Is there a way to compute the volatility of index future based on the underlying index? Is there a website to find the historical volatility of index future?

You want historical volatility or historical implied volatility? If it is the former then just take the prices, stick them in excel and calculate standard deviation of daily log-returns. Then multiply by the square root of 252 to get annualised volatility.

I want historical volatility of index future. Say for example, to compute s&p dec future volatility, you need a set of dec future price, but unlike the near-term future, they are not actively traded, and i don't think it gives you a good estimate of the volatility of the index future when it is close to expiration.

I don't see why you can't use the underlying index to calculate the volatility. The two are closely correlated anyway, true there's basis risk, but it's not that great.

I think we can. I believe the index future volatility should be: = sigma * exp(rt) based on the theoretical value of future price. I just want someone to verify it.

Why do you think the futures have a different volatility than the index? I suppose, on a per-minute basis, they might, but any difference in the daily (or longer) volatility would only have to do with procedural/time differences in the way the closing prices are reported, or singular incidents that cause the FV calculation to change, like a huge, sudden interest rate move. Even then, there has to be a long time to expiration, and the move has to be truly huge to make any significant difference, and it will affect just that one day.

First, I read a lot of academic books (e.g. Future, Options & Swaps from Robert Kolb) that Beta of index future is greater than 1. Second, the fair future value is F = S Exp(rt). It seems that volatility of F should be related to the volatility of S by the same factor ( exp(rt) ). However, t decays as well. t decreases by 1 everyday. So the upper bound of volatility of F = exp(rt) * volatility of S. For small r, exp(rt) is close to 1, and so they are very close. However, with larger t and increasing interest rate, I like to get the exact formula, and I just wonder if anyone knows.

Why do you think the volatility of futures is related to volatility of spot by the same factor exp(rt)? Volatility is calculated based on natural log returns, i.e. ln(S2/S1), so if we substitute S with F we get ln(S2*exp(rt)/S1*exp(rt-1)). So the difference between the spot volatility and futures volatility is that "one day" factor. P.S. Why don't you just get a continuous contract price data and calculate your volatility from there!? Granted, it's not 100% accurate, but so what, historical volatility is just that historical volatility.

MTE, I never traded future, nor future options. I start looking into the possibility of trading future options b/c of higher leverage. My question might be very naive. I just wonder how most traders switch from index option to future option, and how do they compute volatility. Do you trade future options?

No, I don't, yet. I don't use historical volatility though, so I may not have the same problem. On the other hand, my charting software calculates volatility so no need to know the exact formula.