Monte Carlo estimation

01Understand the idea#

Monte Carlo estimation uses random samples to approximate a quantity that is hard to calculate directly. If the target is an expectation, evaluate the target function on each draw and average the resulting numbers. The samples must come from the distribution that defines the expectation.

Monte Carlo estimates $E[g(X)]$ by drawing independently from the target distribution and averaging the transformed draws:

For a transformed quantity, apply its function to each draw first. Draws −2,1,3 give squared payoffs 4,1,9, whose average is 14/3.

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To estimate a probability, turn each draw into an indicator: 1 if the event occurs and 0 otherwise. For outcomes $(1,0,1,1,0)$, the estimate is $3/5=0.6$. To estimate a transformed expectation instead, average the transformed outputs; applying a nonlinear function only after averaging estimates a different quantity.

02Repeated simulated values count each time they appear#

Repeated simulated values count each time they appear. Do not average only the distinct values or multiply each simulated payoff by its model probability again.

For a nonlinear payoff, transforming the sample average can give a different answer. Transform first, then average all simulated payoffs.

03A closer look#

For IID draws with finite variance of $g(X)$, the law of large numbers supports convergence to its expectation under the simulated distribution.

More draws reduce simulation variability. They do not fix sampling from the wrong distribution.

04A closer look#

Simulated model outcomes can estimate expected cost or failure probability under the chosen model. Simulation error can shrink with more independent draws, but a wrong model can still give a precise estimate of the wrong real-world quantity.