Why do we assume asset prices follow a lognormal distribution instead of normal?

This is a foundational concept that connects to derivatives pricing, risk management, and practically every quantitative model in finance.

The Core Logic

If continuously compounded returns r are normally distributed, then the price at time T is:

S_T = S_0 x e^(rT)

Since e raised to any power is always positive, S_T can never be negative — which is exactly what we want for stock prices.

If instead we assumed prices were normal, there would be a nonzero probability of a negative stock price, which is economically impossible for limited-liability equity.

Properties of the Lognormal Distribution

1. Bounded below at zero — no negative prices
2. Right-skewed — a stock can triple in value but can only lose 100%; the distribution has a long right tail
3. Mean > Median > Mode — the right skew pulls the mean above the median

Numerical Illustration — Oakridge Biotech

Oakridge stock trades at $50. Suppose annual continuously compounded return is N(8%, 25%).

What is the probability the stock falls below $0? Under lognormal: exactly zero. Under normal with the same parameters, P(S < 0) would be P(Z < (0 - 50)/?) which could be positive.

The 5th percentile price:

S_5th = 50 x e^(0.08 - 1.645 x 0.25) = 50 x e^(-0.3313) = 50 x 0.7184 = $35.92

The 95th percentile:

S_95th = 50 x e^(0.08 + 1.645 x 0.25) = 50 x e^(0.4913) = 50 x 1.6343 = $81.72

Notice the asymmetry: the upside ($31.72 above current) is larger than the downside ($14.08 below current). This reflects the right skew.

Connection to Black-Scholes

The BSM option pricing model explicitly assumes stock prices are lognormally distributed (equivalently, log returns are normal). If you change this assumption, you get different option prices — which is what volatility smile models try to address.

Exam tip: If a question says "continuously compounded returns are normal," the price distribution is lognormal. If it says "simple returns are normal," the price distribution is normal (but this is less realistic).

Explore more in our CFA Level I quantitative methods course.