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Probability, Statistics and Financial Data / Limit Theorems

Applications — Monte Carlo Estimation, Error, and Confidence Intervals

Every tool this unit built — expectation, variance, the weak law, Chebyshev's inequality, the Central Limit Theorem — comes together in a single, immensely practical technique: Monte Carlo estimation. Whenever E[g(X)]\mathbb{E}[g(X)] is hard to compute analytically (an option price under a complicated model, the probability a basket suffers more than kk defaults, the value of π\pi via Buffon's needle), simulate many independent draws, average, and let the weak law guarantee convergence and the Central Limit Theorem quantify the error. This capstone lesson closes the unit by building that procedure end-to-end: the estimator, its standard error, a confidence interval, and a worked option-pricing example that ties directly into the vault's learn/program/options-pricing/ unit.
Learning objectives. By the end of this note you should be able to:
  • construct a Monte Carlo estimator for E[g(X)]\mathbb{E}[g(X)] and state why it is unbiased and consistent;
  • compute the standard error of a Monte Carlo estimator and use it to build a CLT-based confidence interval;
  • determine the number of simulations needed to achieve a target precision;
  • explain why Monte Carlo error shrinks at the 1/n1/\sqrt n rate, and what this implies about the cost of higher precision.

In this lesson

  1. The Monte Carlo estimator
  2. Standard error and confidence intervals
  3. Worked example: Monte Carlo pricing of a European call
  4. The 1/n1/\sqrt n rate and its consequences
  5. Closing the unit

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