# Numerical Integration

## Document Type

Book Chapter

## Publication Title

Indian Statistical Institute Series

## Abstract

Integration is a very common requirement in quantitative finance. In some cases integrals can be evaluated analytically, or at least in terms of standard functions whose values are known. For example in the case of option pricing under the Black–Scholes model, the integral can be expressed in terms of the normal distribution function. However, very often we encounter problems where the integral cannot be evaluated analytically. In such cases we need numerical techniques that we shall explore in this chapter. For one-dimensional integrals, these deterministic methods work quite well. For evaluating integrals in higher dimensional spaces, it may be more advantageous to use random sampling. We shall see these in the Chap. 8 as Monte Carlo methods. Integrals frequently arise as expectations. For example, in economics one needs to evaluate the expected utility. Option pricing requires computing the expected value of the payoff under the risk-neutral measure. For a random variable X with density fX(x), the expected value of a function h(.) is given by ∫-∞∞h(x)fX(x)dx, which is really an integral.

## First Page

33

## Last Page

46

## DOI

10.1007/978-981-19-2008-0_4

## Publication Date

1-1-2023

## Recommended Citation

Sen, Rituparna and Das, Sourish, "Numerical Integration" (2023). *Book Chapters*. 202.

https://digitalcommons.isical.ac.in/book-chapters/202