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Quantitative Methods — Probability Module · Lesson 8

I. Lesson Positioning

L121 covered the standard normal distribution and z-scores: using the standardization formula z = (X−μ)/σ, we can precisely calculate probabilities for any normal variable. However, the normal distribution has a major flaw: it allows negative values — but stock prices, exchange rates, real estate prices, and other asset prices cannot be negative. That's why CFA uses the lognormal distribution to model asset prices.

II. Core Concepts

1. The Fatal Flaw of Normal Distribution: Allowing Negative Prices

The normal distribution N(μ, σ²) has support (−∞, +∞):

$$P(X < 0) = P\left(Z < \frac{0 - \mu}{\sigma}\right)$$

If μ = 100, σ = 60, then P(X < 0) ≈ P(Z < −1.67) ≈ 4.7% — the normal distribution says this stock has a 4.7% chance of having a negative price. In reality, this is impossible.

🧠 The lognormal distribution solves this problem: the range of X is [0, +∞), perfectly matching the "limited downside, unlimited upside" nature of asset prices.

2. Definition: What Is Lognormal?

Core definition: If Y follows a normal distribution, then X = e^Y follows a lognormal distribution.

Equivalent statement:

$$\ln X \sim N(\mu, \sigma^2)$$

📊 Take the log → becomes normal → the original variable is lognormal. This is the most commonly tested conceptual boundary on the exam.

3. Key Properties of the Lognormal Distribution

Key observation:

$$Median < Mean \quad \text{(always true due to right skew)}$$

$$e^{\mu - \sigma^2} < e^{\mu} < e^{\mu + \sigma^2/2}$$

That is: Mode < Median < Mean (the classic ordering for right-skewed distributions).

Example 1: Difference Between Mean and Median

A stock's log return ln(S₁/S₀) ~ N(0.10, 0.20²), i.e., μ = 0.10, σ = 0.20.

$$E\left(\frac{S_1}{S_0}\right) = e^{0.10 + 0.20^2/2} = e^{0.10 + 0.02} = e^{0.12} = 1.1275$$

$$Median\left(\frac{S_1}{S_0}\right) = e^{0.10} = 1.1052$$

📊 Mean (1.1275) > Median (1.1052). Expected return is 12.75%, but more than half of scenarios have returns below 10.52% — that's what right skew means: a few large gains pull up the mean while most outcomes are moderate.

4. Why Asset Prices Are Modeled with Lognormal: Four Fundamental Reasons

① Prices Cannot Be Negative

A stock falls to ¥0 at worst (bankruptcy), never −¥50. The lognormal distribution is naturally ≥ 0.

② Compounding Generates Lognormal Prices

Continuous compounding over many periods → cumulative return = sum of individual period returns → by the Central Limit Theorem, the sum approaches normality → price = e^(cumulative return) → price approaches lognormal.

$$S_T = S_0 \cdot e^{r_1} \cdot e^{r_2} \cdot ... \cdot e^{r_T} = S_0 \cdot e^{\sum_{t=1}^{T} r_t}$$

Where r_t are independent → ∑r_t is approximately normal → S_T is lognormal.

③ Returns Multiply → Logs Add

Arithmetically: 10-day continuous returns multiply; in log space: 10 log returns add. Addition is easier to handle than multiplication, and the sum tends toward normality.

④ Consistent with Black-Scholes Option Pricing

The core assumption of the B-S model is that stock prices follow a geometric Brownian motion, which results in a lognormal distribution.

5. Continuously Compounded Return

Simple return (discrete): $$R_t = \frac{P_t - P_{t-1}}{P_{t-1}}$$

Continuously compounded return: $$r_t = \ln\left(\frac{P_t}{P_{t-1}}\right) = \ln(P_t) - \ln(P_{t-1})$$

Example 2: Continuous Compounding vs. Simple Return

Stock goes from ¥100 → ¥110 → ¥105:

📊 The two methods produce similar but not identical cumulative results. Continuously compounded returns are more convenient in modeling due to their "additive" property.

6. Stock Price Modeling: From Lognormal to Geometric Brownian Motion

Standard assumption in financial modeling:

$$\ln S_T \sim N\left(\ln S_0 + (\mu - \frac{\sigma^2}{2})T, \;\sigma^2 T\right)$$

Or equivalently:

$$S_T = S_0 \cdot e^{(\mu - \sigma^2/2)T + \sigma \sqrt{T} \cdot Z}$$

Where Z ~ N(0, 1), μ = annualized expected return, σ = annualized volatility.

