What makes a variable lognormal?
A positive variable X is lognormal if its natural logarithm Y = ln(X) is normally distributed. Write ln(X) ~ N(μ, σ²). The parameters μ and σ describe the logarithm of X, rather than the mean and standard deviation of X itself.
A non-degenerate lognormal distribution is right-skewed. It cannot produce zero or negative values, and it can produce very large positive values. This makes it a candidate for multiplicative growth, positive prices and some severity models.
Parameters and formulas
| Quantity | Formula |
|---|---|
| Support | X > 0 |
| Mean | exp(μ + σ²/2) |
| Median | exp(μ) |
| Mode | exp(μ − σ²) |
| Variance | [exp(σ²) − 1] × exp(2μ + σ²) |
The mean exceeds the median because a long upper tail pulls the average upward. The probability density function (PDF) is zero outside the positive support.
Worked example: a positive price
Assume ln(S) is normal with μ = ln(100) and σ = 0.20. These are illustrative horizon parameters, rather than annualised inputs.
| Measure | Result |
|---|---|
| Median price | ₹100.00 |
| Mean price | 100 × exp(0.20²/2) = ₹102.02 |
| Probability price is at most ₹80 | Approximately 13.23% |
| 95th-percentile price | 100 × exp(1.64485 × 0.20) = ₹138.95 |
The log transform converts the ₹80 threshold to a normal z-score: [ln(80) − ln(100)] / 0.20 ≈ −1.1157.
Simple returns, log returns and prices
A log return is ln(S₁/S₀). A simple return is S₁/S₀ − 1. If log returns are normal, the corresponding future price is lognormal and the simple return is a shifted lognormal variable. The three distributions are different.
In a geometric Brownian motion model with constant annual drift g and volatility v, a horizon of T years gives ln(ST) a mean ln(S₀) + (g − v²/2)T and standard deviation v√T. Real-world and risk-neutral drifts serve different purposes.
Excel illustration
| Task | Formula |
|---|---|
| Probability price ≤ ₹80 | =LOGNORM.DIST(80,LN(100),0.2,TRUE) |
| 95th-percentile price | =LOGNORM.INV(0.95,LN(100),0.2) |
| Expected price | =EXP(LN(100)+0.2^2/2) |
| Density at ₹100 | =LOGNORM.DIST(100,LN(100),0.2,FALSE) |
Passing 100 as the log-mean would be a major input error. The required parameter here is LN(100).
Converting a level mean and standard deviation
If the desired level mean is m > 0 and level standard deviation is s > 0, first convert them.
Use these log parameters in the distribution function. Do not mix a level mean with a log standard deviation.
Limitations and checks
Does positive support prove that a price follows a lognormal distribution?
No. Positive support is necessary for a lognormal model, but it does not establish the distributional shape. Jumps, stochastic volatility and changing regimes can make the model unsuitable.
Can a lognormal variable have a zero outcome?
No. If there is a genuine possibility of total loss or zero recovery, a model with a separate point mass at zero may be needed.
Interactive distribution laboratory
Change the parameters and threshold, then switch between cumulative probability and density or mass. Values are numerical approximations for learning.
Horizontal axis: the variable’s value. Vertical axis: density, probability mass, or cumulative probability as labelled. Shading highlights values at or below the threshold. The plot shows a finite viewing window.
Sources and further reading
- National Institute of Standards and Technology: Gallery of Distributions
- Distribution definition and properties
- Microsoft: Statistical functions reference
All numerical finance and project examples are hypothetical teaching illustrations. Probability assumptions require validation before practical use.