Skip to content

FRM Part I · FRM Exam Part I

Simulation and Bootstrapping for FRM Part I

Monte Carlo simulation generates many random scenarios from a model to estimate a quantity such as an expected payoff or a VaR. Bootstrapping resamples your observed data with replacement instead of assuming a distribution. To solve questions, identify the method, apply the standard error formula σ ÷ √N, and judge the limits.

What this chapter covers

This chapter covers two ways to estimate quantities when a closed-form answer is hard or unavailable. Monte Carlo simulation draws random variables from a model you specify, computes an outcome for each draw, and averages the results. Bootstrapping draws from your own observed data, with replacement, and needs no distributional assumption.

The core ideas are few. The estimate is a sample mean, so its error shrinks with 1 ÷ √N. Variance reduction techniques such as antithetic variates and control variates cut that error without more runs. Both methods have limits: bad models, bad data and bad random numbers give bad answers, however many runs you do.

The chapter links to the rest of Part I. It builds on probability, the central limit theorem, standard errors and confidence intervals from Quantitative Analysis. It feeds directly into option pricing, VaR and expected shortfall in Valuation and Risk Models, where simulation and historical (bootstrapped) methods are compared with analytical ones.

The chapter is compact, and its questions are often short and formula-driven, so it rewards a small amount of focused work. GARP does not publish a weighting per chapter, but the ideas recur elsewhere: standard error logic appears in hypothesis testing, and simulation versus historical methods appears in VaR questions. If you master the √N rule, the purpose of each variance reduction method and the main limits of bootstrapping, you can collect marks quickly in a 100-question, 4-hour paper where time per question is limited.

Simulation and Bootstrapping: topics in the order to study them

  1. 1Monte Carlo Simulation BasicsStart here. It defines the steps (model, draw, compute, average) that every other topic modifies or criticises.
  2. 2Sampling Error and Number of SimulationsNext, learn how accuracy depends on N through the standard error σ ÷ √N, since variance reduction is judged against it.
  3. 3Variance Reduction TechniquesOnce you know the error formula, you can see how antithetic and control variates shrink it without extra runs.
  4. 4BootstrappingThis is the second method. It reuses the sampling logic you already know, but draws from observed data instead of a model.
  5. 5Limitations of Bootstrapping and SimulationStudy the weaknesses after both methods are clear, so you can compare them, for example iid data, outliers and model risk.
  6. 6Random Number GenerationFinish with the engine underneath: pseudo-random numbers, seeds and reproducibility, which tie back to simulation quality.

How to prepare Simulation and Bootstrapping

Aim for understanding of why each step works, then practise short calculations. Most marks come from a few formulas and clear conceptual contrasts.

  1. Write out the Monte Carlo steps in your own words: specify the model, draw random numbers, compute the outcome per path, average, and discount if a price is needed.
  2. Practise the standard error: SE = σ ÷ √N. Work out how many runs you need to halve the error (four times as many) and to cut it to a tenth (one hundred times as many).
  3. Build a one-line description of each variance reduction method: what it pairs or uses, and why the variance falls. Note that control variates depend on correlation with a variable whose expected value is known.
  4. Do a small bootstrap by hand: list five returns, resample with replacement, compute the statistic, repeat, and read the spread of results as its sampling variability.
  5. Make a two-column table in your notes comparing simulation, bootstrap and analytical methods on assumptions, data needs and failure modes.
  6. Finish with timed practice questions and review each wrong answer by naming the concept you missed. Do this on your phone in short sessions if needed.

Common mistakes in Simulation and Bootstrapping

  • Thinking the error falls in proportion to N

    Fix: Remember the error falls with √N. Ten times the runs cuts the standard error by about a factor of 3.16 (√10), not 10.

  • Believing more simulations fix model errors

    Fix: Separate the two. More runs shrink sampling error around the model's answer; they do not make the model right.

  • Mixing up antithetic and control variates

    Fix: Antithetic: pair each draw with its opposite. Control: adjust using a correlated variable whose true mean is known.

  • Saying bootstrapping assumes a normal distribution

    Fix: Bootstrapping is non-parametric. Its key assumption is that observations are iid and the sample represents the population.

  • Resampling without replacement

    Fix: Bootstrap draws are with replacement, so each observation can appear several times in a resample.

  • Treating pseudo-random numbers as truly random

    Fix: Know that a fixed seed reproduces the same sequence, which helps testing but means quality depends on the generator.

Last-day revision: Simulation and Bootstrapping

  • Monte Carlo estimate = average of the outcomes from N simulated paths.
  • Standard error of the estimate = σ ÷ √N, where σ is the standard deviation of the outcome.
  • To halve the standard error, you need four times as many simulations.
  • Increasing N reduces sampling error only; it does not fix a wrong model.
  • Antithetic variates use a draw and its mirror image (for example Z and −Z) to induce negative correlation and cut variance.
  • Control variates use a related variable with a known expected value to adjust the estimate; the benefit grows with the correlation.
  • Bootstrapping resamples observed data with replacement and assumes the observations are iid.
  • Bootstrapping cannot create outcomes outside those present in the sample, so tail events may be missed.
  • Simulation is useful for path-dependent and complex payoffs where no closed form exists.
  • Pseudo-random numbers are deterministic given a seed, which allows results to be reproduced.
  • Both methods are only as good as their inputs: model assumptions for simulation, representative data for bootstrapping.

Simulation and Bootstrapping practice questions

Simulation and Bootstrapping in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Simulation and Bootstrapping: frequently asked questions

What is the difference between Monte Carlo simulation and bootstrapping?

Monte Carlo draws random numbers from a model you specify, such as a normal distribution for returns. Bootstrapping draws with replacement from your actual observed data and needs no distribution assumption. Both estimate a statistic by repeating the process many times.

How many simulations do I need?

It depends on the accuracy you want. The standard error is σ ÷ √N, so to halve the error you need four times as many runs. Variance reduction can lower the error without increasing N.

Do I need to memorise formulas for this chapter?

Very few. The key one is the standard error σ ÷ √N, and you should know how it scales with N. The rest is conceptual: what each technique does and where it fails.

When does bootstrapping fail?

It struggles when data are not iid, such as with serial correlation or volatility clustering, and when the sample is small or lacks extreme events. It can only reuse outcomes already seen, so unseen tail losses are not generated.