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 ...