Constant maturity futures price methodology

Question

What is the correct methodology to compute constant maturity futures price.

I've met in some papers that do the following. To create constant maturity synthetic futures prices with maturity $m = 30, 60,...,180$ days. We should take a pair of futures that straddle the chosen maturity $m$ with maturities $s<m<l$ measured in days until expiration.

Then the price is derived using the following formula: $$ p_m = \alpha p_s + (1-\alpha)p_l,\, \alpha = \frac{l-m}{l-s}$$

1. Should the maturity be rounded to days?
2. What happens when shorter futures comes closer to expiration. On which date and how we roll over the pair? Is it recommended to roll over futures several days before expiration. In this case we should have negative $\alpha$.
3. What are the limitations of this methodology? What are general assumptions?
4. We can take only daily closing prices or we can use more frequent data?

Answer 1

What you are doing in the formula is just linear interpolation. This is probably fine, if there was some hindrance such as a seasonality effect, the market would probably contain a contract at that tenor.

1. Just convert to fractions of a year by some day count convention. Say ACT/365
2. I don't see why. The futures price should just approach the spot.
3. It it is very crude, and does not take any financial assumptions or market specific characteristics into consideration. Limitation would be if there's nothing traded at 4.3M but every institution in the world knows that something significant happens at that time which they account for, but which you completely ignore in your simple interpolation.
4. Use any frequency you like. Depends on your application. For risk management purposes, probably you are looking at daily data and create daily closing prices of your constant maturity futures nodes.