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

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

where B_{t} =  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.

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

1 Answer 1


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 $$P(0,T+D) = E^Q[\frac{1}{B_{T+D}}] = \frac{1}{B_{T+D}}$$

In this case your intuition is correct indeed.


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