I've been looking into rate curve interpolation methods and focussing on two basic ones - linear interpolation, and constant forward rate interpolation. In the first one, given a rate curve consisting of pairs of $(t, r)$, one linearly interpolates the $r$ values to achieve the desired rate at any arbitrary $t$ value that lies within the range spanned by the curve. The implication of this are that
- instantaneous forward rates might be non continuous in rate curve points
- forward rates are not constant between nodes
For both of these facts I struggle to prove exactly why they are bad or good - beyond just the basic statements like; "discontinuities in the instantaneous forward rate curve might imply an implausible view of the future", or "its simpler to work with constant forward rates".
Given an application in which all we are interested in is pricing option instruments, American and European, are there any fundamental reasons for choosing linear interpolation over constant forward rate interpolation beyond just simplicity?