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Monte Carlo Methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in simulating physical and mathematical systems. Because of their reliance on repeated computation of random or pseudo-random numbers, these methods are most suited to calculation by a computer and tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithm. 

Monte Carlo methods are useful for modeling phenomena with significant uncertainty in inputs, such as the calculation of risk in business. These methods are also widely used in mathematics: a classic use is for the evaluation of definite integrals, particularly multidimensional integrals with complicated boundary conditions. It is a widely successful method in risk analysis when compared with alternative methods or human intuition. 

Monte Carlo methods are used in 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 their average value over the range of resultant outcomes. 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 in “Risk Analysis in Capital Investment” (Harvard Business Review), discussing their application in Corporate Finance. In 1977, Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal paper “Options: A Monte Carlo Approach” 

Many problems in mathematical finance entail the computation of a particular integral (for instance the problem of finding the arbitrage-free value of a particular derivative). In many cases, these integrals can be valued analytically, and in still more cases they can be valued using numerical integration or computed using a partial differential equation (PDE). However when the number of dimensions (or degrees of freedom) in the problem is large, PDEs and numerical integrals become intractable, and in these cases, Monte Carlo methods often give better results. 

For more than three or four state variables, formulae such as Black Scholes (i.e. analytic solutions) do not exist, while other numerical methods such as the Binomial options pricing model and finite difference methods face several difficulties and are not practical. In these cases, Monte Carlo methods converge to the solution more quickly than numerical methods, require less memory, and are easier to program. For simpler situations, however, simulation is not the better solution because it is very time-consuming and computationally intensive.

Monte Carlo methods can deal with derivatives that have path-dependent payoffs in a fairly straightforward manner. On the other hand Finite Difference (PDE) solvers struggle with path dependence. 

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