Constant maturity indices

Why a perpetual on a fixed contract is ill posed.

A perpetual future has no expiry and is held to its index by periodic funding. Applying that construction to a single option contract fails, and not for an implementation reason.

Why a fixed contract cannot carry a perpetual

Let the index be the price of a contract expiring at T₀. As t → T₀ the index tends to the terminal payoff and time value tends to zero.

The index therefore carries a deterministic drift. Funding would have to offset theta in perpetuity, meaning the funding rate would be dominated by a known quantity rather than by the imbalance it exists to price. Past T₀ the index is undefined. The instrument is ill posed, not merely awkward.

Holding the tenor fixed

Fix τ instead of fixing the contract, and re-derive the index from the current surface at every observation.

Since τ does not depend on t, there is no theta term. The index is a level in volatility points, not a decaying price. The same construction supports a rolling fixed delta call or an at the money straddle, quoted in currency.