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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?

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  • $\begingroup$ I have the feeling that neither interpolation is used in practice much anymore. $\endgroup$ Nov 5, 2023 at 13:50
  • $\begingroup$ @DimitriVulis sure, might be, but do you know for these particular methods what the benefits are of one over the other and why? $\endgroup$ Nov 5, 2023 at 19:51

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Any curve building method that produces non continuous forward rates is subject to being arbitraged by other market participants (assuming you are using it to make decisions about buying and selling things). Let’s say the forward rate for January 2026 is 4.00% and for February 2026 is 4.50% and March 2026 is 4.00%. If you are in a competitive market you will then find yourself paying the 4.50% and receiving the 4.00% , which is an accident of your curve construction. Indeed you will probably find the discontinuities to be unstable which will then produce high pnl volatility on your inventory.

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