Interpolation method for term curve construction

[Olya]

Our  traders for emerging markets chose to use linear (on zero rates) interpolation methodology for term curve construction because they "felt that the local markets worked on linear interpolation". Is there any good example to show that discontinuous forward rates are bad, or does their "argument" make sense?

[diogenes]

Hmmm…I would just be happy they know what linear interpolation is.

However, for local rates they could do worse.

[jungle]

This is one for Graeme... Smiley

[Graeme]

You're too kind.



Let's talk about swaps. A desk might have say liquidity at the 10y and 15y point, and these swaps (along with many others of course) form the set of inputs to the bootstrap of the yield curve. Suppose a client now asks them for a 12y swap.



Now a) the dealer might calculate the fair swap rate from their yield curve and quote that (or a double or whatever. In fact, they might make their double a bit wider because of liquidity concerns). b) Alternatively, they might just perform naive linear interpolation between the 10y and 15y swap rates and quote that.



Whether or not a) dominates over b) in sophisticated markets and b) over a) in less sophisticated markets I cannot say, although it is not an unreasonable hypothesis. Perhaps it will be more a function of how easily the dealer can extract this information from their system and how much they rely on it.



In some (bad) systems the bootstrapper will be doing something like this (linear interpolation) anyway. It's bad because it separates the two processes of bootstrap and interpolation when in fact they really should be viewed as a single process. Pat Hagan and I make this point in our two papers on interpolation both of which you can get at my website.



The good news is that it should not make too much of a difference in pricing, although bad interpolation algorithms lead to implausible or impossible forward curves which could have a dramatic effect on optionality pricing.



The bad news is that its hard to test in a statistical way my claim there shouldn't be too big a difference, or quantify the effect. Suppose in the above scenario I do in fact have a 12y rate which I experiment with to do some quasi-statistical tests by constructing bootstrap curves with i) the 12y rate filled in by interpolation ii) the 12y rate as it truely traded. Then in fact i) informs ii) because of those traders that we referred to already.

[polysena]

Gr. thanks for the pedagogical post and the link to your site.

[arkestra]

Start by thinking about the daily forward rates. Loglinear interpolation on discount factors will give you stepwise constant daily forwards. Linear-on-zero-rates interpolation will give you a "sawtooth" effect.



How much impact the sawtooth has depends on your yield curve shape. If your daily forward rates are changing at roughtly the same rate over the curve, you'll be fine. But the higher the curvature in daily forward rates, the bigger the jump from the point of a sawtooth to the start of the next period, and the more out of whack the surrounding forward rates will be. You can get surprisingly large jumps.



So if you want to get a handle on how much difference this makes then (1) find the currency with the most curvature in daily forward rates (2) identify the area of that curve with the largest sawtooth gaps (3) figure out if this is bad enough to make a genuine difference in quoting or not - assume that people out there will be trying to pick you off, it's a very common pursuit.

[doctorwes]

That rings a bell. It probably depends on the product too. I seem to remember that the sawtooth issue was not too big a problem for swap and swaption pricing, but led to problems in cap/floor pricing.

[arkestra]

Swaps/swaptions are an option on a weighted average of forward rates (roughly speaking). While with caps/floors the individual options are paid off single forwards. So, yes, caps/floors are more sensitive to this kind of quirk.



However you do get people asking for swap spreads/butterflys that are designed to pick out weak areas in yield curves, so some care is advisable on non-option trades too, particularly if the deals concerned are not standard swap maturities.