Interpolation and yield-curve construction

Interpolation constructs a smooth function passing through a finite set of data points. In finance: building a yield curve from a sparse set of observed bond yields, constructing implied-vol surfaces from listed option prices, smoothing risk metrics across time. The choice of interpolation method is a model choice with real economic consequences.

Linear interpolation

Connect successive points with straight lines. Simple, continuous, monotonic-preserving. Discontinuous first derivative. Fine for quick approximations; rarely the right choice for yield curves.

Polynomial interpolation

Pass a single polynomial of degree n-1 through n points (Lagrange or Newton form). Smooth in principle but oscillates wildly at high n (Runge's phenomenon). Never use high-degree polynomial interpolation for noisy or extended data.

Cubic splines

Piecewise cubic polynomials connected so that the function and its first and second derivatives are continuous. Local — changing a node only affects nearby spline segments. Smooth, accurate, the default for most interpolation needs.

Monotone cubic splines (Hyman, Fritsch-Carlson)

Cubic splines that preserve monotonicity of the input data. Critical for yield curves, where the discount factor must be monotonically decreasing in maturity. The Hyman filter or the Fritsch-Carlson scheme adjusts spline slopes to maintain monotonicity.

Yield-curve interpolation

The yield curve is observable at a discrete set of maturities (e.g., 3M, 6M, 1Y, 2Y, 5Y, 10Y, 20Y) but options/swaps may need rates at any maturity in between. Interpolation choices:

• Interpolate yields directly (linear or cubic).
• Interpolate log-discount-factors linearly = piecewise-constant forward rates. Standard practitioner default.
• Interpolate forward rates with monotone splines = smoother forwards.
• Bootstrap zero-coupon yields from coupon-bearing instruments first; then interpolate.

Nelson-Siegel parametric form

Instead of interpolating, fit a 4-parameter functional form to the yields:

y(τ) = β_0 + β_1 (1 - exp(-τ/λ))/(τ/λ) + β_2 [(1 - exp(-τ/λ))/(τ/λ) - exp(-τ/λ)]

Parsimonious, economically interpretable (level, slope, curvature), and forces a smooth shape. Used by central banks (Fed, BoE, BCE, CBK) for official yield-curve reporting. Limitation: cannot fit highly oscillatory curves.

Implied-volatility surface

Listed options have strikes and maturities; trader desks need IVs at any (K, T). Two-dimensional interpolation: typically cubic spline in moneyness (K/S), linear in total variance T·σ² in maturity. Total-variance interpolation prevents calendar-spread arbitrage. SVI (stochastic-volatility-inspired) parameterisations fit smooth functional forms.

Multivariate / scattered-data interpolation

When data isn't on a grid (irregular maturities, sparse vol points), use radial basis functions, kriging, or thin-plate splines. More complex; needed when grid-based methods don't apply.