## futures convexity correction

The

**is a number**

*futures convexity correction***used in the calculation of the theoretical quote of a futures contract**

*c***as implied from a given yield curve, as explained further below**

*FUT(T₁,T₂)*Here

**are the expiry and underlying rate maturity of the futures contract respectively.**

*T₁ , T₂*Typically, for Eurodollar futures

**=**

*T₂***+ 3 months.**

*T₁*In practice, during yield curve construction out of futures quotes, to each futures quote

**one may associate a**

*Q***amount of**

*futures convexity correction***, with the following interpretation:**

*c***implies a so-called**

*Q*

*futures rate***(see below for explanation) defined by the formula**

*f*

*Q = 100(1 - f)*On the other hand, the curve-implied forward rate

**(see below for definition) is related to**

*r***through the formula**

*f*

*f = r + c***can be seen as the fair rate (strike) associated with a ficticious forward contract**

*r***that spans the exact same times**

*FWD(T₁,T₂)***as the futures contract**

*T₁ , T₂*

*FUT(T₁,T₂)*Note

**is supplied in the same units as any other rate. For example, a**

*c***of 1 bp should be entered as 0.0001.**

*futures convexity correction*From

**it follows**

*Q = 100(1 - r - c)***, which means that the same curve-implied forward rate can be associated with the (quote , convexity) pair**

*Q + 100c = 100(1 - r)***or**

*(Q , c)*

*(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

**with respect to the times**

*r***and**

*T₁***is defined by the formula**

*T₂*

*r = (DF₁/DF₂ - 1)/(T₂ - T₁)*But the corresponding futures rate

**that references the same times**

*f***and**

*T₁***, will generally differ from**

*T₂***due to the fact that**

*r***is defined through**

*f***, where the futures price**

*Q = 100(1 - f)***is sensitive on the daily cash flows due to the mark to market of the futures margin account.**

*Q*The difference

**is exactly how the**

*f - r***is defined.**

*futures convexity correction*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