Monte Carlo methods are also often used in data analysis to evaluate a limit on a definite integral whose value is given. A common usage is in a Monte Carlo test. Such a test is performed similarly to the method of any statistical test: a null hypothesis and an alternative are stated; a test statistic is chosen; a ``significance level'' is chosen; a random sample is taken; the test statistic is evaluated from the sample; and the p-value of the test statistic is compared with the significance level.
A Monte Carlo test is conducted in this same manner, except that the p-value is placed into a range determined by simulating many random samples under the null hypothesis and evaluating the test statistic for each simulated sample.
An obvious problem with a Monte Carlo test is that the null hypothesis, together with underlying assumptions must fully specify the distribution. Other Monte Carlo methods involving resampling, such as bootstrap methods, avoid this problem.
Another use of Monte Carlo is to study models of financial data. The most common model for stock prices as a function of time is the geometric Brownian motion model. It is easy to simulate.
First choose a time period, tend; a starting price, s[1]; a drift rate, mu; a volatility, sigma; and a time step and its square root deltat and sqdt. Then the S-Plus command is
for (i in 2:tend) {
s[i]<-s[i-1]*(1+mu*deltat+sigma*z[i-1]*sqdt)
}
The ability to simulate the process allows us to use Monte Carlo methods to study whether the assumptions about the process correspond to observable behavior.
If our assumptions about the process do indeed correspond to reality, we can use Monte Carlo simulation to determine various features of the process. For example, we can determine appropriate prices of other related assets. The way this is done is one of the standard techniques of the Monte Carlo method: Express the quantity of interest as an expected value, simulate the process many times each time computing the outcome, and then estimate the expected value as the mean of the outcomes.