Standard deviation is key to predicting price volatility

Volatility is the most important price driver of option premiums.

We are interested in future volatility. However, this is the only kind of volatility that we cannot know. We are able to calculate historical volatility, but is this a good bias for future volatility?

Every option pricing model tries to evaluate options by ascribing probabilities to several different possible prices of the underlying value at expiry.

Because the distribution of prices occurs in the future, and every underlying value has its own characteristics, there is no clear answer to the question of how probabilities must be allocated. But, as an approximation, most models (some with adjustments) start with the assumption of a normal distribution.

A normal distribution curve is always defined by two things: the average or mean (reflected by the spike in the figure below) and the standard-deviation (the speed of expansion of the curve).

The standard deviation can also be interpreted in terms of a probability of an occurrence. Once you know a certain mean and standard deviation, it is always possible to calculate the probability of an occurrence within a certain range of the mean.

Usually you need a table of standard deviations (SD) to calculate exactly. However, option-traders use the following approximations:
• Plus or minus 1 SD of the mean includes 68.3% (approximately 2/3) of all possible results.
• Plus or minus 2 SD of the mean includes 95.4% (around 19/20) of all possible results.
• Plus or minus 3 SD of the mean includes 99.7% (roughly 369/370) of all possible results.

In other words, we can expect a result that ends further away from the mean in:
• 1 SD in 1 out of 3 occurrences
• 2 SD in 1 out of 20 occurrences
• 3 SD in 1 out of 370 occurrences

The value of an option does not only depend on the probability that an option will expire on-the-money, but also on the height, amount or size at which it expires.

When we assume that possible prices of the underlying value have a normal distribution, the chance for an option to expire deep-in-the-money is much higher when the distribution has a high standard deviation. Therefore, options will be more expensive when prices of underlying values fluctuate a lot. In the same way, options will get cheaper when the market is calm and does not move much.

When we assume that a market is complete and perfect, and there is no change to arbitrage, the mean must be equal to the 'forward price' of the underlying value. The volatility number that we use for the UV is about 1 SD price change, in a percentage, for a period of one year.

Example
When a stock has a forward price (in 1 year from now) of 120.00 and a volatility of 20% then, the SD is 24.00 (20% * 120.00). This means that we can expect that the share in 1 year -2 out of 3 times- will be trading somewhere between 96.00 and 14,00 (120.00 plus or minus 24.00). 19 out of 20 times a price between 62.00 and 168.00 can be expected (120.00 + or - [2*24.00]). And a price between 38.00 and 192.00 (120 + of - [3*24]) can be expected in 369 out of 370 occurrences.