a draw from a standard normal distribution. Let us suppose that a derivative H pays the average value of S between 0 and T then a sample path corresponds to a set and We obtain the Monte-Carlo value of this derivative by generating N lots of M normal variables, creating N sample paths and so N values of H, and then taking the average. Commonly the derivative will depend on two or more (possibly correlated) underlyings. The method here can be extended to generate sample paths of several variables, where the normal variables building up the sample paths are appropriately correlated. It follows from the central limit theorem that quadrupling the number of sample paths approximately halves the error in the simulated price (i.e. the error has order convergence in the sense of standard deviation of the solution). In practice Monte Carlo methods are used for European-style derivatives involving at least three variables (more direct methods involving numerical integration can usually be used for those problems with only one or two underlyings. See Monte Carlo option model. Greeks Estimates for the "Greeks" of an option i.e. the (mathematical) derivatives of option value with respect to input parameters, can be obtained by numerical differentiation. This can be a time-consuming process (an entire Monte Carlo run must be performed for each "bump" or small change in input parameters). Further, taking numerical derivatives tends to emphasize the error (or noise) in the Monte Carlo value – making it necessary to simulate with a large number of sample paths. Practitioners regard these points as a key problem with using Monte Carlo methods. Variance reduction Square root convergence is slow, and so using the naive approach described above requires using a very large number of sample paths (1 million, say, for a typical problem) in order to obtain an accurate result. Remember that an estimator for the price of a derivative is a random variable, and in the framework of a risk-management activity, uncertainty on the price of a portfolio of derivatives and/or on its risks can lead to suboptimal risk-management decisions. This state of affairs can be mitigated by variance reduction techniques. Antithetic paths A simple technique is, for every sample path obtained, to take its antithetic path — that is given a path to also take . Since the variables and form an antithetic pair, a large value of one is accompanied by