Lognormal Distribution
The distribution that best models financial asset prices — positively skewed, incapable of going negative, and the basis of Black-Scholes.
Lognormal = log(Normal)
A variable X follows a lognormal distribution if its natural logarithm ln(X) follows a normal distribution. Put another way: if Y is normal with mean μ and deviation σ, then X = e^Y is lognormal. That has two immediate and fundamental implications: (1) the lognormal always takes positive values, because e^Y > 0 for any real Y; (2) the lognormal is asymmetric with a long tail on the right side (large values). Its parameters are inherited from the log: μ (the mean of the log) and σ (the deviation of the log). But the mean and deviation of the original X (not the log) are different: E[X] = e^(μ + σ²/2) and Var[X] = (e^(σ²) − 1) × e^(2μ + σ²). The probability density is f(x) = (1 / (xσ√(2π))) × e^(−(ln(x) − μ)² / 2σ²) for x > 0. Graphically: it starts at 0, rises to a peak (the mode), and decays to the right with a tail longer and fatter than the normal’s. For small σ (0.1, say) the lognormal looks almost like a shifted normal; for large σ (1.0) the asymmetry is very pronounced.
Why Prices Are Lognormal
The assumption that asset prices follow a lognormal distribution has both theoretical and empirical justification. Theoretically, if continuously compounded (logarithmic) returns are i.i.d. and normally distributed, then prices are lognormal. That follows directly from the fact that the sum of normals is normal, and the exponential of a normal is lognormal. Empirically, the lognormal captures several important properties of real prices: (1) positive prices — a price cannot be negative (leaving aside exceptional cases such as crude futures in April 2020); the normal would allow negative prices, the lognormal does not; (2) asymmetry to the upside: prices can rise without limit but can only fall to 0, which implies positive skew; (3) compounding: if a stock falls 50% and then rises 50%, you do not return to the original ($100 → $50 → $75), behaviour consistent with lognormal rather than normal. The lognormal is the mathematical basis of Geometric Brownian Motion (GBM), the stochastic model Black-Scholes uses to describe the underlying’s dynamics: dS/S = μ·dt + σ·dW, where W is a Wiener process. The solution to that stochastic differential equation is S(t) = S(0) × exp((μ − σ²/2)t + σ·W(t)), meaning S(t) is lognormal at any t.
Positive Skew and Its Consequences
A critical property of the lognormal is its positive skew: the right-hand tail is longer. That has important practical consequences for options trading and return analysis. (1) Mean > Median > Mode: unlike the normal where all three coincide, in a lognormal the mean (expected average) exceeds the median (the 50th percentile), which in turn exceeds the mode (the most frequent value). If the log’s μ is 0 and σ is 1: mode ≈ 0.37, median = 1.00, mean ≈ 1.65. (2) Expected return exceeds the median return: if returns are lognormal, an asset’s expected return exceeds its median. That matters for financial planning: 50% of investors will have returns below the average when distributions have positive skew, which is counterintuitive. (3) The effect of compounding: positive skew implies that small returns compounded consistently can generate large final gains; this is the mathematics behind the "miracle of compound interest". (4) Survivorship in portfolios: with several independent lognormal bets, the portfolio’s mean does not coincide with its median, because most of the return comes from the tail outcomes rather than the central ones. This is the mathematics behind venture capital and long-tail investing.
Implications for Options Pricing
The lognormal assumption has direct implications for options pricing. In Black-Scholes, the probability that a call finishes ITM is not simply 50% when S = K (ATM) — it is slightly higher, because the distribution is skewed to the upside. That is consistent with the empirical observation that ATM calls have a delta of 0.50 to 0.54, not exactly 0.50. The practical implications: (1) ITM probability for calls vs puts — an ATM call has a slightly higher probability of finishing ITM than an equivalent ATM put, because of the lognormal’s positive skew; (2) Asymmetry in expected payoffs: a long call has unlimited upside, because the lognormal admits arbitrarily high prices, while a long put has upside capped at the strike, because price cannot go negative; that is the source of the natural asymmetry between calls and puts; (3) The theoretical prediction about the smile: Black-Scholes with constant σ predicts a completely flat smile, but in real markets OTM puts are more expensive (negative skew in equities). That is consistent with traders knowing the real distribution has fatter downside tails than the pure lognormal predicts. More sophisticated models — Heston, SABR, Merton jumps — incorporate that reality.
Fat Tails: When Even the Lognormal Is Not Enough
Even the lognormal, which significantly improves on the normal for modelling prices, systematically underestimates real tail risk. The logarithmic returns observed in financial markets show excess kurtosis persistently greater than the lognormal predicts. That has motivated the development of models with fatter-tailed distributions: (1) The Student-t distribution, with its degrees-of-freedom parameter ν: at ν=3 kurtosis is infinite and at ν=∞ it is the normal; a ν between 4 and 6 captures observed fat tails well. (2) The generalised hyperbolic, highly flexible: it includes the normal, Student-t, Laplace and normal-inverse Gaussian as special cases. (3) Lévy stable distributions, which admit tails as heavy as you like, though at the cost of variance potentially being infinite: an uncomfortable but realistic property in some markets. (4) Jump-diffusion models: Black-Scholes plus Poisson jumps. Merton (1976) was first and Kou (2002) added asymmetry to the jumps. They capture the discrete events — earnings gaps, news shocks — that pure diffusion cannot. (5) Variance gamma and normal-inverse Gaussian, models based on Lévy processes. In practice, sophisticated market makers use hybrid models combining stochastic volatility and jumps, calibrated to the full implied volatility surface. For the retail trader, the conclusion is pragmatic: models assuming lognormality underestimate the risk, so always add a margin of safety in position sizing.
Practical Use in Analysis and Trading
The lognormal distribution has daily applications in trading and analysis. (1) Calculating the expected move: for a horizon T, the one-standard-deviation move runs at roughly S × σ × √T (assuming lognormal returns with constant σ). It is the basis of the expected move quoted ahead of earnings or events. (2) Probability cones: representations of the probable range of future prices. The cone opens like a logarithmic funnel, reflecting the lognormal’s asymmetry, with more percentage range to the upside than the downside. (3) Monte Carlo for pricing exotics: simulating thousands of lognormal paths and averaging the payoffs is standard technique for exotic options with no closed-form formula. (4) Portfolio projections: Monte Carlo of terminal value assuming lognormal returns is the standard in financial planning software. (5) Risk-neutral pricing: for valuation we use the risk-neutral lognormal, with drift r rather than the real μ, not the real-world distribution. That distinction is technical but central: the risk-neutral density implied by option prices typically has fatter tails than the pure lognormal, reflecting the premium market makers charge for tail risk. (6) Comparing assets: comparing annualised returns of two assets is more meaningful if you assume lognormality, because compounded returns are additive in log space, not in price space.