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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?
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