What is a Monte Carlo simulation?
A Monte Carlo simulation estimates the distribution of an outcome by repeatedly sampling random values for its uncertain inputs, running the model for each draw, and aggregating the results. Instead of one deterministic answer you get the full range of possibilities and their likelihoods. The method was developed at Los Alamos in the 1940s by Stanislaw Ulam, John von Neumann and Nicholas Metropolis, and named after the Monte Carlo casino.
Risk, percentiles and convergence
The output of a simulation is a distribution, summarized by its mean, its percentiles (the P10, P50/median and P90), and the probability of exceeding a threshold — the language of risk analysis, Value-at-Risk and project contingency. The statistical error of a Monte Carlo estimate shrinks in proportion to 1/√N, so quadrupling the number of trials roughly halves the error.
How to use the simulator
Describe each uncertain input with a range or a probability distribution, set the number of trials (more trials = smoother, more accurate tails), and run. Read the histogram, the P10/P50/P90 band and the probability of hitting or missing your target value.
Note: results are only as good as the input distributions and the assumption of how inputs relate. Correlations between inputs, fat tails and rare events can dominate risk and are easy to under-model; treat the output as a structured estimate, not a guarantee.
Frequently asked questions
What is a Monte Carlo simulation?
A technique that samples random values for uncertain inputs many times, runs a model for each sample and aggregates the results into a distribution of outcomes, so you see the full range of possibilities and their probabilities rather than a single estimate.
How many iterations does a Monte Carlo simulation need?
Enough that the results stop changing meaningfully. Because the error falls as 1/sqrt(N), thousands of trials are typical and tens of thousands give stable tail percentiles; quadrupling the trials roughly halves the statistical error.
What does P90 mean in a Monte Carlo result?
P90 is the 90th percentile: 90% of simulated outcomes fall at or below it. Together with P10 and the median (P50), it summarizes the spread of results and the downside/upside risk.
What is Monte Carlo simulation used for?
Estimating risk and uncertainty in finance (Value-at-Risk, option pricing), project cost and schedule, engineering reliability, physics and any problem where inputs are uncertain and an analytic answer is hard.
References
- N. Metropolis & S. Ulam (1949), "The Monte Carlo Method", Journal of the American Statistical Association 44(247):335–341.
- P. Glasserman, Monte Carlo Methods in Financial Engineering (Springer).
- R. Y. Rubinstein & D. P. Kroese, Simulation and the Monte Carlo Method, 3rd ed. (Wiley).