Just occasionally I throw in a bit of finance into these pages to keep my more technically inclined readers happy. This offering is to help clear up the mystery of how to estimate volatility into the option pricing model.
The Black, Scholes and Merton model for pricing options assume that the underlying asset (share price, rate of exchange, NPV) is continuously traded and follows a random process through time along a rising trend of know drift and constant variance. This price process is known as geometric brownian motion and it results in a log-normal distribution of values. The random process is assumed to be continuous and if we take price changes over (theoretically, infinitely) short periods of time the distribution of price changes is normal.
So in order to estimate volatility from past data we need a statistically significant number of price readings (I use 101 daily price observations) and then take a daily return (P1/P-1) -1 on each. If the natural log of this number is taken we have an approximation of the continuously generated daily return. The standard deviation of these 100 returns is the crude historical volatility. A better measure can be developed whereby the volatilities are weighted using a sum of the day's digits approach and a weighted standard deviation calculated. This weighted volatility appears in most cases to give better than 2% accuracy on most actively traded near the money options. More sophisticated techniques are available (GARCH 2 for example) but do not in my experience give notably better results.