will ensure that paths whose probability have been arbitrarily enhanced by the change of probability distribution are weighted with a low weight (this is how the variance gets reduced). This technique can be particularly useful when calculating risks on a derivative. When calculating the delta using a Monte Carlo method, the most straightforward way is the black-box technique consisting in doing a Monte Carlo on the original market data and another one on the changed market data, and calculate the risk by doing the difference. Instead, the importance sampling method consists in doing a Monte Carlo in an arbitrary reference market data (ideally one in which the variance is as low as possible), and calculate the prices using the weight-changing technique described above. This results in a risk that will be much more stable than the one obtained through the black-box approach. Quasi-random (low-discrepancy) methods Instead of generating sample paths randomly, it is possible to systematically (and in fact completely deterministically, despite the "quasi-random" in the name) select points in a probability spaces so as to optimally "fill up" the space. The selection of points is a low-discrepancy sequence such as a Sobol sequence. Taking averages of derivative payoffs at points in a low-discrepancy sequence is often more efficient than taking averages of payoffs at random points. Notes Frequently it is more practical to take expectations under different measures, however these are still fundamentally integrals, and so the same approach can be applied. More general processes, such as Lévy processes, are also sometimes used. These may also be simulated. See also Quasi-Monte Carlo methods in finance Monte Carlo method Historical simulation (finance) Stock market simulator Real options valuation References Notes Articles Boyle, P., Broadie, M. and Glasserman, P. Monte Carlo Methods for Security Pricing. Journal of Economic Dynamics and Control, Volume 21, Issues 8-9, Pages 1267-1321 Rubinstein, Samorodnitsky, Shaked. Antithetic Variates, Multivariate Dependence and Simulation of Stochastic Systems. Management Science, Vol. 31, No. 1, Jan 1985, pages 66–67 Books External links General Monte Carlo Simulation (Encyclopedia of Quantitative Finance), Peter Jaeckel and Eckhard Plateny Monte Carlo Method, riskglossary.com The Monte Carlo Framework, Examples from Finance, Martin Haugh, Columbia University Monte Carlo techniques applied to finance, Simon Leger Derivative valuation Monte Carlo Simulation, Prof. Don M. Chance, Louisiana State University Option pricing by simulation, Bernt Arne Ødegaard, Norwegian School of Management Applications of Monte Carlo Methods in Finance: Option Pricing, Y.