0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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

0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.