Definition of 'Monte Carlo Simulation':

A problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables

What is the Monte Carlo method?

In general terms, the Monte Carlo method (or Monte Carlo simulation) can be used to

describe any technique that approximates solutions to quantitative problems through

statistical sampling. As used here, 'Monte Carlo simulation' is more specifically used to

describe a method for propagating (translating) uncertainties in model inputs into

uncertainties in model outputs (results). Hence, it is a type of simulation that explicitly

and quantitatively represents uncertainties. Monte Carlo simulation relies on the

process of explicitly representing uncertainties by specifying inputs as probability

distributions. If the inputs describing a system are uncertain, the prediction of future

performance is necessarily uncertain. That is, the result of any analysis based on inputs

represented by probability distributions is itself a probability distribution.

Whereas the result of a single simulation of an uncertain system is a qualified statement

("if we build the dam, the salmon population could go extinct"), the result of a

probabilistic (Monte Carlo) simulation is a quantified probability ("if we build the dam,

there is a 20% chance that the salmon population will go extinct"). Such a result (in this

case, quantifying the risk of extinction) is typically much more useful to decision-makers

who utilize the simulation results.

In order to compute the probability distribution of predicted performance, it is necessary

to propagate (translate) the input uncertainties into uncertainties in the results. A variety

of methods exist for propagating uncertainty. Monte Carlo simulation is perhaps the

most common technique for propagating the uncertainty in the various aspects of a

system to the predicted performance.

In Monte Carlo simulation, the entire system is simulated a large number (e.g., 1000) of

times. Each simulation is equally likely, referred to as a realization of the system. For

each realization, all of the uncertain parameters are sampled (i.e., a single random

value is selected from the specified distribution describing each parameter). The system

is then simulated through time (given the particular set of input parameters) such that

the performance of the system can be computed. This results is a large number of

separate and independent results, each representing a possible “future” for the system

(i.e., one possible path the system may follow through time). The results of the

independent system realizations are assembled into probability distributions of possible

outcomes. As a result, the outputs are not single values, but probability distributions.

A Simple Example: Rolling Dice

As a simple example of a Monte Carlo simulation, consider calculating the probability of

a particular sum of the throw of two dice (with each die having values one through six).

In this particular case, there are 36 combinations of dice rolls:

Based on this, you can manually compute the probability of a particular outcome. For

example, there are six different ways that the dice could sum to seven. Hence, the

probability of rolling seven is equal to 6 divided by 36 = 0.167.

Instead of computing the probability in this way, however, we could instead throw the

dice a hundred times and record how many times each outcome occurs. If the dice

totaled seven 18 times (out of 100 rolls), we would conclude that the probability of

rolling seven is approximately 0.18 (18%). Obviously, the more times we rolled the dice,

the less approximate our result would be. Better than rolling dice a hundred times, we

can easily use a computer to simulate rolling the dice 10,000 times (or more). Because

we know the probability of a particular outcome for one die (1 in 6 for all six numbers),

this is simple. The output of 10,000 realizations (using GoldSim software):

How Accurate are the Results?

The accuracy of a Monte Carlo simulation is a function of the number of realizations.

That is, the confidence bounds on the results can be readily computed based on the

number of realizations. The two examples below show the 5% and 95% confidence

bounds on the value for each outcome (i.e., there is a 90% chance the the true value

lies between the bounds):

ADVANTAGES OF MONTE CARLO SIMULATION :

Simplicity.

The chief advantage of Monte Carlo simulation, compared to the other numerical

methods that can solve the same problem, is that it is conceptually very simple. It

does not require specific knowledge of the form of the solution or its analytic

properties.Monte Carlo is also relatively easy to implement on a computer.

Independence of dimension.

The amount of work to obtain the same amount of precision is independent of the

dimension of the underlying random variables.

Unrestricted choice of functions.

The functions to integrate with Monte Carlo can be practically arbitrary. No

smoothness conditions or boundedness conditions are needed, for example,

providing the integral is finite.(However, irregularities in the integrandmay impact

the accuracy of the result.)

Easily parallelizable.

Many computer processors can be participating in a Monte Carlo simulation

simultaneously. Each simulation is independent of another

REFRENECES:

http://www.investopedia.com/terms/m/montecarlosimulation.asp

http://planetmath.org/montecarlosimulation

http://www.goldsim.com/Web/Introduction/Probabilistic/MonteCarlo/