I am using the Straits Time Index (STI) as an example here to compute the historical volatility. I have done some screen captures from different issuers to show what the historical volatility of STI is after market closed today.
Under the column with the heading showing “Percentage Change”, the value in each cell is computed based on the natural logarithm of the prior day closing index and today closing index. Take for example, the percentage change on 18th June 2008 is computed as follow, Ln(3040.09/3028.24) = 0.39%.
Once we have all the various percentage change calculated for the last 30 days, not including the one on 19th June 2008, since this is a historical volatility. To compute the historical volatility for last 30 days, we first find the standard deviation of the percentage change from 8th May 2008 to 18th June 2008 and then multiply it with the square root of 250 or 252; the number of trading days in a year. That is Stdev(-1.78%, -0.31%, 0.57% ….-0.29%, 0.39%)*Sqrt(250) = 15.02%. The reason why we multiply the standard deviation with the square root of the number of trading days is to annualize the historical volatility.
Based on the theory of statistic, if the sample is 30 and above, we can assume the distribution to be normal. This is why I have chosen 30 days and nothing less. Assuming if the STI index does follow the normal distribution, then there is a 68% of the time the STI index will fall within 2992.66 * (1 ± 15.02% * Sqrt (30/250)), which gives us a range of 2836.95 to 3148.37. We multiply the historical volatility with the Sqrt(30/250) to de-annualize it. For those who attended the last WAT gathering, do you find this formula familiar? What happen if we substitute the 2992.66 with half the current equity share price, the 15.02% with the ATM option implied volatility and the 30 with the days to expiration of the option? Effectively, this gives us an idea of how much the share price will move towards the expiration date.