The Lognormal Distribution

Ask someone new to markets to model a share price and they reach for the bell curve. It breaks straight away. A bell curve is symmetric and stretches forever in both directions, so it happily assigns probability to a share price of minus twenty dollars. It also gets the shape of the upside wrong: a stock can double, triple, or twenty-bag, but the very worst it can do is lose everything. Losses are capped at 100 percent; gains are not capped at all. You need a distribution that is pinned against zero on the left and stretched out on the right — and that shape falls out automatically the moment you stop adding returns and start multiplying them.

The analogy: money grows by multiplying

Put $100 in an account paying 10 percent a year. After two years you do not have 100 + 10 + 10 = 120. You have 100 × 1.1 × 1.1 = 121. Returns chain together by multiplication, not addition.

That is awkward, because almost every nice result in probability — averages settling down, the bell curve appearing out of nowhere — is about sums. There is one tool that converts one into the other: the logarithm. Multiplying two numbers is the same as adding their logs. So take logs of a multiplicative process, and you land back in the friendly additive world where the bell curve lives.

That single move is the whole idea. A quantity is lognormal when its logarithm is normal.

Writing it down

Let $S$ be a positive quantity — a share price, a fund value, a house price. Let $\ln S$ mean its natural logarithm. Then $S$ is lognormal when

$$\ln S \sim \mathcal{N}(m,\ s^2).$$

In words: the log of $S$ is a bell curve whose centre is the number $m$ and whose standard deviation is the number $s$. Note the trap already: $m$ and $s$ describe the log, not $S$ itself. They are not the mean and standard deviation of the price.

Turning the log back into the price means exponentiating, $S = e^{\ln S}$, and the exponential function is what bends the symmetric bell into a lopsided hump. Equal steps in log space become unequal steps in price space: from a log-centre of $\ln 100$, one step of $+0.22$ in the log takes you to $125, while one step of $-0.22$ takes you only down to $80. Up-moves get stretched, down-moves get squashed.

Those three centres have clean formulas:

$$\text{mode} = e^{m - s^2}, \qquad \text{median} = e^{m}, \qquad \text{mean} = e^{m + s^2/2}.$$

In words: the most likely value, the middle value, and the average value are all different, and the gap between them grows with $s$ — that is, with volatility. The median is the honest "typical" outcome. The mean is dragged upward by rare, enormous gains that almost never happen but are huge when they do.

Worked example 1: a two-step stock, computed by hand

Forget continuous time. Take a stock at $100 that, in each of two periods, either multiplies by 1.25 or by 0.80, each with probability one half.

Enumerate the four equally likely paths:

path	multiplier	final price	probability
up, up	1.25 × 1.25 = 1.5625	$156.25	0.25
up, down	1.25 × 0.80 = 1.00	$100.00	0.25
down, up	0.80 × 1.25 = 1.00	$100.00	0.25
down, down	0.80 × 0.80 = 0.64	$64.00	0.25

The median. Half the paths end at $100 exactly, one ends higher, one lower. The middle outcome is $100 — after two periods the typical investor has made nothing.

The mean. Weight each ending by its probability:

$$0.25(156.25) + 0.50(100) + 0.25(64) = 39.06 + 50 + 16 = 105.06.$$

So the average ending value is $105.06 while the typical ending value is $100.00. The average is 5 percent higher than anything you are likely to actually experience, and it is not a rounding error — it is driven entirely by the single lucky path to $156.25.

Now look in log space. Take logs of the three outcomes: $\ln 156.25 = 5.0515$, $\ln 100 = 4.6052$, $\ln 64 = 4.1589$. The gaps are $5.0515 - 4.6052 = 0.4463$ and $4.6052 - 4.1589 = 0.4463$. Identical. The lopsided price distribution is perfectly symmetric once you take logs, because $\ln 1.25 = +0.2231$ and $\ln 0.80 = -0.2231$ are equal and opposite.

Add more periods and that symmetric log distribution becomes a bell curve by the The Central Limit Theorem. Exponentiate it back and you have a lognormal. That is the entire origin story.

Worked example 2: a year of a real-looking stock

Now the standard market setup. A stock starts at $100. Over one year its log return is normal with centre $m = 0.06$ and standard deviation $s = 0.20$ (that 0.20 is the annual volatility).

Median price. $100 \times e^{0.06} = 100 \times 1.0618 = 106.18$.

Mean price. Add the half-variance: $s^2/2 = 0.04/2 = 0.02$, so the exponent is $0.06 + 0.02 = 0.08$, giving $100 \times e^{0.08} = 100 \times 1.0833 = 108.33$. The expected price is $2.15 above the median price.

Chance of finishing below $90. Convert to log space first, because that is where the bell curve lives. $\ln(90/100) = \ln 0.9 = -0.1054$. How many standard deviations is that from the centre?

$$z = \frac{-0.1054 - 0.06}{0.20} = \frac{-0.1654}{0.20} = -0.83.$$

From a standard normal table, the area below $-0.83$ is 0.203. So there is roughly a 20 percent chance of being down more than 10 percent after a year.

Chance of finishing above $130. Same recipe: $\ln 1.30 = 0.2624$, so $z = (0.2624 - 0.06)/0.20 = 1.01$, and the area above $1.01$ is 0.156, about 16 percent.

Read those two together. A 10 percent fall and a 30 percent rise are almost equally likely. That asymmetry is invisible if you model returns as normal, and it is exactly what the lognormal is for.

Drag the drift and volatility sliders below and watch the sampled paths fan out. Notice that the fan is not symmetric: the paths above the start spread much further than the paths below, and none of them can cross zero.

Volatility drag, in numbers

Return to the two-step stock. The per-period arithmetic mean return is $0.5(+25\%) + 0.5(-20\%) = +2.5\%$. Yet the median investor made exactly nothing. Where did the 2.5 percent go?

Into the variance. The per-period return has variance $0.5(0.225)^2 + 0.5(-0.225)^2 = 0.0506$, and half of that is $0.0253$ — which is, to within rounding, the entire 2.5 percent arithmetic mean. The relationship

$$\text{median growth} \approx \text{arithmetic mean} - \tfrac{1}{2}(\text{variance})$$

says in words: the growth you actually compound at is the average return minus half the variance. Volatility is a tax on compounding. Two funds with identical average returns but different volatilities do not end up in the same place, and the calmer one wins. This is the same $s^2/2$ that shows up as the drift adjustment in Geometric Brownian Motion and inside The Black-Scholes Model, and it is a direct consequence of Jensen's Inequality.

What this means in practice

• Option pricing. Black-Scholes assumes the terminal price is lognormal. That assumption is what makes the formula produce a strictly positive price and an asymmetric payoff distribution.
• Reporting fund performance. Quoting an arithmetic average return over-promises. A fund that returns +50 percent then −50 percent has an average return of 0 percent and an actual result of −25 percent. Report the compound (geometric) return.
• Simulation. Never simulate prices by adding normal shocks — you will produce negative prices. Simulate the log price additively, then exponentiate.
• Risk limits. A lognormal has a fatter right tail than a normal but a thinner left tail. Real crashes are worse than lognormal predicts, which is why the volatility skew exists.