8.1 Simulation and Bootstrapping

Simulation and Bootstrapping

Risk managers often need expectations, probabilities, or quantiles that have no closed form: a portfolio VaR under nonlinear options, a CVA under stochastic exposure, or a capital number under a multifactor loss model. Monte Carlo simulation and bootstrapping are the two workhorse numerical tools in GARP’s QA–13 reading. Monte Carlo draws scenarios from a model; the bootstrap redraws from data. Both produce sampling error you must manage, and both fail in predictable ways when assumptions break.

Monte Carlo: Core Idea and Steps

A Monte Carlo estimate of θ = E[g(X)] replaces the unknown expectation with a sample average of simulated draws:

θ̂_N = (1/N) Σ_{i=1}^{N} g(X^{(i)})

where X^{(i)} are i.i.d. draws from the model distribution of X, and g is the payoff, loss, indicator, or other functional of interest. By the law of large numbers, θ̂_N → θ as N → ∞ (under standard integrability). By the CLT, √N (θ̂_N − θ) is approximately normal with variance Var(g(X)), so the standard error is roughly σ_g / √N.

Standard Monte Carlo workflow

1. Specify the model: risk-factor dynamics, dependence (correlation, copula), and the mapping from factors to P&L or loss g(·).
2. Calibrate: estimate drifts, volatilities, correlations, mean reversion, or jump intensities from market or historical data.
3. Generate scenarios: draw pseudo-random shocks; transform them into factor paths or end-of-horizon states.
4. Revalue: compute g(X^{(i)}) for each scenario (full reval, Greeks-based approx, or grid).
5. Aggregate: average for expectations; sort for empirical quantiles (VaR); average losses beyond VaR for expected shortfall.
6. Report error: standard errors, confidence intervals, or convergence diagnostics as N grows.

Worked Monte Carlo sketch (expectation)

Suppose a one-period loss L = max(−ΔS, 0) on a long put-like exposure where ΔS ~ N(0, 4) in dollars (σ = 2). You want E[L]. Closed form exists (half-normal style), but simulate for method practice.

Draw N = 10,000 standard normals Z_i, set ΔS_i = 2 Z_i, L_i = max(−ΔS_i, 0). Suppose the simulated mean is θ̂ = 0.798 and the sample SD of L_i is s ≈ 1.20. Then

SE(θ̂) ≈ 1.20 / √10,000 = 0.012.

A rough 95% CI is 0.798 ± 1.96 × 0.012 ≈ [0.774, 0.822]. Doubling precision (halving the SE) requires about 4× as many paths because of the 1/√N law—not 2×.

Sampling Error and Why N Matters

Sampling error (Monte Carlo error) is the randomness of θ̂_N around the true model expectation θ. It is not the same as parameter uncertainty or model risk. Even with a perfect model and infinite data for calibration, finite N leaves noise.

Rules of thumb FRM candidates memorize:

• SE ∝ 1/√N for plain averages.
• Quantile estimators (historical-style VaR from simulated losses) also improve with N, but variance formulas differ and tails need many more paths than means.
• Rare-event probabilities P(L > K) with small p need N on the order of many multiples of 1/p for a usable relative error.

Worked SE scaling

If SE = 0.040 at N = 2,500, then at N = 10,000 (4× paths) SE ≈ 0.040 / 2 = 0.020. At N = 40,000, SE ≈ 0.010. Brute-force path inflation is expensive for production desks; variance reduction is the smarter first move.

Antithetic Variates

Antithetic variates pair each shock U (or Z) with a negatively associated twin. For symmetric uniforms, use U and 1 − U. For standard normals, use Z and −Z. Estimate

θ̂_anti = (1/M) Σ_{j=1}^{M} [g(X(Z_j)) + g(X(−Z_j))] / 2

with M pairs (2M paths of work, but strongly dependent).

Why it helps: if g(X(Z)) is roughly monotone in Z, then g(X(Z)) and g(X(−Z)) are negatively correlated, so the average of the pair has lower variance than a single independent draw. For linear g of a symmetric shock, the antithetic pair can cancel odd noise dramatically.

Limitation: if g is highly asymmetric or non-monotone (digital payoffs, barriers, strong optionality), antithetic correlation may be weak or even harmful. Always measure the empirical variance of the paired estimator versus crude Monte Carlo at equal compute.

Worked antithetic intuition

Crude: two independent losses L₁, L₂ with Var(L) = 4, Corr = 0 → Var((L₁+L₂)/2) = 2. Antithetic pair with Corr(L⁺, L⁻) = −0.6: Var((A+B)/2) = (1/4)(Var A + Var B + 2 Cov). With Var A = Var B = 4 and Corr = −0.6, Cov = −2.4, so Var = (1/4)(4 + 4 − 4.8) = 3.2/4 = 0.80, versus 2 for the independent average—about a 60% variance cut for that pair average.

