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Lesson 1 of 6 · 13 min

The lognormal distribution

A variable is lognormal when its natural log is normal; that makes it never negative and skewed to the right, which is exactly how asset prices behave.

In short

Y is lognormal if $\ln Y$ is normally distributed. Equivalently, if X is normal, then $Y = e^X$ is lognormal.
A lognormal variable is bounded below by zero and skewed to the right (it has a long right tail).
That shape fits asset prices, which cannot fall below zero. Returns are usually modelled as (approximately) normal.
Its two parameters are the mean $\mu$ and variance $\sigma^2$ of the associated normal distribution (of $\ln Y$ ), not of Y itself.
The mean of Y is $e^{\mu + 0.5\sigma^2}$ , which is above $e^{\mu}$ : more volatility pushes the mean up.

Normal and lognormal: "the log is normal"

A lognormal distribution is defined through another distribution. A random variable Y is lognormal when its natural logarithm, $\ln Y$ , follows a normal distribution. Read the name literally: the log of the variable is normal.

The link runs in both directions. Start with a normal variable X and exponentiate it: $Y = e^X$ is lognormal. Start with a lognormal Y and take its log: $\ln Y$ is normal. Logs and exponentials simply undo each other.

Because $e^X$ is positive for every X, a lognormal variable can never be zero or negative, however negative X gets. Very negative values of X are squeezed into a thin region just above zero, while positive values of X are stretched out to the right. That squeezing and stretching is what creates the long right tail.

Normal versus lognormal at a glance
Feature	Normal	Lognormal
Possible values	Any real number	Positive only (bounded below by 0)
Shape	Symmetric bell	Skewed to the right
Parameters	Its own mean and variance	Mean and variance of $\ln Y$ (the associated normal)
Typical use in finance	Returns, especially continuously compounded ones	Asset prices
Where you meet it	Risk models for returns	Underlying price in the Black–Scholes–Merton option model

Exponentiating squeezes every negative X into the thin band between 0 and 1, while positive X is stretched upward. A symmetric normal X therefore becomes a positive, right-skewed lognormal Y.

Why asset prices are modelled as lognormal

A share price cannot fall below zero: with limited liability, the most a shareholder can lose is what was paid. A normal distribution ignores that floor and always assigns some probability to negative values. For a volatile asset over a long horizon that probability is not small, so a normal model of prices can be plainly wrong.

A lognormal distribution has the floor built in, and its right skew matches reality. A price can fall by at most 100%, but it can rise by 200% or 500%. In practice the lognormal has proven a usefully accurate description of the price distribution of many financial assets.

Returns are different. A return can be negative, and a continuously compounded return can take any value from very negative to very positive. So the normal is often a good approximation for returns, while the lognormal is used for prices. Finance needs both, and the next lesson shows they are two views of the same model.

The EUR 50 share with 8% mean and 30% volatility in continuous return. The lognormal price has no mass below zero and a long right tail that pulls its mean (56.66) above its median (54.16); the normal curve is symmetric and spills toward zero.

Take three equally likely continuously compounded returns: −20%, 0% and +20%. They are perfectly symmetric. Applied to a price of 100, they give 81.87, 100 and 122.14. The fall is 18.13 but the rise is 22.14: symmetric returns produce right-skewed prices. Repeat that over thousands of outcomes and you get the lognormal shape.

Two sets of means and variances

The classic trap in this topic is that two means and two variances are in play. The parameters of a lognormal are the mean $\mu$ and variance $\sigma^2$ of $\ln Y$ . The mean $\mu_L$ and variance $\sigma_L^2$ of the lognormal variable Y itself are different numbers, derived from those parameters.

A natural guess is that the mean of $Y = e^X$ is just $e^{\mu}$ . It is not. The correct mean is $e^{\mu + 0.5\sigma^2}$ , which is larger than $e^{\mu}$ by the factor $e^{0.5\sigma^2}$ . That factor is above 1 whenever there is any variance at all.

The intuition: increase $\sigma^2$ and the distribution spreads out. It can spread upward without limit, but it cannot spread downward past zero. The extra spread therefore lands mostly on the right and drags the mean to the right. Higher volatility, higher mean.

The variance formula is heavier. You will rarely compute it by hand; know that it exists, that it uses the same $\mu$ and $\sigma^2$ of the associated normal, and look it up when needed rather than memorising it.

If the lognormal variable is a future price $P_T$ , then $\mu$ and $\sigma^2$ are the mean and variance of the continuously compounded return over the whole horizon to T, not over one period.
If a question gives the mean and standard deviation of $\ln Y$ , square the standard deviation before using it.

Hold the mean continuous return at 8% and raise volatility: the expected price climbs above

50e^{0.08}

by the factor

e^{0.5\sigma^2}

. The extra spread can only go upward, so it drags the mean up.

Formulas

Definition: $Y = e^{X}, \quad X = \ln Y \sim N(\mu, \sigma^2)$
Y is lognormal exactly when ln Y is normal with mean $\mu$ and variance $\sigma^2$ .
Mean of a lognormal variable: $\mu_L = e^{\mu + 0.5\sigma^2}$
$\mu$ and $\sigma^2$ are the mean and variance of ln Y, not of Y.
Variance of a lognormal variable: $\sigma_L^2 = e^{2\mu + \sigma^2}\left(e^{\sigma^2} - 1\right)$
Reference formula: recognise it rather than memorise it.

Worked examples

Example 1 Expected price relative

The natural log of next year's price relative $P_1/P_0$ is normal with mean 5% and standard deviation 20%. What is the expected price relative?

Example 2 Mean and standard deviation of a lognormal variable

$\ln Y$ is normal with mean 0.06 and standard deviation 0.25. What are the mean and the standard deviation of Y?

Example 3 Volatility raises the expected price

A EUR 50 share has a one-year continuously compounded return that is normal with mean 8%. What is the expected price in one year if the standard deviation is 20%, and if it is 30%?

On the exam

Mostly conceptual: the shape of the lognormal (bounded below by zero, right-skewed), why it suits prices rather than returns, and which variable is the normal one. Occasionally a calculation of the lognormal mean from the normal parameters.

Lognormal is skewed to the right, not the left, and it has a lower bound of zero, not an upper bound.
$\mu$ and $\sigma$ in the formulas belong to $\ln Y$ , not to Y.
Forgetting the 0.5 in $e^{\mu + 0.5\sigma^2}$ , or forgetting to square $\sigma$ .
Saying prices are normal and returns lognormal: it is the other way round.

Check yourself

Q1Compared with a normal distribution, a lognormal distribution is most accurately described as:
Q2If X is normally distributed, which variable follows a lognormal distribution?
Q3 $\ln Y$ is normal with mean 0.06 and standard deviation 0.25. The expected value of Y is closest to:
Q4Holding the mean of the associated normal constant, an increase in its variance will most likely cause the mean of the lognormal variable to:
Q5Which feature of share prices makes the lognormal distribution a better model for them than the normal distribution?

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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