The sixth and final input into the Option Pricing Model is volatility.
It is my observation that there is often a bit of confusion about this term. If you listen to any of the financial media, volatility is only ever mentioned when the market is going down. To be sure, a 400-point down day on the Dow is a volatile move, but a 400-point up day is never described as volatile, yet it is equally so.
In the simplest terms, volatility is the relative rate at which the price of a security moves up and down. Market technicians have various methods of measuring volatility, using a variety of formulae, but our option pricing model requires a particular measure of volatility; the annualized standard deviation of logarithmic daily change in price.
Now that's a mouthful, and most option traders view volatility in relative terms without understanding the calculation, but I think it helps to actually understand the mathematics behind it. We can do this with Excel or charting software, which I will give an example of, but let's do it in English first
We start of by calculating for each day's data, today's closing price divided by yesterday's closing price. This will return a number that is today's price as a proportion of yesterday's price. If there is no change, the number will be 1.0, if it is up 2% it will return 1.02, if it is down 5% it will return 0.95 and so on.
The next stage is to find the natural logarithm of the above. This is to reflect the lognormal distribution of stock market returns. Next, multiply this by 100 to express it as a percentage. We can plot this as a scatter chart, which will show the lognormal daily move as a percentage
The next step is to calculate the standard deviation of the above. Normally this is calculated over the last 20 or 30 days of data; it can be any length, but for this example we will use 20 days. This gives us the standard deviation of logarithmic daily change in price, which can be plotted on a chart to see changes in volatility as time goes by. However, Option Pricing Models require that volatility is expressed as an annualized percentage and we do this by multiplying by the square root of the total number of trading days in a year, which is the square root of 252.
This is now the finished volatility calculation, which is called "Historical" or "Statistical" volatility, plotted in the chart below"
This equation can be plotted in charting software to show current and past historical volatility. In Metastock or Amibroker language, (the two platforms I am familiar with) it can be plotted by using the following formula:
The above formula calculates historical volatility based on he last 20 days, the figure in red. Any look-back period can be used and some option traders use various length.
So now we can enter this volatility figure into our Option Pricing Model to get an accurate option price; or can we?
The historical volatility number, depending on the look-back period can vary enormously, and as the name implies, looks at past data, whereas what we really want to know as option traders is what volatility will be in the time left until the option expires. As this cannot be known, this forces the option trader to make a volatility forecast, or at least an idea of where volatility might be relative to the present in order to calculate his or her idea of fair value. This where historical volatility can be used as a tool, but the trader must look forward.
Often the market will disagree with you, which I will discuss in the next section.