FRM Exam Part I · Simulation and Bootstrapping

Monte Carlo Simulation Basics for FRM Part I

Updated 11 October 2026 · Fact-checked

Monte Carlo simulation approximates an expected value, option price or risk measure by drawing many random scenarios from a specified model, computing the outcome in each, and averaging or ranking the results. The steps are: specify the model, generate random draws, build scenarios, compute outcomes, repeat N times, then estimate and check the standard error, which shrinks with √N.

Understand Monte Carlo Simulation Basics

Some problems have no neat formula. A path-dependent option, or a portfolio of nonlinear positions, may be too complex to solve exactly. Monte Carlo simulation gets around this. You describe how the risk drivers behave, let a computer generate many random outcomes, and measure what you care about from those outcomes.

The idea rests on the law of large numbers. If you draw X many times and average g(X), the average converges to the expected value E[g(X)]. An option price is an expected discounted payoff under the risk-neutral measure. So you simulate the underlying price, compute the payoff in each trial, average, and discount. A risk measure like VaR is a quantile of the loss distribution. So you simulate P&L many times, sort it, and read off the quantile.

The usual workflow has six parts. First, choose a model for the risk factors, such as geometric Brownian motion for a stock. Second, generate random numbers, usually uniform draws turned into standard normal draws Z. Third, turn the draws into scenarios, meaning prices or P&L at the horizon. Fourth, compute the quantity of interest in each scenario. Fifth, repeat for N trials. Sixth, summarize and report the estimate with its standard error.

The result is only an estimate. It has sampling error that falls in proportion to 1/√N. It is also only as good as the model: if the distribution or correlations you specify are wrong, a million trials will give a precise but wrong answer. For several correlated risk factors, you must draw correlated random variables, commonly using a Cholesky decomposition of the correlation matrix.

Key formulas to remember

Monte Carlo estimator: Ê[g(X)] = (1 ÷ N) × Σ g(Xᵢ), i = 1 to N; The simple average of the outcome over N independent trials. It converges to the true expected value as N grows.
Standard error of the estimate: SE = s ÷ √N; s is the sample standard deviation of the trial outcomes. To halve the SE you need four times as many trials.
Confidence interval for the estimate: Estimate ± z × SE (z = 1.96 for 95%); Uses the normal approximation, which is reasonable for large N by the central limit theorem.
GBM terminal price: S_T = S₀ × exp[(μ − σ²/2) × T + σ × √T × Z], Z ~ N(0,1); For option pricing under risk-neutral valuation, replace μ with r (or r − q if there is a dividend yield q).
Monte Carlo option price: Price = e^(−rT) × (1 ÷ N) × Σ payoffᵢ; Discount the average payoff at the risk-free rate. For a European call, payoffᵢ = max(S_T,ᵢ − K, 0).
Simulated VaR: VaR at confidence c = minus the (1 − c) quantile of simulated P&L; With N trials at 99%, VaR is roughly the loss at rank 0.01 × N when losses are sorted from worst to best.

How to solve Monte Carlo Simulation Basics questions

Use this sequence for any question on how a simulation is built, what it estimates, or how accurate it is.

1Identify the target: an expected value, an option price (discounted expected payoff), or a quantile such as VaR.
2Identify the model and its parameters: the distribution, drift, volatility, time horizon, and correlations if there are several factors. For pricing, use the risk-free rate as the drift.
3Convert each random draw Z into a scenario using the model, for example with the GBM terminal price formula.
4Compute the outcome in each scenario: payoff for options, P&L or loss for VaR.
5Aggregate: average the outcomes (and discount if pricing), or sort the outcomes and pick the quantile.
6Compute the standard error s ÷ √N if the question asks about accuracy, and the number of trials needed for a target SE.
7Check the answer: a call price must be positive and below S₀, a VaR must be a positive loss figure, and the SE must fall as N rises.

Quickest way: Fast route: label the step, then apply the formula

When to use it: Use this when a question asks which step comes next, what error falls with N, or asks for a quick estimate from a few given draws.

If the question is about accuracy, think √N: four times the trials halves the error, a hundred times the trials cuts it tenfold.
If given draws Z, compute S_T = S₀ × exp(drift + σ√T × Z) for each one, using a calculator exp key.
Take payoffs, average them, then multiply once by e^(−rT). Do not discount each trial separately and then discount again.
For VaR, convert the confidence level to a rank: (1 − c) × N. Pick that worst loss.
Eliminate options that confuse the model with the method. More trials never fix a wrong model.

