Thursday, 19 June 2008

How to compute the historical volatility of index and equities?

In this post, I’ll divert from the discussion of the “Actual Operation of Warrant Trading” and talk about how various issuers of warrant compute the historical volatility of index and equities. In fact, what I am going to illustrate later can also be applied to the US market to compute the historical volatility of index and equities.

Some of my readers may be wondering why we should ever bother about computing the historical volatility since these events had already happened. I have the same thoughts too until my last WAT gathering when Greekman shared with us the expected move formula and it struck me that why did I not think of that? Let me go through the computation before we move on to make some modification to the formula and see where we can go from there.
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.

From the above screen captures, we can see that the historical volatility is around 15.01%. We are not concern about the different terms of warrant chosen as long as they all have the same underlying; their historical volatility should be approximately the same.

The screen capture below is a spreadsheet where I used to compute the historical volatility of STI index. Notice the value I got is quite close to the one shown on the previous screen captures.
Let walk through how each value is being computed. Under the column with the heading showing “Straits Time Index”, the value in each of the cell shows the closing STI index value on that day. Take note that we do not include weekend or any non trading day. For example, we do not include 19th May 2008 as it is Vesak day.

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.

I certainly hope you enjoy this as much as I do. I shall be posting the “Actual Operation of Warrant Trading (Part 4)” soon.


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