Forward-Rate Interpolation and Curve Arbitrage Risk
Summary
The document compares linear interpolation of quoted rates with constant-forward-rate interpolation when constructing a rate curve for pricing American and European options. It highlights that linear interpolation can create discontinuities in instantaneous forward rates and does not keep forwards constant between curve nodes, but the question does not resolve whether those properties are inherently harmful for option pricing.
The answer argues that a curve with discontinuous forward rates may expose a trader to apparent arbitrage: neighboring periods can show an anomalous pattern of borrowing and lending rates created by the interpolation itself. In a competitive market, that pattern may invite trading against the curve and can make inventory profit and loss unstable. This is a qualitative warning rather than a formal proof or a controlled comparison of interpolation methods. The example illustrates a possible market consequence, but does not establish that constant-forward interpolation is always preferable for every instrument, curve, or pricing setup.
Key ideas
- Linear interpolation of quoted rates can produce discontinuous instantaneous forward rates.
- An interpolation-created pattern of neighboring forward rates may invite trades against the curve.
- Such discontinuities can make a trader’s inventory profit and loss unstable.
- The explanation is qualitative and does not prove one interpolation method is universally superior.
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# Answer by dm63 (score 1, accepted) # Are there any structural reasons for choosing constant forward rate interpolation over linear interpolation beyond just simplicity? 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? ## Answer by dm63 (score 1, accepted) https://quant.stackexchange.com/a/77270 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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