Library/CFA (Chartered Financial Analyst)/Quantitative Methods — CFA Program Curriculum, 2026 • LEVEL I • VOLUME 1/Learning Module 6 Simulation Methods

Learning Module 6 Simulation Methods

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Lognormal distribution and continuous compounding5 min

Lognormal distribution and continuous compounding: A random variable Y is lognormal if ln(Y) is normally distributed. The lognormal distribution is bounded below by zero and typically right skewed, which makes it a useful model for asset prices. If X ~ N(mu, sigma^2) and Y = exp(X), then E[Y] = exp(mu + 0.5 sigma^2) and Var[Y] = exp(2 mu + sigma^2) [exp(sigma^2) - 1]. Continuously compounded returns aggregate additively: the T-period continuously compounded return r0,T equals the sum of periodic continuously compounded returns and, under i.i.d. assumptions with mean mu and variance sigma^2, E[r0,T] = mu T and Var[r0,T] = sigma^2 T. If periodic continuously compounded returns are (approximately) normal, then r0,T is normal and PT = P0 exp(r0,T) is lognormal. Volatility is commonly annualized by multiplying the periodic standard deviation by the square root of the number of periods per year (for daily returns, sqrt(250)).

Key Points

Y is lognormal if ln(Y) is normal.
Lognormal is nonnegative and right skewed—useful for prices.
If X ~ N(mu, sigma^2), E[exp(X)] = exp(mu + 0.5 sigma^2).
Continuously compounded returns add across subperiods.
Annualize periodic sigma by sqrt(number of periods).

Monte Carlo simulation: concept and uses5 min

Monte Carlo simulation: Monte Carlo simulation generates many random samples from specified probability distributions for key risk factors and converts those sampled risk factors into simulated paths for underlying variables (for example, asset prices). Typical Monte Carlo steps: (1) specify the quantity of interest and model, (2) set a time grid of K subperiods and horizon T, (3) specify distributions and draw random numbers for each risk factor for each subperiod, (4) map draws to variable paths using the chosen dynamic model, (5) compute payoffs or outcomes per trial and discount if needed, and (6) repeat for I trials and aggregate results (for example, compute the mean present value). Monte Carlo is used to value complex contingent claims and to estimate risk measures (VaR, expected shortfall) when closed-form solutions are unavailable. The variance reduction, sensitivity analysis, and stress testing are typical complementary uses. Monte Carlo provides statistical estimates (with simulation error) rather than closed-form exact results.

Key Points

Monte Carlo draws random samples from specified distributions.
Convert draws to paths using a model, compute payoffs, and repeat I trials.
Used for path-dependent payoffs and complex instruments.
Results are statistical estimates subject to simulation error.
Useful for sensitivity analysis and VaR estimation.

Monte Carlo example: path-dependent contingent claims5 min

Example: Pricing a path-dependent contingent claim such as an Asian option: simulate K monthly steps for each trial, compute the average path price and the final price, compute the payoff max(final - average, 0), discount to present, and average across I trials. The histogram of simulated payoffs illustrates the payoff distribution; many trials may produce zero payoff if final <= average. Steps include specifying drift and volatility inputs, drawing standard normal variates Zk for each subperiod, mapping to price changes (for example, geometric model increments or discrete approximations), and computing path-dependent outcomes. Aggregate the present values across trials to estimate the claim value and examine the distribution to assess risk.

Key Points

Simulate monthly (or K) steps and compute path statistics (average, min, max).
Payoff for Asian-like claim = max(final price - average price, 0).
Discount each trial payoff to present and average across trials.
Histogram of payoffs shows probability of zero and tail outcomes.
Check sensitivity to drift, volatility, and time-step choices.

Bootstrap resampling: method and applications5 min

Bootstrap resampling: Bootstrap is a nonparametric resampling technique that treats the observed sample as an empirical population. A bootstrap draw samples with replacement from the observed data to create a resample of the same size as the original dataset. Repeat B times to obtain B resample estimates of the statistic of interest (mean, median, regression coefficient, etc.). The distribution of the B resample statistics approximates the sampling distribution of the statistic and can be used to estimate standard errors, confidence intervals, and bias. The bootstrap is useful when analytical formulae for standard errors are unavailable or when the statistic is complicated; it rests on the idea that the original sample is a reasonable approximation of the population. Jackknife is an alternative resampling method that systematically leaves out one observation at a time to estimate bias and variance.

Key Points

Bootstrap samples with replacement from the observed sample.
Use many resamples (B large) to approximate sampling distribution.
Applies to means, medians, regression coefficients, and complex stats.
Jackknife leaves out one observation at a time; bootstrap resamples.
Bootstrap accuracy depends on representativeness of original sample.

Practical strengths, weaknesses, and implementation notes5 min

Strengths and weaknesses: Monte Carlo strengths include flexibility to model complex dynamics and path-dependent features and to test sensitivity to distributional assumptions; weaknesses include computational cost and that results are approximate with simulation error. Bootstrap strengths include simplicity, relying on observed empirical distributions rather than parametric assumptions, and usefulness to estimate sampling distributions and standard errors for complicated estimators; weaknesses include dependence on the representativeness of the observed sample and that bootstrap gives statistical estimates rather than exact inference. Implementation notes: When using Monte Carlo, ensure appropriate choice of distributions and risk factor correlation structure, use sufficient trials to reduce simulation error (I large), and consider variance-reduction techniques where relevant. For bootstrap, choose B large enough (often thousands) to stabilize estimates and be aware of any serial dependence in time-series data, which may require block bootstrap methods. Both methods complement analytical techniques; where closed-form solutions exist, those should be used for insight and validation.

Key Points

Monte Carlo: flexible but computationally intensive and approximate.
Bootstrap: simple and data-driven but depends on sample representativeness.
Use many trials/resamples (I, B large) to reduce Monte Carlo/bootstrap error.
Consider block bootstrap for serial dependence in time series.
Use analytical solutions when available to validate simulations.

Questions

Question 1

Which statement correctly links normal and lognormal variables?

Learning Module 6 Simulation Methods

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