Numerical integration is a standard topic in numerical analysis, and in the previous chapter we have hinted at the link between integration and Monte Carlo methods. In this chapter we have a twofold objective:

On the one hand, we want to insist on the link between numerical integration and Monte Carlo methods, as this provides us with the correct framework to understand some variance reduction methods, such as importance sampling, as well as alternative approaches based on low-discrepancy sequences.

On the other hand, we want to outline classical and less classical approaches to numerical integration, which are deterministic in nature, to stress the fact that there are sometimes valuable alternatives to crude Monte Carlo; actually, stochastic and deterministic approaches to numerical integration should both be included in our bag of tricks and can sometimes be integrated (no pun intended).

In many financial problems we are interested in the expected value of a function of random variables. For instance, the fair price of a European-style option may be evaluated as the discounted expected value of its payoff under a risk-neutral probability measure. In the one-dimensional case, the expected value of a function *g*(·) of a single random variable ...

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