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Monte Carlo simulation methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.

Many times it is not possible to determine an exact solution to a problem because there are random(ish) possible inputs, and therefor random outcomes.

The typical "Monte Carlo Method" goes something like:
1. Determine the possible inputs.
2. Generate the random inputs based on the probability distribution of your inputs.
3. Compute the output based on that particular random input.
4. Compile the results.

Monte Carlo simulation in the context of Quantitative Finance refers to a set of techniques to generate artificial time series of the stock price overtime, from which option prices can be derived. There are several choices available in this regard.The first choice is to apply a standard method such as the Euler, Milstein, or implicit Milstein scheme The advantage of these schemes is that they are easy to understand, and their convergence properties are well-known. The other choice is to use a method that is better suited, or that is specifically designed for the model.These schemes are designed to have faster convergence to the true option price, and in some cases, to also avoid the negative variances that can sometimes be generated from standard methods