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Interpolation & Curve Fitting

Yield curves and volatility surfaces require interpolation between discrete market pillars (e.g., 1M, 3M, 6M, 1Y, 5Y, 10Y, 30Y).

Interpolation Schemes

1. Linear Interpolation (`LinearInterpolator`)

Continuous piecewise linear function ( $C^0$ continuous). Fast and monotonic, but non-smooth at knot points.

2. Natural Cubic Spline (`CubicSpline`)

Twice continuously differentiable ( $C^2$ continuous). Minimizes curvature across the knot points:

$S''(x_0) = S''(x_n) = 0$

3. PCHIP Hermite (`PchipInterpolator`)

Piecewise Cubic Hermite Interpolating Polynomial. Shape-preserving and strictly monotonic—prevents the artificial oscillatory overshoots that standard cubic splines introduce into discount factors and forward rates.

Rust Code Example

use sabiquant::math::interpolation::{LinearInterpolator, PchipInterpolator, Interpolator1D};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    // Zero curve pillars: [Maturity in years, Zero rate %]
    let maturities = vec![0.5, 1.0, 2.0, 5.0, 10.0, 30.0];
    let zero_rates = vec![0.035, 0.038, 0.041, 0.045, 0.048, 0.051];

    // Monotonic PCHIP Hermite curve
    let pchip = PchipInterpolator::new(&maturities, &zero_rates)?;

    // Query rate at 3.5 years
    let rate_3_5y = pchip.interpolate(3.5)?;
    println!("Interpolated 3.5Y Zero Rate: {:.4}%", rate_3_5y * 100.0);

    // First derivative (forward rate proxy)
    let slope = pchip.derivative(3.5)?;
    println!("Curve Slope (dRate/dt):      {:.6}", slope);

    Ok(())
}