Control Variates

A control variate uses a related quantity Y whose expectation E[Y] is known analytically (or to high accuracy). For each path compute g(X) and Y, then form

g*(X) = g(X) − β (Y − E[Y])

Choose β to minimize Var(g*). The optimal β* = Cov(g, Y) / Var(Y), estimated from a pilot sample or the same run with care. Then E[g*] = E[g], so the estimator stays unbiased (for known E[Y] and fixed β), but variance falls when Y tracks g.

Classic risk/finance example: price a Asian option by Monte Carlo while using the geometric-Asian or European closed-form price as the control. In market risk, a linear delta-normal P&L can control a full-reval nonlinear P&L if they move together.

Bootstrapping Versus Monte Carlo

Bootstrapping treats the observed sample {x₁, …, xₙ} as an empirical distribution. Draw B resamples of size n with replacement, recompute the statistic T on each resample, and use the distribution of T₁, …, T_B to approximate the sampling distribution of T (standard errors, bias, confidence intervals, VaR of a historical portfolio, etc.).

Worked bootstrap sketch

Daily returns (n = 5 for toy math): 0.01, −0.02, 0.00, 0.03, −0.01. Sample mean x̄ = 0.002. One bootstrap resample might be {0.01, 0.01, −0.01, 0.03, 0.01} with mean 0.010; another {−0.02, 0.00, −0.02, −0.01, 0.00} with mean −0.010. After B = 1,000 such means, the SD of the bootstrap means estimates SE(x̄). With real n = 250, the same idea estimates uncertainty of VaR, Sharpe ratios, or regression coefficients without assuming normality.

Parametric bootstrap: fit a model to data, then simulate from the fitted model (a hybrid of bootstrap spirit and Monte Carlo mechanics).

Pseudo-Random Numbers

Computers do not produce true randomness for standard Monte Carlo. They use pseudo-random number generators (PRNGs): deterministic algorithms that emit sequences Uᵢ ∈ (0,1) designed to mimic i.i.d. Uniform(0,1). From uniforms, inversion or Box–Muller / Ziggurat methods produce normals and other laws.

Properties that matter in production risk systems:

• Reproducibility: same seed → same scenarios → auditability and debugging.
• Period: long cycle before the sequence repeats (modern generators have enormous periods).
• Statistical quality: pass tests of uniformity and independence; poor generators create spurious patterns in high dimension.
• Dimension: a path with many time steps and risk factors consumes many uniforms; correlations across dimensions must remain well-behaved.

Quasi-Monte Carlo (low-discrepancy sequences such as Sobol) is a related topic: not “random,” but designed for even coverage; FRM focuses more on pseudo-random MC plus variance reduction than on QMC theory depth.

When the Bootstrap Fails

The classical i.i.d. bootstrap assumes that redrawing independent observations from the empirical distribution mimics the true sampling process. It breaks or misleads when:

1. Dependence: returns are serially correlated, clustered in volatility, or cross-sectionally dependent. Naive i.i.d. resamples destroy the dependence structure → SEs too small or intervals wrong. Fixes: block bootstrap, stationary bootstrap, or model the dependence then resample residuals.
2. Heavy tails / insufficient n: extreme quantiles (99.9% VaR) estimated from a short sample are noisy; bootstrap cannot invent tail mass that the sample never showed.
3. Non-smooth statistics: indicators, max functions, or some quantile functionals can make bootstrap consistency fail or converge slowly.
4. Biased sampling frame: if the historical window omits crises, every bootstrap resample still omits crises—resampling does not fix selection bias.
5. Tiny samples: with n very small, the empirical distribution is a poor proxy for the population; bootstrap confidence intervals become fragile.

Worked failure sketch

GARCH-style returns with volatility clustering: an i.i.d. bootstrap of daily P&L understates the chance of long high-vol streaks. A block bootstrap that keeps consecutive weeks intact better preserves clustering for VaR uncertainty.

Disadvantages of Simulation

Simulation is indispensable—but costly and risky as a process:

• Computational cost: full revaluation of large books across tens or hundreds of thousands of paths is expensive; overnight batches and approximations proliferate.
• Model risk: garbage dynamics in → precise-looking garbage out. Sampling error bars do not capture wrong volatility surfaces or broken correlations in stress.
• Calibration ambiguity: many parameter sets fit history; different calibrations shift simulated capital.
• Rare events: plain MC is inefficient for far-tail probabilities without importance sampling or stratified designs.
• Opacity: pathwise results can be hard to explain to non-quants versus a transparent analytical formula.
• False precision: reporting VaR to many decimals when SE is large misleads governance.

Exam Discipline

When a stem says “reduce Monte Carlo variance,” think antithetic and control variates before “always multiply N by 100.” When it contrasts bootstrap and Monte Carlo, ask whether randomness comes from a model or from resampling data. When it asks when bootstrap fails, lead with dependence, tails, and nonrepresentative samples. Simulation estimates a model expectation; it does not remove the need for judgment about whether the model deserves trust.