Monte Carlo Simulation
A probabilistic technique that simulates thousands of random scenarios to estimate the distribution of outcomes.
Definition
Named after the famous casino, Monte Carlo simulation repeatedly samples random inputs from a model's input distributions to generate a distribution of outcomes. In finance it is used for option pricing, risk management (VaR, CVaR), retirement planning, and forward-looking scenario analysis. Crucially, it produces a full probability distribution — not a single point estimate — making it ideal for risk reporting. The accuracy improves with sqrt(N) of simulations, so 10,000 paths are typically enough for stable percentiles.
Formula
For i in 1..N:
Sample inputs X_i ~ joint distribution
Compute output Y_i = f(X_i)
Estimate quantiles, mean, stdev from {Y_i}
Example for prices (GBM):
S_{t+dt} = S_t * exp( (mu - 0.5*sigma^2)*dt + sigma * sqrt(dt) * Z )
Z ~ N(0,1)Worked example
Starting capital $100k, expected return 7% per year, volatility 15%, 30-year horizon. Run 10,000 Geometric Brownian Motion paths: the 5th-95th percentile range of final wealth is roughly $180k-$2.4M, with median ~$760k. The wide dispersion is the real lesson — point estimates lie.
How ARIA Analyst uses it
ARIA runs 10,000-path Monte Carlo on every portfolio analysis and on individual stocks, with optional fat-tail (Student-t) and correlated multi-asset modes.
Related terms
Value at Risk (VaR)
The maximum loss expected over a given horizon at a given confidence level.
Conditional VaR (Expected Shortfall)
The average loss in the worst (1 - alpha)% of cases — a coherent tail-risk measure.
Volatility (σ)
Standard deviation of returns — the most common dispersion-based risk measure.
Walk-Forward Analysis
A backtesting procedure that retrains the model on a rolling window and tests on the next out-of-sample period.
See Monte Carlo Simulation in action on any asset
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