Monte Carlo Simulation for Derivative Pricing › Summary

This chapter developed Monte Carlo simulation as a powerful numerical method for derivative pricing. The core principle is simple: simulate many possible futures under risk-neutral dynamics, compute payoffs, and average.

We saw that for European options depending only on terminal prices, a single simulation step using the exact GBM solution suffices. For path-dependent options like Asians, barriers, and lookbacks, we simulate full trajectories using the log-Euler discretization scheme.

The Monte Carlo estimator converges to the true price as the number of simulations grows, with standard error proportional to $1/\sqrt{N}$. Confidence intervals quantify uncertainty in our estimates. This slow convergence motivates variance reduction techniques covered in the next chapter.

Practical implementation requires vectorized code for efficiency, proper random number generation, and sometimes batched processing for memory management. Multi-asset options extend naturally through correlated random variables generated via Cholesky decomposition.

Monte Carlo's great strength is flexibility: it handles virtually any payoff structure. Its weakness is speed: when closed-form solutions exist, they are vastly faster. You must know when each tool applies and match method to problem.