Learning Module 6
Simulation Methods
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices when using continuously compounded asset returns.
Describe Monte Carlo simulation and explain how it can be used in investment applications, including valuing complex securities.
Describe the use of bootstrap resampling in conducting a simulation based on observed data in investment applications.
1 · Lognormal Distribution & Continuous Compounding
The lognormal distribution is fundamental to asset price modelling. Its two defining properties make it ideal for stock prices: it is bounded below by zero (prices cannot go negative) and positively skewed (long right tail, unlimited upside).
Y is lognormal if and only if ln(Y) is normally distributed
Memory aid: "the LOG is NORMAL" → lognormal
If X ~ Normal(μ, σ²) and Y = exp(X), then Y is lognormal.
Conversely: if Y is lognormal, then ln(Y) is normal.
Mean of lognormal: μ_L = exp(μ + 0.50σ²)
Variance of lognormal: σ_L² = exp(2μ + σ²) × [exp(σ²) − 1]
Note: μ_L > exp(μ) because the distribution can only spread rightward (bounded at 0).
Future stock price at time T given current price P₀:
P_T = P₀ × exp(r₀,T)
Continuously compounded return to T is the sum of sub-period returns:
r₀,T = r_{T-1,T} + r_{T-2,T-1} + … + r₀,₁
If sub-period returns are i.i.d. with mean μ and variance σ²:
E(r₀,T) = μT and σ²(r₀,T) = σ²T
So: σ(r₀,T) = σ√T (volatility scales with √T)
Because r₀,T is normal (or ~normal by CLT), P_T = P₀exp(r₀,T) is lognormal.
Volatility = standard deviation of continuously compounded returns
Annualized from daily observations (250 trading days/year):
σ_annual = σ_daily × √250
Steps to estimate:
1. Compute daily log returns: r_t = ln(P_t / P_{t-1})
2. Find sample standard deviation s of the log returns
3. Annualize: σ̂_annual = s × √250
Example: daily σ = 0.01 → annual σ = 0.01 × √250 = 15.81%
Why not normal for prices? Normal distributions allow negative values (P < 0), which is impossible for stock prices with limited liability. Lognormal distribution enforces P ≥ 0.
i.i.d. assumption: independently and identically distributed returns. Independence = cannot predict future returns from past (no momentum/mean reversion). Identical = stationarity (constant mean and variance over time).
Central Limit Theorem basis: even without normality assumption, the sum of many i.i.d. returns r₀,T is approximately normal by CLT — so stock prices are approximately lognormal regardless of the exact return distribution.
Black-Scholes-Merton model: explicitly assumes the underlying asset price is lognormally distributed, making the lognormal distribution central to all of options pricing theory.
2 · Monte Carlo Simulation
Monte Carlo simulation generates a very large number of random samples from specified probability distributions to estimate the likelihood of a range of outcomes. It is the primary tool for valuing complex securities where no closed-form pricing formula exists.
Key investment applications: (1) Estimating risk and return of a portfolio over a time horizon — simulate P&L, compute VaR, expected shortfall; (2) Valuing complex securities with embedded options (Asian options, mortgage-backed securities, lookback options) where analytical formulas are unavailable; (3) Sensitivity analysis — testing model outputs under different distributional assumptions for key risk factors.
SIX-STEP PROCESS
Define the quantity of interest (e.g., contingent claim value) and the underlying variable (e.g., stock price). Set starting values.
Split the horizon T into K sub-periods. Time increment Δt = T/K. E.g., 1 year with K=12 gives monthly steps Δt = 1 month.
Choose distributional assumptions for key risk factors. E.g.: ΔPrice = μ×Price×Δt + σ×Price×Z_k where Z_k ~ N(0,1).
Draw K random values of Z_k from the specified distribution. Convert to K stock prices using the Step 3 model for one simulation trial.
Compute the average stock price over the trial. Calculate the contingent claim payoff, then discount to present value. One trial complete.
Repeat Steps 4–5 for I total trials (e.g., I = 1,000). The average of all discounted payoffs is the Monte Carlo estimate of the security's value.
Asian option example: Payoff = max(Final price − Average price, 0). In 1,000 trials: 654 trials (65.4%) paid zero (final price ≤ average). The remaining 346 trials paid positive amounts up to USD 11. The average of all discounted payoffs = Monte Carlo value of the claim. The final price distribution is wider than the average price distribution because averaging smooths out price fluctuations.
Limitation — statistical not exact: Monte Carlo provides statistical estimates with sampling error. Analytical methods (when available) give exact solutions and clearer cause-and-effect insight.
Strength — flexibility: can handle path-dependent payoffs (Asian, lookback, barrier options), multiple correlated risk factors, complex distributions, and evolving market conditions that analytical models cannot capture.
Sensitivity testing: a major strength is the ability to easily change distributional assumptions and re-run to see how outputs change — "what-if" analysis across assumptions on key risk factors.
Number of trials matters: more trials → more accurate estimate but higher computational cost. Typical simulations run 1,000 to 100,000+ trials depending on required precision.
3 · Bootstrapping (Bootstrap Resampling)
Bootstrapping is a resampling method that treats the observed sample as a proxy for the true population. Instead of drawing from a theoretical distribution (as in Monte Carlo), bootstrap draws randomly from the observed historical data, with replacement.
Draw random numbers from a specified (assumed) theoretical distribution (e.g., normal). Analyst defines the distribution parameters (μ, σ). Generates synthetic data that may not have occurred historically. Used when the distributional form is known or assumed.
Draw randomly from theobserved historical data sample(empirical distribution). No distributional assumption required. Treats the sample as the population. Used when the true distribution is unknown. Each drawn item isreplacedbefore the next draw.
Given observed sample of size n:
1. Draw a resample of size n with replacement from the original sample
2. Compute the statistic of interest (mean, σ, skewness, kurtosis, etc.)
3. Repeat B times (e.g., B = 1,000 bootstrap samples)
4. The distribution of B estimates = bootstrap sampling distribution
5. Use this to estimate standard errors, confidence intervals, etc.
Key: drawn items are replaced → same observation can appear
multiple times in one resample; others may not appear at all.
Dimension | Monte Carlo | Bootstrap |
|---|---|---|
Data source | Random draws from specified theoretical distribution | Random draws from observed historical sample |
Distribution assumption | Required — analyst specifies (e.g., normal, lognormal) | Not required — uses empirical distribution |
When to use | When distributional form is known or can be reasonably assumed | When true population distribution is unknown |
Replacement | N/A — draws from theoretical distribution | Yes — with replacement (same obs. can appear multiple times) |
Outputs | Simulated distribution of portfolio returns or security values | Bootstrap distribution of estimated statistical parameters |
Limitation | Results depend on distributional assumptions; statistical, not exact | Only as good as the original sample; statistical, not exact |
Common bond | Both are statistical complements to analytical methods, not replacements | |
Bootstrap strengths: simple to implement; makes no distributional assumptions; can infer population parameters (mean, variance, skewness, kurtosis) from a single sample; good representation when sample is drawn randomly from population.
Bootstrap weakness: only provides statistical estimates, not exact results; accuracy depends on the quality and representativeness of the original sample. Poor or unrepresentative samples lead to poor bootstrap estimates.
Replacement rule is critical: drawing WITH replacement means some observations appear multiple times in a resample while others may not appear at all. This mimics random sampling from the population where each draw is independent of past draws.