# Convexity adjustment for futures/FRA under T+D measure

In an internal document in my company, the convexity adjustment for Futures is defined as:

$ConvAdj&space;=&space;E^Q(L(T,T,T+D))&space;-&space;E^{Q^{T+D}}(L(T,T,T+D))=&space;&space;E^{Q^{T+D}}(L(T,T,T+D)*(B_{T+D}*P(0,T+D)-1))$

where $B_{t}&space;=&space;&space;exp(\int_{0}^{t}r_s*ds)$ and P(0,T+D) is the ZC bond maturity at T+D.

I don't understand why $B_{T+D}*P(0,T+D)$ is not equal to 1 as I thought they were the same except B has a positive sign in the integral while P has a negative sign.

• I think it’s because $B_{T+D}$ is a random variable whereas $P(0,T+D)$ is known at time 0.
– dm63
Commented Sep 9, 2022 at 3:39

I think this can't hold unless interest $$B_{T+D}$$ is deterministic. Here's why:
You're imlpying that $$E^Q[L(T,T,T+D)] = B_{T+D}*P(0,T+D)*E^{Q^{T+D}}[L(T,T,T+D)]$$ Because all the terms in the equation are deterministic except $$B_{T+D}$$, this term itself must be deterministic.
In the case $$B_{T+D}$$ is deterministic we have the following relationship $$P(0,T+D) = E^Q[\frac{1}{B_{T+D}}] = \frac{1}{B_{T+D}}$$