Editing Monte Carlo methods in finance (section)

===Mathematically===
The [[fundamental theorem of arbitrage-free pricing]] states that the value of a derivative is equal to the discounted expected value of the derivative payoff where the [[expected value|expectation]] is taken under the [[risk-neutral measure]] <sup>[1]</sup>. An expectation is, in the language of [[pure mathematics]], simply an integral with respect to the measure. Monte Carlo methods are ideally suited to evaluating difficult integrals (see also [[Monte Carlo method]]).

Thus if we suppose that our risk-neutral probability space is <math>\mathbb{P}</math> and that we have a derivative H that depends on a set of [[underlying instruments]] <math>S_1,...,S_n</math>. Then given a sample <math>\omega</math> from the probability space the value of the derivative is <math>H( S_1(\omega),S_2(\omega),\dots, S_n(\omega)) =: H(\omega) </math>. Today's value of the derivative is found by taking the expectation over all possible samples and discounting at the risk-free rate. I.e. the derivative has value:

:<math> H_0 = {DF}_T \int_\omega H(\omega)\, d\mathbb{P}(\omega) </math>

where <math>{DF}_T</math> is the [[discount factor]] corresponding to the risk-free rate to the final maturity date ''T'' years into the future.

Now suppose the integral is hard to compute. We can approximate the integral by generating sample paths and then taking an average. Suppose we generate N samples then

:<math> H_0 \approx {DF}_T \frac{1}{N} \sum_{\omega\in \text{sample set}} H(\omega)</math>

which is much easier to compute.