Futures are marked to market daily while forwards settle once. For an asset whose price is positively correlated with the short rate, which contract is priced higher, and how large is the difference on a one year contract with correlation 0.3, asset volatility 20% and absolute rate volatility 1%?

Futures are marked to market daily while forwards settle once. For an asset whose price is positively correlated with the short rate, which contract is priced higher, and how large is the difference on a one year contract with correlation 0.3, asset volatility 20% and absolute rate volatility 1%?

Approach: Trace the cash flows of daily settlement against the rate at which variation margin is financed, then apply the standard approximation for the correction in terms of the covariance and maturity.

The futures price is higher, by about 3 basis points. Daily settlement pays variation margin into an account earning the short rate, so a long futures position receives cash exactly when the asset has risen, which under positive correlation is when the rate being earned is high, and it funds losses when rates are low. That asymmetry is worth money, so the futures price has to exceed the forward. The convexity correction is approximately rho * sigma_S * sigma_r * T^2/2, which here is 0.3 * 0.20 * 0.01 * 0.5, or 0.0003 of the price and 3 basis points. For an equity index over one year that sits inside the bid-ask spread, while for contracts written on the rate itself the same covariance term is large enough that traders quote an explicit convexity adjustment.

Follow-up: For a contract on the short rate itself, what is the sign of the convexity adjustment and why does it grow with the square of maturity?

Key concepts: convexity correction, daily settlement, variation margin, covariance, futures forward difference.