Generally speaking there are more inputs that are required to precisely specify the multicurve structure, and they are potentially more important.
For example consider constructing a EUR interest rate curveset for 3 years, in the indexes EONIA, 3M EURIOBOR, 6M EURIBOR.
The information you have available are:
- Some outright EONIA quotes in generic tenors; 1M, 2M, 3M, 6M, 12M,
- Some EONIA/3M basis prices; 1Y, 2Y, 3Y, and some sporadic IMM basis instruments,
- The 2Y and 3Y 6M-Swap prices,
- The 2Y and 3Y 6s3s IBOR basis prices.
- The Euribor futures (3M) quotes (and potentially some marketable convexity quotes)
You broadly recognise there is a general quadratic shape to convexity.
Your task is to produce a curveset that minimises error to all of the given prices that you are privy to knowing. This is not a science but an art, since the variability in all the prices, the liquidity and the ability to 'discover the outlier' is needed.
For example suppose that the Euribor strip is well bid (implying lower overall yields), but the 2Y 6M-Swap was well bid (implying higher rates) then you might expect that convexities need to tighten (go closer to zero) but if the 6s3s basis is well bid also then adjusting that might correct your curveset, without any thought of convexity.
Alternatively you might find that the illiquidity of some of these specific products has systematic arbitrage inescapable - you simply cannot find a curveset that simultaneously falls inside market bid-offers. But still the prices are not immediately arbitraged away due to transaction costs of various kinds.
Practitioners operate on the basis on having convexities set and unchanged from day to day. But convexities are updated when sporadic market trades are reported or there is strong price evidence to suggest that they are in demand/supply.
Whilst the theoretic models will describe futures convexity as being dependent upon volatility and correlation between basis (OIS/IBOR) and curve structure, this describes a relatively small window of movement. I have personally seen convexity prices move significantly based on completely exogenous factors. Such as trading a large amount of futures on one exchange (EUREX) versus a large amount of swaps on another exchange (LIFFE) to mitagate maintenance margin in either venue. The on-going capital savings will dwarf the one or two basis points of convexity that the trade costs on entry by a skewed convexity market.