Financial Mathematics

Chapter 18

Numerical Applications to Derivative Pricing

18.1 Overview of Deterministic Numerical Methods

18.1.1 Quadrature Formulae

One of the fundamental problems in numerical analysis is the evaluation of integrals. Consider a one-dimensional definite integral of a function f : [a, b] → ℝ,

$I(f)\equiv I(f;[a,b]):={\displaystyle {\int }_a^{b}f(x)dx\text{with}-\infty <a<b<\infty .}$

The function f is said to be integrable (and the integral of f is defined) if the integral I(|f|) exists and is finite. Assuming that a closed-form expression for I(f) is unavailable or intractable, we rely on a numerical evaluation. Any explicit formula that is suitable for providing an approximation of I(f) is called a quadrature formula, or a quadrature rule, or a numerical integration formula. A typical quadrature ...