Adaptive Quadrature

Approximating the integral of a function numerically usually means chopping the interval into small pieces and summing up simple approximations (like trapezoids) over each piece — but a fixed, uniform grid wastes effort: it needs very fine subdivisions everywhere just to handle the one region where the function wiggles sharply, even though most of the interval might be nearly flat and could be approximated coarsely with no loss of accuracy. Adaptive quadrature fixes this by checking, region by region, how much the approximation changes when a region is split in two; if splitting barely changes the estimate, that region is smooth enough and the coarser estimate is kept, while if splitting changes the estimate substantially, the algorithm recurses and keeps subdividing that region further until the estimate stabilizes within a target tolerance.

The practical payoff is that adaptive quadrature concentrates computational effort exactly where the function needs it — near a kink, a spike, or a fast-changing region — and spends almost nothing on the smooth parts, reaching a target accuracy with far fewer function evaluations than a uniform grid would need for the same accuracy. This matters whenever a quant needs to numerically integrate something like an option's expected payoff under a non-standard distribution, where closed-form integration isn't available.

A worked feel: suppose you're integrating a function over $[0,10]$ that's nearly flat except for a sharp spike between 4 and 5. A uniform-grid method might need 1,000 evenly spaced points to resolve the spike accurately, wasting most of them on the flat regions. Adaptive quadrature instead starts with one coarse estimate over $[0,10]$, splits it in half, and compares the sum of the two half-estimates to the original whole estimate. Over $[0,4]$ and $[5,10]$ the two estimates will nearly agree (function is flat there), so the algorithm stops subdividing those regions immediately. Over $[4,5]$ the estimates disagree sharply because of the spike, so the algorithm recurses — splitting $[4,5]$ again and again — until the estimate there stabilizes too. The end result might use only 50 points total: a handful spread thinly over the flat regions and a dense cluster concentrated right on the spike.