FRM Part I · FRM Exam Part I

Simulation and Bootstrapping: formula sheet

Full chapter guide

Key formulas

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.
Standard error of a simulated mean: SE = σ ÷ √N; σ is the standard deviation of the simulated outcomes (use the sample standard deviation s if σ is unknown). N is the number of independent replications.
Confidence interval for the estimate: x̄ ± z × σ ÷ √N; z is 1.645 for 90%, 1.96 for 95% and 2.576 for 99% two-sided confidence, based on the normal approximation.
Required number of simulations: N = (z × σ ÷ E)²; E is the target half-width of the confidence interval, the maximum acceptable error at the chosen confidence level.
Scaling rule: SE₂ ÷ SE₁ = √(N₁ ÷ N₂); To cut error by a factor k, multiply N by k². Holds when σ is unchanged.
Standard error of a Monte Carlo estimate: SE = σ ÷ √N; Quadrupling N halves SE. This is the baseline that variance reduction improves on.
Antithetic pair variance: Var[(X₁ + X₂) ÷ 2] = (σ² ÷ 2) × (1 + ρ); X₁ and X₂ have the same variance σ². ρ is their correlation. ρ < 0 helps; ρ = 0 gives no gain over two independent draws.
Control variate adjusted estimator: X* = X − b(Y − E[Y]); E[Y] must be known exactly. The adjusted estimator stays unbiased because E[Y − E[Y]] = 0.
Optimal control coefficient: b* = Cov(X, Y) ÷ Var(Y) = ρ × σX ÷ σY; This is the regression slope of X on Y.
Variance after optimal control: Var(X*) = Var(X) × (1 − ρ²); Gain depends on the size of ρ, not its sign. ρ = 0.9 removes 81% of the variance.
Bootstrap sample: Draw n values from n observations, with replacement, each draw equally likely (probability 1/n); Each resample has the same size as the original sample. Repeat B times to get B statistics θ*₁, …, θ*_B.
Bootstrap standard error: SE(θ̂) ≈ √[ Σ (θ*_b − mean of θ*)² ÷ (B − 1) ], summed over b = 1 to B; This is the sample standard deviation of the B bootstrap estimates. It is not the standard deviation of the raw data.
Percentile confidence interval: For a (1 − α) interval, use the α/2 and (1 − α/2) percentiles of the B bootstrap estimates; For a 95% interval, use the 2.5th and 97.5th percentiles.
Chance an observation is left out of one resample: (1 − 1/n)ⁿ, which approaches e⁻¹ ≈ 0.368 as n grows; So about 63.2% of the original observations appear at least once in a large resample.
Bootstrap bias estimate: Bias ≈ mean of θ* − θ̂; θ̂ is the statistic from the original sample. Mean of θ* is the average of the bootstrap statistics.
Bootstrap resampling rule: Draw n observations with replacement from the original n observations; compute the statistic; repeat B times; Each draw is from the empirical distribution. The bootstrap standard error is the standard deviation of the B statistics.
Bootstrap i.i.d. condition: Observations independent and identically distributed, and sample representative of the population; If either part fails, the bootstrap can be unreliable.
Simulation sampling error: Standard error of a simulated mean ≈ s ÷ √N; Quadrupling the number of simulations N roughly halves the standard error. It does not fix model risk.
Bootstrap standard error: SE = √[ Σ(θ*ᵢ − θ̄*)² ÷ (B − 1) ]; θ*ᵢ is the statistic from resample i and θ̄* is the average across B resamples.
Inverse transform method: X = F⁻¹(U), where U ~ Uniform(0, 1); Works for any distribution whose CDF can be inverted. X then has CDF F.
Standard normal from uniform: Z = N⁻¹(U); Excel: NORM.S.INV(RAND()). U must lie strictly between 0 and 1.
General normal draw: X = μ + σ × Z; Z is a standard normal draw. Use σ, not σ².
Lognormal price draw: S_T = S_0 × exp[(μ − σ²/2)T + σ√T × Z]; Standard geometric Brownian motion over time T. Under risk-neutral pricing, use the risk-free rate (net of any yield) for μ.
Correlated normals (two variables): Z₂ = ρ × ε₁ + √(1 − ρ²) × ε₂; ε₁ and ε₂ are independent standard normals and Z₁ = ε₁. Then Corr(Z₁, Z₂) = ρ.
Standard error of a simulation estimate: SE = s ÷ √N; s is the sample standard deviation of the outputs and N is the number of trials. Changing the seed does not change the expected SE.

Quick revision

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.

Common mistakes

Using the real-world expected return μ as the drift when pricing an option. 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. 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).
Thinking that doubling N halves the standard error. Fix: Error falls with √N. Doubling N cuts error by about 29% (a factor of 1 ÷ √2). Halving error needs 4 times N.
Using σ ÷ N instead of σ ÷ √N. Fix: Either use the standard deviation σ ÷ √N, or the variance σ² ÷ N. Never σ ÷ N.
Saying variance reduction works by increasing the number of draws. Fix: Remember the point: same N, lower variance. The techniques change what is averaged, not how many draws are used.
Using a control variate whose expected value is unknown or estimated. Fix: The control needs a known E[Y], often from a closed-form price. Otherwise the adjustment adds bias or noise.
Resampling without replacement Fix: Remember that replacement is what makes each resample different. Without it, a full-size resample would just reproduce the original data.
Using the data's standard deviation as the bootstrap standard error Fix: The bootstrap standard error is the standard deviation of the statistic across resamples, for example of the B resampled means.
Saying more Monte Carlo draws fix a wrong model Fix: More draws shrink sampling error only. A misspecified distribution or correlation stays wrong.
Claiming bootstrapping needs a normality assumption Fix: Standard bootstrapping is nonparametric. Its key assumptions are i.i.d. data and a representative sample.

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.
Memorise the √N rule as a scaling fact: error ÷ 2 means N × 4, error ÷ 10 means N × 100. Many questions are solved in seconds this way.
Read carefully whether the given standard deviation is for one run or for the average. Only divide by √N for a single-run σ.
If a question asks which change reduces error most efficiently, a variance reduction technique that lowers σ is often the better answer than raising N.