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