Monte Carlo in Finance

The questions a quant actually gets paid to answer rarely have a formula: Will my retirement pot survive 30 years of random markets? What’s a barrier option worth when its payoff depends on the whole path? How fat is the tail when ten correlated positions crater together? When the pencil runs out of road, you stop solving and start rolling dice — Monte Carlo simulation: generate ten thousand random futures, measure each, and let the law of large numbers converge the average to an answer no equation can give you.

Here’s what we build, from the ground up:

• What Monte Carlo is — the law of large numbers, why random sampling converges, and the dart-throwing trick that estimates π with no circle formula.
• Sampling from distributions — turning uniform random numbers into normal returns, fat-tailed shocks, anything you can describe.
• Geometric Brownian motion — the price engine behind the widening fan of simulated paths under half of modern finance.
• Portfolio & retirement outcomes — the cone of futures, sequence-of-returns risk, and the odds your money outlives you.
• Path-dependent options — pricing Asians and barriers by simulating the whole journey and averaging the payoff.
• Convergence & error — the brutal $1/\sqrt{M}$ rule, confidence intervals on a simulated number, and variance-reduction tricks (antithetic and control variates) that buy the same accuracy from a quarter of the paths.

This is the workhorse of quantitative finance — under VaR engines, option desks, retirement calculators and risk dashboards alike. By the end you can take a problem with no clean solution, build a simulator for it, and read its convergence honestly — including when a beautiful ten-thousand-path answer is just a precise rendering of a garbage assumption.