Gaussian Quadrature

The trapezoid and Simpson rules both sample a function at evenly-spaced points and only get to choose the weights applied to those samples. It turns out that if you're also allowed to choose where to sample — not just evenly spaced, but at specially-computed locations — a small number of cleverly-placed points can integrate a function exactly as well as many more evenly-spaced points. Gaussian quadrature is that idea taken to its logical extreme: for a given number of function evaluations, it's provably the most accurate possible choice of sample points and weights, for a wide class of smooth functions.

An analogy: weighing a seesaw with the fewest measurements

Imagine estimating the total weight and balance point of an oddly-loaded seesaw by taking a small number of measurements at chosen spots. If you're forced to measure at evenly-spaced points, you'll need many measurements to characterize a complicated loading. But if you're allowed to choose where to measure, there's a small set of specific spots — not evenly spaced, clustered more densely toward the edges — that captures far more information about the true loading per measurement taken. Gaussian quadrature is the mathematical version of finding those optimal spots, for integrating a function instead of weighing a seesaw.

The math, one piece at a time

Gaussian quadrature approximates an integral as a weighted sum of function values at $n$ specially chosen points $x_i$, called nodes, with specially chosen weights $w_i$:

$$\int_{-1}^{1} f(x)\,dx \approx \sum_{i=1}^{n} w_i \, f(x_i) .$$

In words: instead of evenly spacing $n$ points and giving them simple weights (like Simpson's alternating 4s and 2s), Gaussian quadrature solves for the $n$ node locations and $n$ weights that make this formula exactly correct for every polynomial up to degree $2n-1$ — nearly double the degree that $n$ evenly-spaced points could match. The nodes turn out to be the roots of the $n$-th Legendre polynomial, a specific family of polynomials that are mutually orthogonal over $[-1,1]$; the reader doesn't need to derive this, only to know that these root locations are tabulated once and reused for any smooth integrand.

The practical payoff: with $n$ points, Gaussian quadrature exactly integrates any polynomial up to degree $2n-1$, whereas Simpson's rule (also using roughly $n$ points) only reaches about degree 3 exactness locally per pair of intervals. For smooth (non-polynomial) integrands, this translates into Gaussian quadrature reaching a target accuracy with dramatically fewer function evaluations than evenly-spaced rules — often an order of magnitude fewer for smooth pricing integrands. Integrals over a general range $[a,b]$ instead of $[-1,1]$ use a simple linear rescaling of the nodes and weights, so the tabulated $[-1,1]$ values work for any interval.

Worked example 1: exactness on a polynomial

2-point Gaussian quadrature uses nodes $x_{1,2} = \pm\frac{1}{\sqrt{3}} \approx \pm 0.5774$ with weights $w_1 = w_2 = 1$, and is exact for any polynomial up to degree 3. Check on $f(x) = x^3 + 2x^2 + 1$ over $[-1,1]$ (true integral: the odd $x^3$ term vanishes by symmetry, leaving $\int_{-1}^1 (2x^2+1)\,dx = \frac{14}{3} \approx 4.667$). Evaluating at the two nodes and summing with exact fractions confirms the 2-point rule reproduces $\frac{14}{3}$ exactly, using only 2 function evaluations for a cubic that Simpson's rule would need several evenly-spaced points to match.

Worked example 2: pricing a smooth expected payoff

Estimating $\mathbb{E}[\max(S-K,0)]$ under a smooth density with a 5-point Gaussian quadrature rule (rescaled to the relevant price range) typically matches the accuracy of a Simpson's-rule integration using 20-30 evenly-spaced points, because the payoff and density together form a smooth-enough function that Gaussian quadrature's higher polynomial exactness pays off directly — cutting a pricing library's per-integral evaluation count by roughly 4-6x for the same accuracy target, which matters when the same integral is re-evaluated thousands of times during calibration.

What this means in practice

Gaussian quadrature is the workhorse behind fast numerical integration in pricing libraries — computing expected payoffs, integrating characteristic functions in Fourier-based option pricing (like the Carr-Madan method), and any repeated, performance-sensitive integral over a smooth function where evaluation count matters. It's less useful when the integrand has kinks, jumps, or is only known at fixed pre-determined points (like historical data), since its main advantage — choosing where to sample — requires being able to evaluate the function anywhere.