Common mistakes in Monte Carlo Simulation Basics

Using the real-world expected return μ as the drift when pricing an option.
Candidates mix up simulating for risk measurement with simulating for pricing.
Fix: For pricing, simulate under the risk-neutral measure with drift r (or r − q), then discount at r. For VaR, use the real-world drift (often set near zero over short horizons).
Forgetting the −σ²/2 term in the GBM exponent.
It is easy to remember S₀ × exp(μT + σ√T Z) from log returns and drop the correction.
Fix: Write the exponent as (μ − σ²/2)T + σ√T Z every time. The correction keeps the expected terminal price at S₀ × e^(μT).
Believing that doubling the trials halves the standard error.
People assume error falls linearly with N.
Fix: SE = s ÷ √N. Halving the error needs four times the trials. Doubling N cuts SE by a factor of about 1.41.
Assuming more simulations fix model error.
A large N feels like it makes the answer 'right'.
Fix: More trials reduce only sampling error. Wrong distributions, volatility or correlations bias the result no matter how large N is.
Discounting the payoff twice or not at all.
Candidates discount inside the loop and again at the end, or skip it after averaging.
Fix: Average the undiscounted payoffs, then multiply by e^(−rT) once.
Reading VaR from the wrong tail of the sorted outcomes.
Sorting from best to worst, or mixing up profit and loss signs.
Fix: Sort P&L from worst to best. For 99% VaR, use the 1% worst outcome and report it as a positive loss.

Worked examples

Example 1

A stock trades at $100. The risk-free rate is 5% a year, volatility is 20%, and there is no dividend. A one-year European call has strike $100. A simulation draws four standard normal values: Z = −1, −0.5, 0.5, 1. Estimate the call price with these four draws (illustration only; real runs use thousands).

Show the solution

Use risk-neutral GBM: S_T = 100 × exp[(0.05 − 0.20²/2) × 1 + 0.20 × 1 × Z] = 100 × exp(0.03 + 0.2Z).
Z = −1: exp(−0.17) = 0.8437, so S_T = 84.37 and payoff = 0.
Z = −0.5: exp(−0.07) = 0.9324, so S_T = 93.24 and payoff = 0.
Z = 0.5: exp(0.13) = 1.1388, so S_T = 113.88 and payoff = 13.88.
Z = 1: exp(0.23) = 1.2586, so S_T = 125.86 and payoff = 25.86.
Average payoff = (0 + 0 + 13.88 + 25.86) ÷ 4 = 9.936.
Discount: e^(−0.05) = 0.9512, so price = 9.936 × 0.9512 = 9.45.

Answer: About $9.45. With only four draws the estimate is rough and would have a large standard error.

Example 2

A simulation of a one-day portfolio P&L uses 10,000 trials. Sorted from worst to best, the 100th worst outcome is a loss of $2.4 million and the 500th worst is a loss of $1.6 million. (a) Estimate the 99% one-day VaR. (b) The pricing simulation for another product has a sample standard deviation of payoffs of 12 and N = 10,000. Find the standard error, and the N needed to reach a standard error of 0.03.

Show the solution

(a) 99% confidence leaves 1% in the tail: 0.01 × 10,000 = 100 trials.
The 100th worst loss marks the 1% quantile, so VaR ≈ $2.4 million. (The 500th worst, $1.6 million, would be the 95% VaR, so it is not the answer.)
(b) SE = s ÷ √N = 12 ÷ √10,000 = 12 ÷ 100 = 0.12.
For SE = 0.03: 12 ÷ √N = 0.03, so √N = 400.
N = 400² = 160,000.

Answer: (a) 99% one-day VaR ≈ $2.4 million. (b) SE = 0.12; you need N = 160,000 trials, which is 16 times as many, to get SE down to 0.03.

Exam tips

Questions often test the √N rule. Know that four times the trials halves the error, and be ready to compute the N needed for a target SE.
Know the ordered steps: specify the model, generate random numbers, build scenarios, compute outcomes, repeat N times, summarize. Expect a question asking which step comes first or is missing.
Check the drift in any pricing question. Risk-neutral drift is r (or r − q), and the discount factor uses r.
For VaR, convert confidence to a rank with (1 − c) × N, and state VaR as a positive loss.
Know the limits: Monte Carlo is slow for large portfolios, depends on the model assumed, and needs correlated draws for several risk factors.

Practice questions from Simulation and Bootstrapping

Monte Carlo Simulation Basics in other exams

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

Monte Carlo Simulation Basics: frequently asked questions

How does Monte Carlo simulation work in finance?

You specify a model for risk factors, such as geometric Brownian motion, and draw many random scenarios from it. In each scenario you compute a payoff or a P&L. You then average the results to get an expected value or price, or sort them to get a quantile such as VaR.

What are the steps of Monte Carlo simulation in FRM?

Specify the stochastic model and parameters, generate random draws, convert them into scenarios, compute the outcome in each scenario, repeat for N trials, and aggregate. Finally, assess accuracy with the standard error.

How is Monte Carlo used to calculate VaR?

You simulate the risk factors over the VaR horizon, revalue the portfolio in each trial, and record the P&L. Sort the results from worst to best. The VaR at confidence c is the loss at the (1 − c) quantile, for example the 1% worst outcome for 99% VaR.

How many simulations do I need?

It depends on the accuracy you want. Standard error equals s ÷ √N, so you can solve for N from a target SE. Because error falls only with √N, each extra digit of accuracy is costly, which is why variance reduction techniques matter.