ABSTRACT
One of the fundamental problems in numerical analysis is the evaluation of integrals. This chapter considers two main approaches to the construction of a quadrature formula, namely, the Newton–Cotes quadrature formula and the Gaussian quadrature formula. Finite differences are used to numerically solve (initial-)boundary value problems for differential equations by reducing them to systems of linear or nonlinear algebraic equations. The finite-difference approach can be applied to solve numerically a partial differential equation (PDE) that involves more than one independent variable. The stability of a computational method is closely associated with numerical errors. Valuation of a European derivative is probably the most elementary problem of computational finance. In the case of a single asset price model, this problem reduces to the numerical evaluation of a one-dimensional integral. The PDE formulation of the derivative price problem allows for the application of finite-difference schemes.
