Lognormal Stock Prices: Assumption, Mathematics, and Limits

For educational purposes only; not investment advice.

Direct answer

Classic option models often assume that a future stock price is lognormally distributed, which means its continuously compounded log return is normally distributed. The price itself is not assumed normal, and a return is not “lognormal.” Under geometric Brownian motion:

S_T=S_0 exp[(μ-σ²/2)T+σ√T Z], where Z∼N(0,1).

The exponential keeps modeled prices positive and makes multi-period log returns additive. It also leads to tractable option formulas. Real markets do not obey the assumption exactly: returns show skew, fat tails, volatility clustering, jumps, trading interruptions, and changing distributions. Lognormality is a transparent benchmark, not a market law.

Distribution and pricing mechanics

Simple return is R=S_T/S_0-1; log return is g=ln(S_T/S_0)=ln(1+R). Log returns add through time, while simple returns compound multiplicatively. A rise from $100 to $110 followed by a fall to $99 gives ln(1.10)+ln(0.90)=ln(0.99)=-1.005%. Adding +10% and -10% as simple returns would incorrectly suggest zero.

If ln S_T∼N(m,v), the price median is e^m, mean is e^(m+v/2), and mode is e^(m-v). The right-skewed price distribution has a zero lower boundary and no finite upper boundary. It cannot reach exactly zero in finite time, so default and other absorbing events are absent unless modeled separately.

In Black-Scholes risk-neutral pricing, expected log return uses (r-q-σ²/2)T, not the stock’s estimated real-world expected return. This distribution prices contingent claims under no-arbitrage assumptions; it is not a forecast of physical frequencies. Constant σ would also imply one implied volatility across strikes for a maturity, contrary to the observed volatility skew or smile.

A one-year distribution example

Let S_0=$100 and assume one-year log return is normal with mean 5% and standard deviation 20%. The future price median is $100e^0.05=$105.13. Its mean is $100e^(0.05+0.20²/2)=$107.25, while its mode is $100e^(0.05-0.20²)=$101.01. “Expected price” is therefore not the same as the middle or most likely region.

Using Z=-1 and Z=+1 as approximate 16th and 84th percentiles gives $100e^(0.05-0.20)=$86.07 and $100e^(0.05+0.20)=$128.40. The dollar range is asymmetric even though the log-return interval is symmetric.

A separate compounding example shows why return definitions matter. A 50% fall from $100 to $50 followed by a 50% gain reaches only $75, a cumulative simple return of -25%. The log returns sum exactly to ln(0.5)+ln(1.5)=ln(0.75)=-28.77%. Recovering from $50 to $100 requires a 100% gain.

Testing and model-risk checklist

Use total-return or correctly adjusted price data; splits and ordinary distributions can create false jumps in unadjusted series.
Align timestamps, currencies, market closes, missing observations, and non-trading days before calculating returns.
Report simple and log returns explicitly; they are close only for small changes.
Compare empirical skew, excess kurtosis, 1%/5%/95%/99% quantiles, histograms, and Q-Q plots with a normal benchmark.
Check calm and crisis periods separately. Mixing changing volatility regimes can itself create fat-tailed unconditional returns.
Test serial dependence in returns and squared returns; volatility clustering violates constant independent increments.
Do not apply √T scaling mechanically around earnings, scheduled events, long memory, or regime changes.
Separate physical parameters used for forecasting from risk-neutral parameters inferred for pricing.
Use the current implied-volatility surface for cross-strike option valuation rather than forcing one at-the-money volatility onto every strike.
Stress jumps, default, trading halts, liquidity loss, and volatility repricing outside the smooth diffusion.
Validate parameters and hedge errors out of sample; fitting historical extremes does not prove future stability.
For path-dependent options, simulate intermediate paths and discretization error, not only lognormal terminal prices.

Common misconceptions

“Stock returns are lognormal.” The classic statement is that prices are lognormal and log returns are normal.
“Positive modeled prices prove realism.” Positivity does not capture default, jumps, fat tails, or volatility regimes.
“Mean price is the most likely price.” A lognormal distribution’s mean, median, and mode differ.
“A symmetric log-return range is symmetric in dollars.” Exponentiation creates an asymmetric price range.
“Risk-neutral probability is the actual probability.” It includes pricing adjustments and risk premia and serves a different purpose.
“Annual volatility is always daily volatility times √252.” That scaling needs stable independent increments.
“One historical volatility prices every option.” Market option prices generally imply strike- and maturity-dependent volatility.
“Rare events should be deleted as outliers.” After data checks, extremes may be essential evidence against the assumed tails.
“Lognormal terminal sampling prices every derivative.” Barriers, Asians, and American options depend on paths or exercise decisions.