Monte Carlo Pricing
Price derivatives by simulating many risk-neutral paths and averaging discounted payoffs.
Monte Carlo pricing simulates many paths of the underlying under the risk-neutral measure.
For each path it computes the payoff at maturity, discounts it back, and averages across paths.
The law of large numbers ensures convergence to the arbitrage-free price, but sampling error shrinks only as 1/√N.
This visual shows Monte Carlo pricing of a European option: many simulated risk-neutral paths for S_t, with the option value given by the discounted average payoff at maturity.
Each blue line is one simulated risk-neutral path for the underlying. The option payoff depends only on S_T and whether it finishes above or below the strike K; the Monte Carlo price is the discounted average payoff across all paths.
As you increase the number of paths, the Monte Carlo estimate fluctuates less and moves closer to the analytic Black–Scholes price, but convergence is slow: the noise falls only like 1/√N. Monte Carlo is flexible and model-agnostic but computationally expensive, which is why variance reduction and good random generators matter in production systems.