When Futures and FRA Convexity Adjustments Disappear

Summary

The document examines a claimed relationship between a futures rate and a forward rate agreement under a T-plus-D measure. The questioner expects the convexity adjustment to equal one, reasoning that the bank-account accumulation and zero-coupon bond terms appear to have opposite signs in their integrals. The included answer challenges that equality when the accumulation factor at the payment date is stochastic.

The argument writes the expected payoff under the money-market measure in terms of the terminal bank account, a zero-coupon bond price, and an expectation under the corresponding forward measure. It concludes that the proposed identity would require the bank account term to be deterministic. In that deterministic case, the bond price is the reciprocal of the bank account and the intuition about equality holds. The post gives only a short conditional argument; it does not define all notation, derive a general convexity adjustment, or discuss stochastic-rate models in detail.

Key ideas

• The proposed equality between futures and FRA quantities depends on whether the terminal bank account is deterministic.
• A stochastic bank account prevents the expectation from being simplified by treating that factor as a constant.
• Under deterministic interest accumulation, the zero-coupon bond price equals the reciprocal of the bank account.
• The answer supports the equality intuition only in the deterministic case and does not derive a general adjustment.

Tags

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


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












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

where and P(0,T+D) is the ZC bond maturity at T+D.

I don't understand why 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.

## Answer by Xman (score 0, accepted)

https://quant.stackexchange.com/a/72164

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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