The Black-Scholes formula for estimating the value of a stock option is rather elegant. It estimates the value of the stock option by assuming a particular random distribution of future stock price movements, and averaging over all of these to come up with the current value of a stock option as an expectation value over a random variable. A second way of deriving the Black-Scholes formula is to come up with a strategy for fully hedging an options position by buying and/or selling short stock, in which case the Black-Scholes price of the option is the price at which there are no arbitrage gains to be had by trading options against stock positions.
Underlying the calculation of the Black-Scholes price is the assumption that price variations are log-normally distributed. That is to say, if P2 is the price of the stock at time t2, and P1 is the price of the stock at time t1 = t2 - dt, then for all different values of t2, the variable x = log( P2/P1 ) is a random variable with normal (or Gaussian) distribution.
In 2005 I analyzed stock prices for a real stock to see if they did fit a log-normal distribution after all. I took the daily closing prices for a tech company stock for a 14 year period.
|14 years of closing prices|
|Price ratios for 1 day spacing between prices|
Finally, we will show plots like this with a logarithmic y-axis so that tail behavior can be seen clearly. The daily volatility plot above then looks like this:
|Daily Volatility with y-axis on logarithmic scale|
|Price Ratios with about 1 Month between Prices|
|About half a year between prices|
|Price Ratios with about 1 year between prices|