# Convexity in interest rate curve bootstrapping

Whether the bootstrapping is a multicurve one or not, one can use futures quotes. One link these quotes to corresponding (synthetic) forwards (that can be expressed as known functions of zero-coupons) plus convexity.

It is often said that this convexity is negligible. If one still wants to take it precisely into account, one has to use a model (often Hull-White) that will have to be calibrated on ... a spot discount curve at least (in a non multicurve setting) as well as on a spot "forward" curve (in the multicurve setting), curve(s) that one is constructing : it is a vicious circle.

How to do a simultaneous boostrap and calibration of model used to compute the convexity ?

Is there a kind of iterative procedure to put in place to handle this ? (How do practitioners do if they want to take convexity into account ? (In fact I know how they do, but I don't want to do it this way, I want a joint bootstrap and calibration of model used to calculate the convexity.))

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.

• "Your task is to produce a curveset that minimises error to all of the given prices that you are privy to knowing" --> yes but no : I don't want a best fit (I want to exactly reprice the instruments used to bootstrap the curve), I want an exact fit. Regarding what you wrote, I knew what practitioners do, but I am more interested in what is done if one wants to do a joint boostrap with model used for convexity calibration. Jun 13, 2019 at 7:07
• Ok then I'm afraid I misunderstand your question; firstly bootstrap is no longer employed, a non-linear solver is used to account generalised complexities of curveset construction. Secondly, a non-linear solver uses the concept of a loss function or objective function which is the error to the supplied inputs. In the case your solver is underconstrained it will have zero error and solve to all prices simultaneously, in the case it is over-constrained it won't. Either way you have the problem of deriving inputs, which might be easy: transparent mid markets or subjective/model based assessments.
– Attack68
Jun 13, 2019 at 7:16
• I am sorry I wasn't clear, plus I have the bad habit of using the verb bootstrap in the broader meaning "finding Zero-coupons/rates such that", when usually it means "solving explicitely, forward or not". Ok, so at the end, what is done is A) an optimization (non-linear solver) with targets the prices/quotes of 1) instruments used to calibrate the OIS + IBOR of various tenors + basis curves 2) swaptions used to calibrate let's say Hull-White models used for the various tenors convexities B) maybe more with constraints the initial targets of repricing the input two categories of instruments. Jun 13, 2019 at 7:42
• For IBOR futures it is more common to use caplet/floorlets volatilities for 2). See for instance bloomberg's ICVS screen. Jun 16, 2019 at 9:49