Monotone Convex Interpolation for Curves

A yield curve is only directly observable at a handful of quoted maturities — say, 1, 2, 5, and 10 years — and every rate in between has to be interpolated. A standard cubic spline, chosen to fit those points as smoothly as possible, often overshoots between them, and because a curve's forward rates are derived from the slope of the discount curve, a smooth-looking spline in yields can imply forward rates that dip negative or oscillate in economically implausible ways, even when the input quotes look perfectly sensible.

Monotone convex interpolation, developed specifically for curve construction, fixes this by working with the forward-rate curve directly rather than the yield or discount curve, and constraining the interpolation so that forward rates stay well-behaved (no manufactured wiggles, no sign flips) between input points while still exactly repricing every quoted instrument used to build the curve. It sacrifices a small amount of the spline's raw smoothness in exchange for forward rates that a trading desk can actually use to price forward-starting instruments without absurd artefacts.

This matters day to day because much of fixed-income and derivatives pricing depends on forward rates, not just the yields at the quoted maturities — pricing a forward rate agreement or a swaption struck between two liquid maturities means trusting whatever the curve construction method implies about the shape in between, and a method that introduces a spurious negative forward rate there will misprice the instrument.