Math Atom · M-C07

The lognormal distribution

8 minMath AtomProbability

Know these first

By the end you will be able to

  1. 1Define a lognormal variable as e raised to a normal, and recognise one from its shape.
  2. 2Explain why a price built from normal log returns is lognormal, and write S(T) = S₀ exp((μ − σ²/2)T + σ√T Z).
  3. 3Compute the median, mean, quantiles and probabilities of a lognormal by working with its log.
  4. 4Simulate lognormal prices from normal draws, as Monte Carlo VaR does.

Why you need it

A share can double, but it cannot fall by more than 100%. A model for prices has to respect that, and the lognormal does: it is the normal distribution, seen through an exponential.

These Core Path lessons use it:

You need logarithms and e from M-A04: Logarithms and M-A05: The number e and continuous growth, and the normal distribution from M-C06: The normal distribution. How far real returns stray from the normal is M-C08: Skewness and fat tails.

The idea

Multiply, don't add

Prices move by percentages. A 10% rise then a 10% fall leaves you at 99, not 100, and a run of rises compounds into a big gain while a run of falls can never take the price below zero. Add up normal log returns and exponentiate, and you get exactly this lopsided shape.

One set of draws, two views

S = 100 e^(σZ), 2,000 seeded draws

Show

share of draws lognormal curve median (solid), mean (dashed)
050100150200300
Median price100
Mean price 100 e^(σ²/2)108.33
Lowest of the draws32.13
Highest of the draws323.96

Prices pile up on the left and trail off to the right, and none is below zero. A fall can at most wipe out the price; a rise has no ceiling.

The price is lognormal: skewed to the right as a price, a plain bell as a log. Every question about it is answered by moving to the log, using the normal, and moving back.

The rules

The definition. A lognormal is e raised to a normal. Its two parameters belong to the log.

Prices. Log returns add up, so a price is today's price times e to a normal.

Median, mean and quantiles. Quantiles pass through the exponential; the mean does not.

The rules, for reference

ln⁡X∼N(m,s2):median=em,E[X]=em+12s2,qα=em+sΦ−1(α)\ln X \sim N(m, s^2): \quad \text{median} = e^{m}, \quad E[X] = e^{m + \frac12 s^2}, \quad q_\alpha = e^{m + s\Phi^{-1}(\alpha)}
P(X≤x)=Φ ⁣(ln⁡x−ms)P(X \le x) = \Phi\!\Big(\frac{\ln x - m}{s}\Big)
ST=S0exp⁡ ⁣((μ−12σ2)T+σT Z),E[ST]=S0eμTS_T = S_0 \exp\!\Big(\big(\mu - \tfrac12\sigma^2\big)T + \sigma\sqrt{T}\,Z\Big), \qquad E[S_T] = S_0 e^{\mu T}

In the calculator, work with the log: exp(m + norminv(p) * s) gives a quantile and normcdf((ln(x) - m) / s) a probability.

Try it

The UK equity index of the Module 3.2 trading book stands at 8,000 (illustrative). Choose a volatility, a growth rate and a horizon, and see where the index can end up. Things to try: at 1 day the lognormal and normal curves are almost the same; at 5 years with σ = 40% they are not, and the normal gives the index a real chance of falling below zero.

Try itWhere can the index end up?

Open the distribution calculator →

Horizon

lognormal S(T) (shaded: below K) normal with the same mean and sd (dashed)
4,0006,0008,00010,00012,000

Index level after the horizon.

Lognormal: ln S(T) ~ N(m, s²), s = 0.16

Median S₀e^((μ − σ²/2)T)7,898
Mean S₀e^(μT)8,000
1% quantile5,444
P(S(T) < 6,400)9.43%

Normal, same mean and sd (1,288)

1% quantile5,003
P(S(T) < 6,400)10.71%
P(S(T) < 0): impossible for a price0.0000%

Worked examples

1. Median, mean and a quantile

Step 1 of 5
  1. 1

    The set-up

    ln X is normal with mean m = 0.05 and standard deviation s = 0.2. Find the median, the mean, the 5% quantile, and P(X < 1).

2. The UK index: one day, one year, five years

Step 1 of 4
  1. 1

    The set-up

    In the Module 3.2 book the UK index stands at 8,000 and its daily log return has standard deviation 1%, with mean taken as zero. The desk is long £60,000,000 of index futures. Where is the index on the worst 1% of outcomes?

3. Simulating prices, as Monte Carlo VaR does

Step 1 of 4
  1. 1

    The recipe

    Draw a standard normal Z, set the log return r = 1% × Z, and the price S = 8,000e^r. Repeat 10,000 times (seeded, so the page always shows the same draws). This is Monte Carlo simulation.

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

ln X is normal with mean 0.1 and standard deviation 0.3. What is the median of X? (4 decimal places)

Question 2 · objective 3

ln X ~ N(0, 0.5²). What is the mean of X? (4 decimal places)

Question 3 · objective 3

A share is at 50. Its log return over the horizon is normal with mean 0 and standard deviation 10%. What is the 5% quantile of the price? (2 decimal places)

Question 4 · objective 2

Why are prices modelled as lognormal rather than normal?

Question 5 · objective 4

To simulate a lognormal price S = S₀e^(m + sZ), what do you draw?

Glossary

Lognormal distribution
X is lognormal when ln X is normal: X = e^(m + sZ) with Z standard normal. Always positive, with a long right tail.
Log return
ln(S₁/S₀): the log of the price ratio. Log returns add up over time; prices multiply.
Parameters m and s
The m and s of a lognormal are the mean and standard deviation of ln X, not of X. The median of X is eᵐ.
Right skew
Leaning to one side. A lognormal is right-skewed: a long tail of large values and a bunching of small ones.
Monte Carlo simulation
Monte Carlo simulation: drawing many random scenarios from a model and reading risk measures off the results.

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