Example 3: Distribution of Stock Price After One Year

S₀ = ¥100, μ = 12% (annualized), σ = 25% (annualized), T = 1 year.

ln S₁ ~ N(ln 100 + (0.12 − 0.25²/2)×1, 0.25²×1)

ln S₁ ~ N(4.6052 + 0.08875, 0.0625) = N(4.6939, 0.25²)

$$\begin{aligned} E(S_1) &= 100 \cdot e^{0.12} = 112.75 \ 95\% \text{ CI for } \ln S_1 &= 4.6939 \pm 1.96 \times 0.25 = [4.2039, 5.1839] \ 95\% \text{ CI for } S_1 &= [e^{4.2039}, e^{5.1839}] = [66.97, 178.30] \end{aligned}$$

📊 The 95% confidence interval [¥66.97, ¥178.30] — the interval is asymmetric! Upside potential (+78.3%) > downside risk (−33.0%), perfectly reflecting the right-skewed nature of the lognormal distribution.

7. Summary of Lognormal Applications in the Exam

III. Key Formula Quick Reference

IV. Common Pitfalls

V. Practice Test

【Questions】

Q1 (Concept) Regarding the lognormal distribution, which of the following is correct?

A. If X follows a lognormal distribution, then X follows a normal distribution
B. The mean of a lognormal distribution equals e^μ
C. The shape of a lognormal distribution is symmetric
D. If ln X ~ N(0, 1), then the median of X is 1

Q2 (Identification) Which of the following variables is most appropriately modeled by a lognormal distribution?

A. Daily temperature change (can be positive or negative)
B. Annual return of a stock (lower bound −100%, no upper bound)
C. S&P 500 index level
D. Sampling distribution of z-scores

Q3 (Calculation) Given ln X ~ N(2, 0.5²), the value of E(X) is closest to:

A. e² = 7.389
B. e^(2.125) = 8.372
C. e^(2.25) = 9.488
D. e^(2.5) = 12.182

Q4 (Concept) In option pricing, the stock price is assumed to follow which distribution?

A. Normal distribution
B. Uniform distribution
C. Binomial distribution
D. Lognormal distribution

Q5 (Integrated) The daily log return of a stock ~ N(0.001, 0.02²). Current price is ¥50. Evaluate the following statements:

I. The stock's simple daily return follows a normal distribution
II. The probability of tomorrow's price being ¥48 is greater than the probability of it being ¥52
III. The lognormal distribution is right-skewed, therefore the probability of a price increase > 50%
IV. When converted to continuously compounded returns, multi-period cumulative returns can be calculated by direct addition

Which statements are correct?

A. IV only
B. I and III
C. I and IV
D. II and IV

【Answers & Explanations】

A1: D

• A ❌ If X is lognormal, then ln X follows a normal distribution, not X itself
• B ❌ E(X) = e^(μ + σ²/2); it equals e^μ only when σ² = 0
• C ❌ Normal is symmetric; lognormal is right-skewed
• D ✅ Median = e^μ = e^0 = 1 (when μ = 0, σ = 1)

🧠 Remember: for the lognormal, "median = e^μ" while "mean = e^(μ + σ²/2)" — the extra σ²/2 adjustment term.

A2: C — S&P 500 index level

• A ❌ Temperature changes can be negative; normal distribution is more appropriate
• B ❌ Returns can be modeled using normal approximation; lognormal is not required (negative returns are normal)
• C ✅ The key features of an index level: lower bound of 0, no upper bound, long-term compounding growth → lognormal is a perfect match
• D ❌ z-scores themselves can be negative; use normal distribution

A3: B — e^(2.125) = 8.372

Calculation steps: - μ = 2, σ = 0.5, σ² = 0.25 - E(X) = e^(μ + σ²/2) = e^(2 + 0.25/2) = e^(2 + 0.125) = e^(2.125) ≈ 8.372

⚠️ Common exam trap: directly choosing e^μ = e² = 7.389 (forgetting to add σ²/2).

A4: D — Lognormal distribution

The core assumption of the Black-Scholes option pricing model: the underlying asset price follows geometric Brownian motion, which is equivalent to the price following a lognormal distribution.

• A ❌ Normal distribution allows negative values, unsuitable for prices
• D ✅ Lognormal is the fundamental assumption of the B-S model

A5: A — IV only

Analysis of each statement: - I ❌ Simple daily returns ≈ continuously compounded returns (for small movements), but strictly speaking, the distribution of simple returns is a transformed version of the lognormal, not strictly normal - II ❌ The distribution is right-skewed → higher prices have greater probability. ¥50 → ¥48 (down 4%) vs ¥50 → ¥52 (up 4%); under right skew, the probability of ¥52 is slightly higher - III ❌ Right skew ≠ probability of price increase > 50%. Right skew describes the shape of the distribution (long tail to the right), not the direction of probability. If μ is small or negative, the probability of a decline can still exceed 50% - IV ✅ Core advantage of continuously compounded returns: r₁ + r₂ + ... + r_T directly gives the multi-period cumulative return; no multiplication needed

🧠 Statement III is the most commonly missed: right skew refers to "shape," not "direction." It's like the income distribution being right-skewed (a few people have very high incomes), but that doesn't mean most people's incomes are growing — the median may be stagnant.

📚 Next Lesson L123: Monte Carlo Simulation — From Lognormal Distribution to Stochastic Simulation