Here is the method for estimating future volatility of a stock. Remember though that the method is not perfect, but by the law of averages, in the long run, it is a good estimate. The estimation of future volatility is essential for the Black and Scholes model
Computing Historical Volatility:
Suppose we want to compute the historical volatility of the last 10 trading days in the S&P 500. We first need to compute the log of back to back returns. Then we can sum the square log of the returns with this formula
Where . Better known as the average
Here is an hypothetical example of the last 5 trading days closing prices:
Day 1 = 100, Day 2 =100.8, Day 3 =100.3, Day 4 = 100.2, Day 5= 100.03
Log of returns:
0.007861, -0.004972,-0.001,-0.001698
Average:
The Volatility
To get the annualized volatility estimation, just multiply by the square of 248, which are the actual trading days excluding the weekend. So in our case
If you are sampling the weekly volatilities, make sure to mulitply by . Scale accordingly to the sampling you make!
Common Erros In Computation
The most common error is not scaling for the time. If you take the annualized volatility, make sure to also take the annualized continously compounded interest rate. For example, suppose we know that the monthly interest rate is 0.003%, the volatility 15%, and the time to expiration to be 2 months. We can just plug in these in the excel file and put in the time to expiration, 2 months. This is because each unit is 1 month. If we wanted to scale it to 1 year, we would have to scale the variables in this following way:
Where is the new time to expiration
What volatility should you take
Should you sample the last 10 days? 20 days? There is no straight answer. The disagreements create the market.
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Great blog.
I enjoy all the posts.
It would be nice if you were to elaborate on the “10 mistakes I make” and offer a more proper method.
Thanks again.
TimK
I love reading your blog, it is a great help to my learning process. I was just trying to work through the about calculations, and I noticed a typo? the log (dayi/dayi-1) should perhaps read ln instead of log
Yes.. I didn’t write ln on purpose because I thought some people might not know what it is..Most of the people when they see log they assume it’s ln :) But you are right.. the meanin is for the natural log.