In a lognormal distribution, the natural logarithm ln⁡X\ln X of a value XX follows a normal distribution1. A normal distribution also takes negative values, but XX, obtained by undoing the logarithm, only takes values from 0 to infinity1. For example, if ln⁡X=0\ln X = 0 then X=1X = 1, and even if ln⁡X=−2\ln X = -2, which is negative, XX is about 0.14 and never reaches 0 or below.

In price models, if the log return (the continuously compounded rate of return) is assumed to follow a normal distribution, the price follows a lognormal distribution and never becomes negative. A lecture note lists this as one reason log returns are preferred (consistency with limited liability), and says it is also consistent with the lognormal model used by the Black-Scholes model and others2.

However, the same note points out that actual equity returns are more likely than a normal distribution to take extreme negative values, and have a thicker tail (extreme values occur with larger probability)3. The lognormal distribution is an assumption made by the Black-Scholes model and others, not the actual distribution itself.

Notes

  1. NIST/SEMATECH e-Handbook of Statistical Methods, "Lognormal Distribution" (1.3.6.6.9): the definition at the top (X is lognormal if Y = ln(X) is normal) and the Range in Common Statistics (0 to infinity). ↩ ↩2

  2. Tokuo Iwaisako, "Lecture 2: ランダム・ウォーク仮説のテスト" (Lecture 2: Testing the random walk hypothesis), Institute of Economic Research, Hitotsubashi University (in Japanese), p. 2, "なぜ対数リターンの方が好ましいのか？" (Why are log returns preferable?), items (1) (consistency with limited liability) and (3) (consistency with the lognormal model used in Black-Scholes and others). Page numbers are PDF pages. ↩

  3. The same source, p. 3, "株式収益率の分布について，対数正規性を仮定することの問題点" (Problems with assuming lognormality for the distribution of equity returns), items (1) (extreme negative values) and (2) (thick tail). ↩