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Key Takeaways
- A variable is log-normally distributed when its natural logarithm follows a normal distribution, so the variable itself can never fall to zero or below.
- For a fixed positive starting price, normal log return ln(ST / S0) is equivalent to a lognormal terminal price.
- A log-normal distribution is right-skewed, so its mode sits below its median and its median sits below its mean.
- Black-Scholes and Black-76 use lognormal price dynamics. Other pricing and simulation models use different dynamics.
- Use the implied-volatility surface to compare option quotes across strikes and expiries, and the selected pricing model to calculate the corresponding values and probabilities.
What is a log-normal distribution?
A log-normal distribution is the distribution of a variable whose natural logarithm is normally distributed. Because the variable is the exponential of a normal value, it stays strictly positive and, whenever its log has positive variance, is right-skewed, with its mode below its median and its median below its mean. In finance, a stock price is log-normal when its continuously compounded return is normal, as in the Black-Scholes model.
For a fixed positive starting price S0, write ln(ST / S0) as a normal variable with mean m and standard deviation s. Exponentiating gives the terminal price.
Log-normal mean, median and mode
Median price is S0 × exp(m). Mean price is S0 × exp(m + s² / 2). The density’s mode is S0 × exp(m − s²).
For the example, S0 is $100, proportional drift is 2%, volatility is 25% and the horizon is one year. Under geometric Brownian motion, m = −0.01125 and s = 0.25.
Median price is $98.88, mean price is $102.02 and mode is $92.89. The mode is the peak of the continuous density; probability belongs to intervals rather than individual exact prices.
Formula
S_T is log-normal when ln(S_T / S_0) ~ Normal(m, s²); median(S_T) = S_0 × exp(m), mean(S_T) = S_0 × exp(m + 0.5 × s²), mode(S_T) = S_0 × exp(m − s²)- S_0 is the price today and S_T is the price at horizon T measured in years.
- ln(S_T / S_0) is the continuously compounded return over the horizon, and that log return is the quantity assumed to be normal.
- m is the mean of the log return and s is its standard deviation. Under geometric Brownian motion, m = (μ − 0.5 × σ²) × T and s = σ × √(T).
- μ is the proportional price drift in dS = μ × S × dt + σ × S × dW. The expected log-return rate is μ − σ² / 2.
- For positive s, the formulas give mode below median below mean. Convexity of exp explains mean exceeding median; the density formula locates the mode.
Worked Example
Model a $100 stock over one year with proportional price drift mu = 2% and annualized volatility sigma = 25%. The mean log return is 0.02 − 0.5 × 0.25² = −0.01125 and its standard deviation is 0.25.
- Median price: $100 × exp(−0.01125) = $98.88, the level with equal probability above and below it
- Mean price: $100 × exp(−0.01125 + 0.5 × 0.25²) = $100 × exp(0.02) = $102.02
- Mode price: $100 × exp(−0.01125 − 0.25²) = $92.89, the peak of the density curve
- One standard deviation down: $100 × exp(−0.01125 − 0.25) = $77.01; one standard deviation up: $100 × exp(−0.01125 + 0.25) = $126.97
- Using unrounded prices, the upper endpoint is approximately $28.08 above the median and the lower endpoint is $21.87 below it.
The central log-return interval maps to approximately $77.01 through $126.97 and contains 68.27% of the terminal probability. The median is $98.88 and the mean is $102.02. Approximately 54.97% of outcomes finish below the mean. Equal log distances produce different dollar distances.
Log-normal price ranges and probabilities
The exponential function preserves order. Transform m minus s and m plus s to obtain the central 68.27% price interval: $77.01 to $126.97.
Using unrounded values, those endpoints are $21.87 below and $28.08 above the median. The chart’s $10 bars show the probability in each displayed terminal-price interval.
At $120, the standardized log threshold is 0.7743 and the probability of finishing above it is 21.94%. At $80, the lower-tail probability is 19.83%. 54.97% of terminal prices are below the mean.
Price drift vs log drift
For geometric Brownian motion, m = (mu − sigma² / 2) × T and s = sigma × √T. mu is the proportional price drift, while mu − sigma² / 2 is the expected log-return rate.
Risk-neutral stock pricing uses r − q in place of mu, with r the continuously compounded risk-free rate and q the continuous dividend yield. For positive strikes, the terminal above-strike probability is N(d2). Call delta is exp(−qT) × N(d1).
Log-normal prices and option payoffs
A terminal distribution prices terminal events. Touching a barrier or remaining within a range requires path dynamics as well.
Independent normal log increments sum to a normal log return. A finite sum of arbitrary independent increments need not be normal. Monte Carlo can simulate this model or other specified dynamics.
For a strictly positive underlying and a nonpositive strike, a European call payoff is ST − K and a put payoff is zero. Price these directly. Underlyings that can cross zero require a distribution with the matching support.
Inspect the option’s strike, expiry and implied-volatility input together, and use the Options Profit Calculator to turn modeled price scenarios into each position’s payoff.
Limitations of the log-normal distribution
- One volatility fixes the whole shape. Equity returns show fatter tails and a different skew than that shape allows, which is one reason implied volatility differs across strikes, so price the tails from the surface.
- Use a path model for barrier and range events. A terminal distribution describes the end price, not the path to it.
- An unshifted log-normal variable cannot reach zero or go negative. Underlyings that can cross zero need a model with matching support.
- Probabilities move with the inputs. Real-world and risk-neutral drifts give different odds for the same price level, so match the drift to the question before you quote a probability.
Bottom line
The log-normal distribution is the standard model for prices that compound and cannot fall below zero, and it underlies Black-Scholes. Use it to turn a drift, volatility and horizon into price ranges and probabilities, and check those outputs against the implied-volatility surface, which shows how option prices depart from a single-volatility model.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- NIST/SEMATECH e-Handbook of Statistical Methods: Lognormal Distribution
- OIC (Options Industry Council): Technical Information Reference Library
- Cboe: The Cboe SKEW Index White Paper
- Federal Reserve (FEDS 2011-45): Risk, Uncertainty, and Expected Returns
How entries are sourced, calculated and updated: editorial methodology.