Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions (sources of uncertainty) of the problem increase. Monte Carlo methods were first introduced to finance in 1964 by David B. Hertz through his Harvard Business Review article, discussing their application in Corporate Finance. In 1977, Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal Journal of Financial Economics paper. This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of so-called "quasi-random" methods such as the use of Sobol sequences. Overview The Monte Carlo method encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. Essentially, the Monte Carlo method solves a problem by directly simulating the underlying (physical) process and then calculating the (average) result of the process. This very general approach is valid in areas such as physics, chemistry, computer science etc. In finance, the Monte Carlo method is used to simulate the various sources of uncertainty that affect the value of the instrument, portfolio or investment in question, and to then calculate a representative value given these possible values of the underlying inputs. ("Covering all conceivable real world contingencies in proportion to their likelihood.") In terms of financial theory, this, essentially, is an application of risk neutral valuation; see also risk neutrality. Some examples: In Corporate Finance, project finance and real options analysis, Monte Carlo Methods are used by financial analysts who wish to construct "stochastic" or probabilistic financial models as opposed to the traditional static and deterministic models. Here, in order to analyze the characteristics of a project’s net present value (NPV), the cash flow components that are (heavily) impacted by uncertainty are modeled, incorporating any correlation between these, mathematically reflecting their "random characteristics". Then, these results are combined in a histogram of NPV (i.e. the project’s probability distribution), and the average NPV of the potential investment – as well as its volatility and other sensitivities – is observed. This distribution allows, for example, for an estimate of the probability that the project