# Interpolating cross-currency basis curve

Just wondering how do people "interpolate" between different "pillar dates" on a cross-currency basis curve? So say for example, if the observed spot is 1.5, observed CC basis for 9 months is -1.25 and CC basis for 1 year is -1.35, and I am trying to work out, say, the cross-currency basis for, say, 10 months, how should I do that?

A few ideas I can think of:

1. Interpolate between -1.25 and -1.35 (which ended up say, around -1.30) and call that my CC basis.

2. Interpolate between the outright rates derived from the basis and spot (i.e., in the example above, between 1.5-1.25/100 = 1.4875 and 1.5-1.35/100 = 1.4865) and then subtract the interpolated value (which is the forward rate) with the spot?

3. Interpolate on the IR curves of r_f and r_d and use parity to derive the forward rates and subtract that with spot?

Or maybe something else?

What is the "correct" way of doing it and how does people do that in general in the industry?

Finally, for the "correct" way, what type of interpolator is the conventional one to be used? Linear? Cubic spline?

The most common way I have seen in front office systems is to interpolate/bootstrap the CC basis zero curve $D^{CC}(T)$, defined from the following representation $$FX^{fd}(T)=FX^{fd}(0) \frac{D^f_{OIS}(T)}{D^d_{OIS}(T)} D^{CC}(T)$$ where $FX^{fd}(T)$ is the forward FX, $D^d_{OIS}(T)$ is the domestic OIS discount factor and $D^f_{OIS}(T)$ is the foreign OIS discount factor.
$D^{CC}(T)$ is bootstrapped on market FX swaps and XCCY. Interpolation of $D^{CC}(T)$ tends to be along the same lines than that of other curves, i.e. linear on yields or linear on log discount. Splines are always a problem as they do not form a local interpolation method, and thus generates nonsensical sensitivities (e.g. a 5 years deal having non zero sensitivity to a 10 year instrument).