chart 2.1, one can see that 95% of the observations are within 2 standard deviations on either side of the mean (on the chart at 0.00), totalling 4 standard deviations. So 0.80 m (the difference between 1.90 and 1.10) represents 4 standard deviations, resulting in a standard deviation of 0.20 m.

Chart 2.1 Normal probability distribution

With a mean of 1.50 m and a standard deviation of 0.20 m one could say that there is a likelihood of 68% for the people to have a height between 1.30 and 1.70 m, a high likelihood of 95% for people to have a height between 1.10 and 1.90 m and almost certainty, around 99.7%, for people to have a height between 0.90 and 2.10 m. Or to say it differently; hardly any person is taller than 2.10 m or smaller than 0.90 m.

STANDARD DEVIATION IN A FINANCIAL MARKET

The same could be applied to the daily returns of a Future in a financial market. According to its volatility ...