Per the suggestion of a reader (Thanks JB), today’s blog discusses a mathematical topic – in simple, layman’s terms, and how it relates to the world of option trading.
You might want to include “normal curve” and a brief description vis-a-vis its usage in the options world.
In our world, random variation tends to conform to a specific probability distribution – the ‘normal distribution.’ It’s known as the Gaussian distribution in the scientific community.
For readers interested in more details, try this entry from Wikipedia.
The shape of the graph resembles a bell. Thus, it’s often referred to as a bell curve.
The 68/95/99.7% Rule:
All normal density curves satisfy the following property (the Empirical Rule).
68% of the observations fall within 1 standard deviation (sd) of the mean*
95% of the observations fall within 2 standard deviations of the mean
99.7% of the observations fall within 3 standard deviations of the mean
Thus, for a normal distribution, almost all values lie within 3 standard deviations of the mean.
* Below, I’ve assumed the ‘mean’ is zero.
OK, what does all this have to do with options?
When buying or selling options, the investor must be concerned with how far the asset that underlies the option is likely move (in either direction) during the lifetime of the option. To determine that number, we must calculate the standard deviation. The math can be done using a simple calculator:
sd = S * V * SQRT (t/252)
S = Stock price in $/share
V = Implied volatility of underlying (as a decimal; i.e., 0.25 for a volatility of 25)
t = number of days until expiration arrives
NOTE: I prefer to use 252 as the number of trading days in one year. Others use 365, and that makes a significant (20%) difference in the sd.
The normal distribution tells us:
· Approximately 2/3 of the time, that asset moves less than 1.0 sd by the time the option expires
· The asset moves more than 2 sd only 5% of the times
· A huge move of 3 sd is expected only 3 times per 1,000 events. For a monthly stock or index move, that’s once every 28 years
This normal distribution tells us that events such as the massacre of 1987 should never occur [a decline of 20 sd (find the phrase ‘1987’ on the linked page)]. Some observers (Talib, Mandelbrot) believe that the normal curve does not truly represent events as we observe them, and explain that the ‘tails of the curve’ occur much more frequently than predicted by the normal distribution curve. The occurrence of these so-called ‘black swan’ events is important for those of us who trade options.
It tells us that selling far out of the money options is not as safe as predicted by the normal distribution curve and that the reward for selling such inexpensive options is not worth the risk. It also suggests that buying such options as insurance for a portfolio exposed to a large loss if that black swan event occurs, is probably worthwhile over the long-term.