Cubic Spline Interpolation

Cubic spline interpolation is a numerical method that uses piecewise third-order polynomials to connect data points smoothly. Unlike linear interpolation, which creates sharp corners, cubic splines ensure that both the first and second derivatives are continuous at each node.

This property makes them highly desirable in finance for building smooth yield curves or volatility surfaces. By ensuring a smooth second derivative, traders avoid unrealistic jumps in forward rates or option greeks.

However, cubic splines can sometimes oscillate excessively if the underlying data points are clustered too closely or contain outliers. They require careful parameterization to ensure the resulting model remains economically intuitive.

This method is a staple in the algorithmic construction of financial curves.