CFA L1  /  Quant  /  M6. Simulation Methods
Module 6 · Quantitative Methods

Simulation Methods

EN: The lognormal distribution for asset prices, Monte Carlo simulation, and bootstrap resampling.
VN: Phân phối lognormal, mô phỏng Monte Carlo, bootstrap resampling.

In this module
  1. Lognormal Distribution of Prices
  2. Continuously Compounded Return Distribution
  3. Price Path with Stochastic Term
  4. Monte Carlo Simulation Steps
  5. Bootstrap Resampling

1. Lognormal Distribution of Prices Core

About: Lognormal distribution is bounded at 0 and right-skewed — the standard model for asset prices. Comes from continuously-compounded returns being normally distributed.Tóm tắt: Phân phối lognormal không âm, lệch phải — mô hình tiêu chuẩn cho giá tài sản. Suy ra khi log-return phân phối chuẩn.

EN: If continuously compounded returns are normally distributed, then prices follow a lognormal distribution — bounded below at zero and right-skewed.
VN: Nếu log-return phân phối chuẩn, giá tài sản phân phối lognormal — không âm, lệch phải.

\[ P_T = P_0 \cdot e^{r_{cc} \cdot T},\qquad r_{cc} \sim N(\mu, \sigma^{2}) \implies P_T \sim \text{Lognormal} \]

Properties / Tính chất

  • ≥ 0 Lognormal is bounded at zero — useful for prices.
  • Skew > 0 Right-skewed; mean > median.
  • No upper bound Long right tail allows for large gains.
Practice problem

Stock price S = $100 today. Continuously compounded annual return is N(μ=8%, σ=20%). What is E(P_T) for T = 1 year?

Show solution
\(E(P_T) = S_0 \cdot e^{\mu + \sigma^{2}/2}\)
\(= 100 \cdot e^{0.08 + 0.02}\)
\(= 100 \cdot e^{0.10} = 100 \times 1.1052\)
≈ $110.52 (note > 100·e^0.08 because of variance term)

2. Continuously Compounded Return Distribution Core

About: Log-returns aggregate (sum) over time and the variance scales linearly with time → volatility scales with √T. The 'square-root-of-time' rule for annualizing volatility.Tóm tắt: Log-return cộng dồn theo thời gian; phương sai scale tuyến tính → volatility scale theo √T. Quy tắc √T.

EN: Continuously compounded returns aggregate (sum) over time — the T-period mean and variance scale linearly.
VN: Log-return cộng dồn theo thời gian — mean và variance scale tuyến tính theo T.

\[ r_{cc,\,0 \to T} \sim N(\mu \cdot T,\ \sigma^{2} \cdot T) \] \[ \sigma_{0 \to T} = \sigma \cdot \sqrt{T} \quad \text{(volatility scales with } \sqrt{T}\text{)} \]
Practice problem

Daily continuously compounded return has μ = 0.04% and σ = 1%. What is the annualized volatility (252 trading days)?

Show solution
\(\sigma_{annual} = 1\% \times \sqrt{252}\)
≈ 15.87%

3. Price Path with Stochastic Term Core

About: One step of geometric Brownian motion. Drift term + a normal random shock scaled by σ√Δt. The kernel of Monte Carlo simulation.Tóm tắt: Một bước Brownian motion hình học. Drift + cú sốc chuẩn × σ√Δt. Hạt nhân của Monte Carlo.

EN: Standard Monte Carlo step for one period — draw a standard-normal shock and translate to the log-return.
VN: Bước Monte Carlo: rút mẫu \(Z\) chuẩn và quy đổi sang log-return.

\[ P_{t+1} = P_t \cdot e^{\,(\mu - \tfrac{1}{2}\sigma^{2})\,\Delta t \,+\, \sigma\sqrt{\Delta t}\, Z} \]

Components / Thành phần

  • \(\mu\) Drift / expected continuously compounded return.
  • \(\sigma\) Volatility.
  • \(\Delta t\) Time step length.
  • \(Z\) Standard normal random draw, \(Z \sim N(0, 1)\).
Practice problem

S = $50, μ = 6%, σ = 25%, Δt = 1/252 (daily), Z = +1.5 drawn. Compute next-day price.

Show solution
Drift = (0.06 − 0.5(0.0625))(1/252) = 0.0287/252 ≈ 0.000114
Diff = 0.25 × √(1/252) × 1.5 = 0.25 × 0.0630 × 1.5 ≈ 0.0236
P+1 = 50 × e^(0.000114 + 0.0236) ≈ 50 × 1.024
≈ $51.21

4. Monte Carlo Simulation — Steps Concept

About: Monte Carlo answers complex what-if questions when closed-form solutions don't exist (path-dependent options, retirement plans, multi-input risk). Speed-accuracy trade-off via number of trials.Tóm tắt: Monte Carlo giải các bài toán không có công thức đóng (option phụ thuộc đường, kế hoạch hưu trí, rủi ro nhiều biến). Đánh đổi tốc độ-độ chính xác qua số lần thử.

Procedure / Quy trình

  • 1. Specify the model (DGP) and assumptions about input distributions.
  • 2. Generate random draws for each input.
  • 3. Compute the model output for each draw.
  • 4. Repeat for many trials (10,000+).
  • 5. Summarize the output distribution (mean, std dev, percentiles, VaR, etc.).

Use cases: option pricing (path-dependent), VaR, retirement planning, complex CF models.

5. Bootstrap Resampling Concept

About: Bootstrap estimates the sampling distribution of any statistic by resampling the observed data with replacement — no distribution assumption needed. Key advantage over Monte Carlo for unknown distributions.Tóm tắt: Bootstrap ước lượng phân phối mẫu của thống kê bằng cách rút mẫu có hoàn lại — không cần giả định phân phối. Ưu điểm so với Monte Carlo.

EN: Estimate the sampling distribution of a statistic by repeatedly drawing samples (with replacement) from the observed dataset.
VN: Ước lượng phân phối mẫu của một thống kê bằng cách rút mẫu có hoàn lại từ dataset đã có.

Procedure / Quy trình

  • 1. Take the original sample of size n.
  • 2. Draw a bootstrap sample of size n with replacement.
  • 3. Compute the statistic of interest (mean, median, etc.).
  • 4. Repeat thousands of times.
  • 5. The empirical distribution of these statistics estimates its sampling distribution and standard error.

Advantage over Monte Carlo: No distribution assumption — directly resamples observed data.