Brute-Force Pricing
Monte Carlo simulation is the Swiss army knife of quantitative finance. Instead of deriving an elegant formula, you simply simulate the future thousands of times and average the results. Generate random stock price paths, calculate the payoff for each path, discount back to today, and take the mean. With enough paths, the answer converges to the true price — guaranteed by the law of large numbers.
Simulating Stock Prices
Each price path follows geometric Brownian motion: at each time step, the stock price is multiplied by a random factor drawn from a log-normal distribution. The drift is set to the risk-free rate (risk-neutral pricing), and the volatility determines how wildly the paths spread. Watch the simulation draw thousands of paths fanning out from the initial price — the visual distribution of final prices reveals the option's probability landscape.
Convergence and Accuracy
The Monte Carlo estimate improves with more paths, but convergence is slow: halving the error requires quadrupling the paths. The standard error quantifies uncertainty in the estimate. Variance reduction techniques — antithetic variates (pairing each random draw with its negative), control variates (adjusting using a known analytical price), and importance sampling — can accelerate convergence by orders of magnitude.
Beyond Vanilla Options
Monte Carlo's real power emerges for exotic options that lack closed-form solutions. Asian options (averaging the price path), barrier options (knocked in or out at threshold prices), lookback options (depending on the path extremum), and multi-asset basket options all yield naturally to simulation. This flexibility makes Monte Carlo the dominant pricing method for structured products in modern investment banks.