## futures convexity correction

The futures convexity correction is a number c used in the calculation of the theoretical quote of a futures contract FUT(T₁,T₂) as implied from a given yield curve, as explained further below
Here T₁ , T₂ are the expiry and underlying rate maturity of the futures contract respectively.
Typically, for Eurodollar futures T₂ = T₁ + 3 months.
In practice, during yield curve construction out of futures quotes, to each futures quote Q one may associate a futures convexity correction amount of c, with the following interpretation:
Q implies a so-called futures ratef (see below for explanation) defined by the formula Q = 100(1 - f)
On the other hand, the curve-implied forward rate r (see below for definition) is related to f through the formula f = r + c
r can be seen as the fair rate (strike) associated with a ficticious forward contract FWD(T₁,T₂) that spans the exact same times T₁ , T₂ as the futures contract FUT(T₁,T₂)
Note c is supplied in the same units as any other rate. For example, a futures convexity correction of 1 bp should be entered as 0.0001.
From Q = 100(1 - r - c) it follows Q + 100c = 100(1 - r), which means that the same curve-implied forward rate can be associated with the (quote , convexity) pair (Q , c) or (Q + 100c , 0)
For example, the following two specifications are fully equivalent:
A) Quote = 98 and Convexity Correction = 0.0001
B) Quote = 98.01 and Convexity Correction = 0

The curve-implied forward rate r with respect to the times T₁ and T₂ is defined by the formula r = (DF₁/DF₂ - 1)/(T₂ - T₁)
But the corresponding futures rate f that references the same times T₁ and T₂, will generally differ from r due to the fact that f is defined through Q = 100(1 - f), where the futures price Q is sensitive on the daily cash flows due to the mark to market of the futures margin account.
The difference f - r is exactly how the futures convexity correction is defined.
In the particular case of Eurodollar futures, Bloomberg calculates and displays a convexity correction, apparently based on a formula derived by Kiriko and Novak in a '97 Risk Magazine article titled Convexity Conundrums, as described at
here
The Risk article and some related information at
here