# Lognormal Distribution

> A lognormal distribution describes values whose log is normal, like prices that cannot go negative. Learn log returns, volatility drag and its option uses.

Source: https://learn.tradelabsai.com/math/lognormal-distribution/  
Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Lognormal Distribution", https://learn.tradelabsai.com/math/lognormal-distribution/

A variable is lognormally distributed if its natural logarithm follows a normal distribution. In finance, the lognormal distribution is the standard simple model for asset prices: if log returns are normal, prices are lognormal. This captures two real features of prices: they cannot fall below zero, and their potential upside is open ended while the downside is limited to losing everything. The lognormal assumption underpins the Black Scholes model and many simulations of price paths.

## Simple vs log returns

```
simple return R = (P_t / P_(t-1)) - 1
log return r = ln(P_t / P_(t-1)) = ln(1 + R)
```

| Feature | Simple returns | Log returns |
|---|---|---|
| Add across time | No (they compound) | Yes |
| Add across assets in a portfolio | Yes (weighted) | No (approximately) |
| Lowest possible value | minus 100% | Minus infinity (price goes to zero) |
| Typical use | Portfolio returns, reporting | Time series models, option pricing |

For small returns, the two are nearly equal: a 1% simple return is a 0.995% log return.

## Why prices are modelled as lognormal

If each period's log return is normal and independent, the log of the price after many periods is the sum of many normal variables, which is normal. So the price itself is lognormal:

```
ln(P_T) ~ Normal(ln(P_0) + (μ - σ²/2) × T, σ² × T)
```

## The shape of a lognormal distribution

- **Always positive.**
- **Right skewed:** a long tail to the upside.
- **Mean > median > mode.**

**Example: Mean vs median of future prices**
A stock at $100 has expected log return drift of 7% a year and volatility of 30%. After 10 years, the median price is about $100 × e^(0.07 × 10 minus 0.5 × 0.09 × 10) = $100 × e^(0.25) ≈ $128, while the mean price is $100 × e^(0.07 × 10) ≈ $201. The average is pulled up by a small chance of very large gains; the typical outcome is much lower. This gap grows with volatility and time. See [Mean, Median and Mode](https://learn.tradelabsai.com/math/mean-median-and-mode/).

## Volatility drag

The difference between the mean and median growth rate is about σ²/2 per year, often called volatility drag. High volatility lowers the typical compound return even if the average simple return stays the same.

```
median growth rate ≈ arithmetic mean return - σ² / 2
```

With an arithmetic mean return of 10% and volatility of 40%, the median growth rate is about 10% minus 8% = 2% a year. This explains why highly volatile assets and leveraged products can disappoint over time. See [Compounding and Geometric vs Arithmetic Returns](https://learn.tradelabsai.com/math/compounding/).

## Lognormal and option pricing

The Black Scholes model assumes the underlying price follows geometric Brownian motion, which makes future prices lognormal. This is why its formulas involve ln(S/K) and normal distribution functions. See [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/) and [Monte Carlo Option Pricing](https://learn.tradelabsai.com/options/monte-carlo-option-pricing/).

## Where the lognormal model falls short

| Limitation | Reality |
|---|---|
| Thin tails in log returns | Real returns have fatter tails. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/) |
| Constant volatility | Volatility changes and clusters. See [GARCH](https://learn.tradelabsai.com/math/garch/) |
| No jumps | Prices jump on news and earnings |
| No negative prices | Some assets, like oil futures in April 2020 and some interest rates, went negative. See [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/) |

The volatility smile in option markets is partly a sign that real price distributions differ from lognormal. See [Volatility Smile and Skew](https://learn.tradelabsai.com/volatility/volatility-smile-and-skew/).

## Practical tips

Use log returns when adding returns over time or building time series models, and simple returns when combining assets in a portfolio. When simulating prices, generate normal log returns and exponentiate them, which keeps prices positive. When reporting performance, convert back to simple returns, since that is what investors experience. See [Measuring Returns and CAGR](https://learn.tradelabsai.com/portfolio/measuring-returns-and-cagr/).

## Frequently asked questions

### What is a lognormal distribution?

A distribution of a variable whose natural logarithm is normally distributed; it is always positive and skewed to the right.

### Why are stock prices modelled as lognormal?

Because prices cannot go below zero and compound over time, and assuming normal log returns produces lognormal prices.

### What is the difference between simple and log returns?

Simple returns measure percentage change; log returns are the natural log of the price ratio, which add up neatly over time.

Next, learn a fat tailed alternative in [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/).

## Continue learning

- Next lesson: [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/)
- Previous lesson: [Normal Distribution](https://learn.tradelabsai.com/math/normal-distribution/)
- Related: [Normal Distribution](https://learn.tradelabsai.com/math/normal-distribution/): The normal distribution is the bell curve behind many financial models. Learn its properties, the 68 95 99.7 rule, where traders use it and why markets break it.
- Related: [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/): The Black Scholes model prices European options from five inputs. Learn the formula, its assumptions, a step by step example and where the model breaks down.
- Related: [Compounding and Geometric vs Arithmetic Returns](https://learn.tradelabsai.com/math/compounding/): Compounding means returns earn returns over time. Learn the formulas, why losses hurt more than gains help, volatility drag and how it shapes position sizing.
- Related: [Mean, Median and Mode](https://learn.tradelabsai.com/math/mean-median-and-mode/): The mean, median and mode measure the centre of data in different ways. Learn when each is best for trading data, how outliers distort averages and geometric means.
- Related: [Monte Carlo Option Pricing](https://learn.tradelabsai.com/options/monte-carlo-option-pricing/): Monte Carlo pricing simulates many random price paths and averages the discounted payoffs. Learn the method, a Python sketch, accuracy, variance reduction and uses.