Adapting to Direct Forward Curves

In interest rate pricing direct forward curves are defined on forward rates for a specific tenor as opposed to the more common discount factor representation, or the instantaneous forward curve representation. A simple example of a direct forward curve for 3M LIBOR would consist of a set of points and an interpolator, the points would not be zero rates or discount factors, but be (direct) forward rates for 3M LIBOR. Such a curve could not provide discount factors, nor forward rates for any other index tenor, it could only produce forward rates for 3M LIBOR at any forward start date.

Initially, one might wonder why a more restrictive representation might be considered, however in the modern two-curve pricing world it is natural to have a curve dedicated to determining forward rates for a specific index tenor, and the ability to produce an explicit discount factor (or zero rate) is merely an artefact from the old world. Some aspects of curve construction, particularly at the shorter end, can be handled more conveniently with a direct forward curve.

One of the first practical issues that arise with such curves is their integration with existing or legacy valuation and risk management systems. Most systems can accept curves represented by zero rates/discount factors and an interpolator, or simply a list of daily zero rates/discount factors, but the new direct forward curves can produce neither, or can they?

Let us consider a trivial example, a direct forward curve with only three carefully chosen points,

Fwd Start Fwd End Rate
2016-02-10 2016-05-10 0.4%
2016-05-10 2016-08-10 0.6%
2016-08-10 2016-11-10 0.7%

A discount factor representation of these three points would require four discount factors for the dates 2016-02-10, 2016-05-10, 2016-08-10 and 2016-11-10. So four unknowns, and only three equations of the form $$\frac{D_i}{D_{i+1}} = 1 + \Delta_i F_i $$ linking them, in which case we have many possible discount factor curves that give rise to the same three forward rates; three such examples are,

Date Curve1 Curve2 Curve3
2016-02-10 1.0 1.00432829015037 0.5
2016-05-10 0.999000999000999 1.00332496518519 0.4995004995005
2016-08-10 0.997471542635624 1.00178888888889 0.498735771317812
2016-11-10 0.995690363208108 1.0 0.497845181604054

Let us suppose a more general setting in which, for a specific index tenor such as 3M LIBOR, we are given a forward rate for each business day (and so all forward rates we need for pricing our interest rate products), we can actually produce many discount factor curves which agree on these daily direct forwards. A relatively simple approach is to identify ‘chains’ of forward rates (those rates in which the end date is the start date of another forward) and decide on an arbitrary value for the missing discount factor, much like the example above.

Identifying chains of forwards that partition the complete set of forwards is not as easy as it might first seem, there may be many forwards with the same end date but different start date because of date generation rules; for example, there are three 3M LIBOR fixings with forward end date 2016-05-03.

Fixing Date Reset Date (Fwd Start) Maturity Date (Fwd End)
2016-01-28 2016-02-01 2016-05-03
2016-01-29 2016-02-02 2016-05-03
2016-02-01 2016-02-03 2016-05-03

Although this aspect complicates the matter, it is still possible to partition the forwards into distinct chains with \(n\) forward rates and \(n+1\) discount factors, and produce a discount factor curve consistent with the direct forwards.

Another aspect of the date generation rules is that two different fixing dates can have precisely the same forward start and end dates; for example,

Fixing Date Reset Date (Fwd Start) Maturity Date (Fwd End)
2016-02-11 2016-02-16 2016-05-16
2016-02-12 2016-02-16 2016-05-16

in which case, it is more natural to interpolate on the forward start dates as opposed to the fixing date.

A final date generation issue that can arise in this context is the ‘End-End’ rule on LIBOR;

if a forward start date is on the final business day of a month, then the forward end date should also be the final business day of the relevant month

This rule is not implemented in all systems.

Although we have concentrated on the coercing of the new direct forward curves into the existing discount factor curve representation, it is of course much better if systems can be updated to support direct forward curves, and the conversion of the curves be a temporary step, if needed at all.

Further Reading

The abcd of Forward Rate Bootstrapping by F. Ametrano
Interest Rate Modelling in the Multi-Curve Framework by M. Henrard.

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