Log-Normal Distribution
A log-normal distribution is a continuous probability distribution where the natural logarithm of the random variable follows a normal distribution. In finance, asset prices are commonly modeled as log-normally distributed, ensuring that prices remain positive and that continuously compounded returns (log returns) are normally distributed—a theoretical foundation of the Black-Scholes option pricing model.
Key takeaways
- If a random variable X is log-normally distributed, then ln(X) ~ N(μ, σ²), where μ and σ are the mean and standard deviation of the log-normal random variable's logarithm.
- Log-normality ensures asset prices are always positive—a key requirement for equity prices and most financial assets—unlike the normal distribution, which has support on the entire real line.
- The mean of a log-normal distribution is e^(μ + σ²/2), which exceeds the median e^μ; the distribution is right-skewed, consistent with the observation that asset returns have fat right tails (large gains are possible but not symmetric with large losses).
- The Black-Scholes model explicitly assumes that stock prices follow geometric Brownian motion, implying log-normally distributed prices—this assumption underlies the model's closed-form solution.
- Log-normal distributions understate fat tails observed in actual markets (leptokurtosis), motivating extensions such as stochastic volatility models (Heston), jump-diffusion models (Merton), and variance-gamma models.
Explanation
The log-normal distribution arises naturally in financial modeling from the multiplicative structure of asset price dynamics. If a stock price S_t evolves by small multiplicative increments—each day's price is yesterday's price multiplied by a random growth factor—then the natural logarithm of the price ratio (the log return) is the sum of many small independent increments. By the Central Limit Theorem, this sum converges to a normal distribution, implying that log prices are normally distributed and therefore levels are log-normally distributed.
Formally, if log(S_t/S_0) ~ N(μt, σ²t), then S_t follows a log-normal distribution with parameters depending on the drift μ and volatility σ of the continuous-time process. The price at time t has expected value S_0 × e^(μt + σ²t/2), where the extra σ²/2 term reflects Jensen's inequality—the expected value of the exponential of a normal random variable exceeds the exponential of its expected value due to the distribution's asymmetry.
The log-normal model has several desirable properties for asset price modeling. First, prices are always positive: if log(S_t) is normally distributed (which is unbounded below), S_t = e^{log(S_t)} is always strictly positive. Second, the model is scale-independent: the percentage return over any period is the same regardless of the starting price level. Third, log returns over non-overlapping periods are independent and identically distributed under the geometric Brownian motion assumption, which greatly simplifies mathematical treatment.
Black, Scholes, and Merton's seminal 1973 paper derived the famous option pricing formula by modeling stock prices as following geometric Brownian motion—a continuous-time process consistent with log-normally distributed prices at any future date. The Black-Scholes formula for a European call option is C = S·N(d₁) − K·e^{-rT}·N(d₂), where d₁ and d₂ are functions of S, K, r, T, and σ—all derivable under the log-normality assumption. This analytical tractability is the primary reason log-normality became the canonical model despite its empirical limitations.
Empirical evidence is mixed on log-normality. Short-horizon (daily, weekly) returns of equities exhibit excess kurtosis (fat tails relative to normal) and negative skewness—both inconsistent with the log-normal model. Options markets consistently price implied volatility surfaces with skew and term structure that deviate from the flat surface predicted by Black-Scholes, reflecting the market's own assessment that log-normality is inadequate. Long-horizon returns (annual) are closer to log-normal but still exhibit statistically significant departures. This has motivated a rich literature on alternative models: local volatility (Dupire), stochastic volatility (Heston, SABR), jump processes (Merton, Kou), and regime-switching models.
Formula
If X ~ LogNormal(μ, σ²): E[X] = e^(μ + σ²/2); Var[X] = (e^(σ²) − 1) × e^(2μ + σ²); f(x) = (1/(xσ√(2π))) × exp(−(ln x − μ)²/(2σ²)) for x > 0
Example
An equity option trader prices a one-year at-the-money call on a stock currently trading at $100 using the Black-Scholes model. The stock's historical volatility is 25% per annum, the risk-free rate is 5%, and the stock pays no dividends. Under the log-normal model, the stock price in one year has a log-normal distribution with μ = ln(100) + (0.05 − 0.5 × 0.25²) × 1 = 4.636 and σ = 0.25. The 10th percentile of the price distribution is e^(4.636 − 1.282 × 0.25) = e^{4.315} ≈ $74.6, and the 90th percentile is e^{4.636 + 1.282 × 0.25} = e^{4.957} ≈ $142. The Black-Scholes call price is approximately $12.34. In practice, the actual call price in the market might be $13.00–13.50 because traders price in fat tails (higher probability of extreme moves than log-normal implies) through an implied volatility slightly above 25%.
Related terms
At The Money Black Scholes Model Brownian Motion Call Option Central Limit Theorem Equity Fat Tails Forward Rate Formula Geometric Brownian Motion Historical Volatility Implied Volatility Jensens Inequality