Running a Monte Carlo simulation, step by step

In short

• Specify: (1) the quantity of interest and starting values, (2) a time grid of K sub-periods of length $\Delta t$, (3) the model and the distributions of the key risk factors.
• Run: (4) draw random values and convert them into a price path, (5) compute the payoff and discount it to today, (6) repeat for I trials and average.
• A common price model: $\Delta P = \mu P_{\text{prior}}\Delta t + \sigma P_{\text{prior}} Z_k$, where $Z_k$ is a standard normal draw, the key risk factor.
• One trial = one path = one discounted payoff $C_{i,0}$. The Monte Carlo value is the mean of all the $C_{i,0}$.
• Change the payoff rule in step 5 (average price, minimum price) and the same engine values a different claim.