🎲 Monte Carlo Estimation: From π to Financial Risk

Geometric Brownian motion (GBM) models an asset price S as dS = μS dt + σS dW, where μ is the annualised drift, σ the annualised volatility, and dW a Wiener process increment (Gaussian noise). It is the standard building block of the Black–Scholes framework because it keeps prices positive and gives log-returns that are Normally distributed, matching the rough shape of real return data reasonably well over short-to-medium horizons.

Value-at-Risk at the 5% level is the loss that should not be exceeded with 95% confidence over a given horizon. This simulator estimates it empirically: it simulates many independent GBM price paths, collects their final prices, and reads off the 5th percentile of that empirical distribution — the loss at that percentile is the simulated VaR. A theoretical VaR is also shown, computed in closed form from the known lognormal distribution of GBM's terminal price.

VaR is one of the most widely used risk metrics in finance, but it has well-known weaknesses: it says nothing about how bad losses beyond the threshold can be (it is not "coherent" in the technical sense, unlike Expected Shortfall/CVaR), it can encourage risk concentration in the tail, and real markets exhibit fat tails, volatility clustering and jumps that a simple GBM model understates. It remains standard because it is intuitive and simple to compute, but risk managers usually pair it with other measures.

Grid quadrature evaluates a function at deterministic, evenly spaced points and combines them with fixed weights; its error typically shrinks polynomially in the number of grid points per dimension, but the total number of points needed to cover a d-dimensional grid grows exponentially in d. Monte Carlo instead evaluates the function at random points; its error shrinks as O(1/√N) independent of dimension, so it becomes the more efficient choice once a problem has more than roughly 3–4 dimensions — a big reason it dominates in finance and physics, where models often have dozens or thousands of dimensions.

It is both rigorous and practical. Its convergence properties are proven by the law of large numbers and the central limit theorem, giving quantifiable error bars (the standard error shown live in this simulator), not just a plausible-looking answer. Monte Carlo methods are used throughout physics (particle transport, statistical mechanics), engineering (reliability analysis), and finance (derivative pricing, risk management) precisely because their accuracy can be estimated and controlled by choosing N.