When a closed-form formula breaks down (path-dependent payoffs, fat-tailed returns, complex portfolios) you simulate. The exam tests three flavors and the trade-offs between them.
Why simulate at all. A closed-form formula (Black-Scholes, mean-variance, duration) gives one number from one set of inputs. Simulation generates thousands of plausible outcomes and lets you read percentiles, tail probabilities, and confidence bands directly off the empirical distribution.
Historical simulation applies the actual sequence of past returns to today's portfolio. If you have 1,000 daily returns for each holding, you compute 1,000 hypothetical one-day portfolio profit and loss (P&L) values by repricing today's positions at each past return vector. The 5th percentile of that empirical distribution is the 95% one-day VaR.
KEY: Historical simulation makes NO distributional assumption. It does not assume normality, lognormality, or any parametric shape. The distribution IS the data.
Strengths. Captures actual fat tails, skewness, and cross-asset correlations as they really occurred. No estimation of variance-covariance matrices. Easy to explain to non-quants.
Common mistakes
- Calling bootstrap "without replacement." Bootstrap is DEFINED by sampling with replacement. Without replacement is just a permutation of the original sample and produces no new information about variance.
- Claiming Monte Carlo is assumption-free. Monte Carlo requires a distributional assumption. Misspecifying the distribution (normal when reality is fat-tailed) produces wrong tail estimates no matter how many paths you run.
- Saying historical simulation requires a normality assumption. It does not. Historical simulation uses the empirical distribution directly. Confusing it with the variance-covariance (parametric) VaR method is a frequent trap.
Bottom line
- Historical simulation reuses actual past returns in their original time order. No distributional assumption, but limited to events that actually happened.
- Bootstrap resampling draws WITH REPLACEMENT, generating many synthetic samples from one dataset. Each draw is independent and equally likely.
- Monte Carlo generates random draws from an ASSUMED distribution (often calibrated to historical moments) and is the only method that can simulate events outside the historical record.
- Trade-off: historical = realism, no assumptions, but tied to the past. Monte Carlo = flexibility, scenario design, but assumption-dependent.
Exam shortcut
For method identification: "actual past returns, used once" = historical; "with replacement" = bootstrap; "assumed distribution + random draws" = Monte Carlo. For "which can model a scenario that never happened" the answer is Monte Carlo, always. For confidence intervals around a statistic (Sharpe, drawdown, regression coefficient) the default tool is bootstrap.
The full lesson (about 2,092 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- simulation of financial asset prices and returns
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