# bootstrapping a basis curve to get a forward basis curve

suppose i have a trade whose payoff underlying is 3m libor minus 1m libor. the standard approach is to bootstrap seperately 2 projection curves: a) a 3m projection curve, b) a 1m proj curve.
however, that gives rise to a big potential for excessive fluctuations in results due to having risk on both factors and when each seperate curve is interpolated, the result is not the same as if the spread curve were interpolated. Now, since the market gives quotes of the basis curve directly (there is a) the cash rates , b) the basis swap out to various tenors: 6m 1y 2y ...), why not just bootstrap the basis = 3mlibor minus 1m libor? That would give a much smoother resulting forward basis curve and so P&L volatility would be in line with the variability of the market quoted basis curve

## 2 Answers

The results are the same as long as you use the same data and common interpolation methods.

For instance 12M Libor vs fixed swaps are less liquid than 12M Libor vs 3M Libor basis swaps so one usually uses the latter to bootstrap the 12M projection curve (after the 3M projection curve has been bootstrapped).

Also it is straightforward to see that for common interpolation schemes such as linear on log discount or linear on zero yield the interpolated difference curve is the same as the difference of interpolated curves.

So in the end bootstrapping separate curves or bootstrapping directly a basis curve will give the same trade value as long as the same data is used.

• "Also it is straightforward to see that for common interpolation schemes such as linear on log discount or linear on zero yield the interpolated difference curve is the same as the difference of interpolated curves." hmmm. well typically on the 3m curve you have futures , whereas on the 1m curve, you dont , so each curve will have different Shape , and this will cause the spread of the interpolated forwards to be different from the forward of the interpolated spread - this is the problem! – Randor Jun 11 '18 at 11:02
• Couple of issues here: 1/ to value the trade you still need to compute both 3M and 1M forwards because either the payments frequencies in both legs of the basis trade are not the same or at least one leg is compounded. You can't simply settle on the spread. 2/ whether you choose to represent the 2nd curve as "absolute" or as a "spread" to the 1st one will not change the 2d leg forwards and thus the valuation as long as you use the same input data and consistent interpolation methods (if $X(t)$ and $Y(t)$ are both piecewise linear then $X(t)+Y(t)$ is also piecewise linear). – Antoine Conze Jun 11 '18 at 11:23
• 1/ yes - but the impact of that could be done as some convexity adjustment – Randor Jun 11 '18 at 12:20
• Convexity adjustment is a separate issue and would arise if one leg has unnatural payment dates, for instance 3M leg paid every month. It would still have to be done on a leg by leg basis, thus requiring the forwards for each leg. Note that on vanilla tenor basis swaps the bid/ask does not exceed a couple of bps so you can't have 2 methodologies that give significantly different results. – Antoine Conze Jun 11 '18 at 12:38
• the market quotes 1m short swaps: are you saying to ignore them and use the 1s/3s basis swaps instead ? – Randor Jun 11 '18 at 13:45

PnL variability and risk reliability should not be a problem if you have well designed control (knot) points and enough data. See Darbyshire: Pricing and Trading Interest Rate Derivatives.

I would advise trying to build the fundamental constructs, 1M and 3M for the following reasons;

• You probably have a better subjective opinion on the interpolation of individual IBORs than you do on the basis.
• I know the market maker desks I have contact with all build in this manner so deviating might produce curves inconsistent with the market consensus.
• In a multi curve environment it becomes more difficult to adapt to other models. E.g. say you built an OIS curve and then decided that actually the 1M-OIS spread is actually more stable at the short end and you want to factor that it, how would you marry an interpolated 1M-OIS with an interpolated 1M-3M construction?

Practically choosing your instruments and knot points is subjective. If you choose too many market data prices you will overfit your curve and create kinks. If you choose too little you leave useful market information on the table. But you can certainly incorporate information about the 3s1s basis and 1m prices if you want to.

The scenario you mention in a previous comment about inconsistent pricing is common. You cannot always satisfy the market pricing. I'll highlight the example of the 2Y-6M-IBOR IRS (market quoted) versus the 2Y-3M-IBOR IRS (priced from the 3M-IBOR curve generated by futures) and the 2Y 6s3s Basis (market quoted). In all currencies these are frequently inconsistent with each other upto a price which is too small to warrant the additional brokerage and multi-execution risk, say 0.15 to 0.3bps. So there is, practically, no arbitrage to capture but the difference is significant enough to make the curve build difficult, but there are solutions and reasons why the 6M is generally derived in the first 3Y from the 3M + 6s3s basis prices.