# Bootstrapping the zero-curve/spot-curve from incomplete swap curve par-rates

TL;DR: I have an incomplete set of swap rates and want to bootstrap the zero-rate curve, what can I do?

I'm trying to construct a spot-rate/zero-rate curve from a swap curve (i.e. par-rate quotes) based on ESTR OIS swaps for tenors (OverNight; 1week; 2w; 1month; 2m; 3m, 6m; 1year; 2y; 3y; 4y; 5y; 6y; 7y; 8y; 9y; 10y; 11y; 12y; 15y). I make the assumption that all tenors up to 1y (inclusive of 1y) already are the zero-rate, since ESTR OIS swaps pay annually and thus only have one pay-out for all tenors up to one year. If it helps, here is the BBG description of the instruments I'm using:

In order to construct the spot-rate/zero-rate curve, I naturally apply bootstrapping. I.e. I use the following approach to calculate the spot-rates/zero-rates, ignoring day-count conventions for now:

First, we extract the discount factors $$df$$ with $$s_n$$ being the fixed par-rate of the swap as per the market quote

$$df_n=\frac{1-s_n\times\sum_{i=1}^{n-1}df_i}{1+s_n}$$

then we find the spot-rates/zero-rates from the discount factors by applying:

$$zeroRate_n=\sqrt[n]{\frac{1}{df_n}}-1$$

The above is of course done in an iterative way, progressing forward on the curve. My starting rate, as implied earlier, is the 1y-tenor par-rate, which I take to be a spot-rate as is.

However, as the given tenors are missing the 13y- and 14y-tenor, I am at a loss as to what I should do when trying to calculate the 15y spot-rate/zero-rate... Upon browsing the web, little can be found w.r.t this issue. All articles and papers I found assume a complete information set. Of course I have tried linearly interpolating between the 12y- and 15y-tenor, but this is unsatisfactory as it results in a kink and seems overly simplistic and will heavily influence the result.

Ultimately, I'm trying to construct a panel dataset of zero-rate curves to experiment with various Dynamic Nelson Siegel (DNS) models.