Interpolation: splines, Runge and building a curve

NUM · Chapter 211 min readAsked at Optiver, Citadel Securities, DRW, Jump

Assumes Root finding, with implied volatility as the worked case.

After this lesson you should be able to

  • Say why high-degree polynomial interpolation fails.
  • Choose an interpolation scheme for a yield curve or a volatility surface.
  • State what shape constraints a financial curve must satisfy.

A yield curve is quoted at a dozen maturities and needed at every maturity; a volatility surface is quoted at listed strikes and needed everywhere between. Interpolation fills the gaps, and the choice of scheme is a modelling decision because it determines quantities — forward rates, implied densities — that must not come out negative.

Proposition 2.1

Runge’s phenomenon

A single polynomial through nn points has degree n−1n-1, and high-degree polynomials oscillate violently between the knots — worst near the ends of the range. Passing through every point exactly is easy; behaving sensibly between them is not, and the two are in tension.

Holds when

  • The oscillation grows with the degree, so adding data makes it worse rather than better.
  • Chebyshev-spaced knots reduce it substantially, but you rarely choose where market quotes sit.
  • The fix in practice is to use many low-degree pieces instead of one high-degree curve.

Definition 2.2

Splines

Cubic spline — A separate cubic on each interval, joined so that the value, the first derivative and the second derivative all match at the knots. That gives a curve which is visually smooth, passes through every quote, and cannot oscillate because no individual piece has enough degrees of freedom to.

Why cubic and not something else. Count the conditions. Each interval’s cubic has four coefficients; matching the value at both ends uses two, and matching the first and second derivatives with the neighbours uses the other two. A quadratic has too few to match curvature, and a quartic has one spare per interval that must be pinned down arbitrarily. Cubic is the lowest degree that produces a curve with continuous curvature, which is why it became the default — and the two remaining free conditions at the ends are what the "natural" and "clamped" variants choose differently.

0153000.51At 4 per centAt 6 per centYearsDiscount factor
Figure 2.3 · What you are actually interpolating. Interpolate discount factors and the curve stays monotone and positive whatever you do. Interpolate zero rates and a cubic can produce a forward rate that is negative between two perfectly sensible quotes, which is the standard argument for choosing the variable carefully before choosing the spline.

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Interpolation: splines, Runge and building a curve · QuantMax