Explain numerical integration in numerical method

Question

Explain numerical integration in numerical method.

Answer 1

Infrequently one can write down the solution of an option-pricing problem within form of a numerous integral. It is as you can interpret the option value like an expectation of a payoff, and also an expectation of payoff is mathematically only the integral of the product of which payoff function and a probability density function. It is only possible in particular cases. The option has to be European, the underlying stochastic differential equation should be explicitly integrable (therefore the lognormal random walk is ideal for that) and the payoff shouldn’t generally be path dependent. So when this is possible then pricing is simple... you have a formula. The only complexity comes in turning this formula in a number. And which is the subject of numerical integration or quadrature.