Why asset prices are lognormal

Why not model prices as normal?

Because a normal variable can take any value, including negative ones, and a share price cannot fall below zero. A continuously compounded return, $\ln(P_T \div P_0)$, has no such floor: it heads toward minus infinity as the price heads toward zero. So the normal fits returns, and prices need a distribution that stops at zero.

	Normal	Lognormal
Range	−∞ to +∞	0 to +∞
Shape	symmetric	skewed right
Used for	returns (continuously compounded)	prices
Parameters	its own mean and variance	the mean and variance of $\ln Y$

The normal distribution is the bell curve you already know. The lognormal distribution is defined through it: a variable $Y$ is lognormal when its natural log, $\ln Y$, is normal, and the reverse also holds. The name tells you the test: take the log and you get a normal.

Note the last row of the table. A lognormal distribution is described by the mean and variance of $\ln Y$, the associated normal variable, not by the mean and variance of $Y$ itself. Keep track of both sets of numbers.

Drag the volatility up and watch the price curve. The left edge stays at zero while the right tail stretches, and the mean drifts above the median.

Price after T periods$P_T = P_0\,e^{r}$

• Continuously compounded return Memorize normal, can be negative without limit
  • Sum of period returns adds across time
  • Normal or close to it central limit theorem
• Price Memorize lognormal: bounded at 0, skewed right
  • Mean above median the long right tail pulls the mean up

How do continuously compounded returns turn into prices?

A price equals today’s price times $e$ raised to the continuously compounded return, and the return is the log of the price ratio.

$$P_T = P_0\,e^{r_{0,T}}, \qquad r_{0,T} = \ln\left(\frac{P_T}{P_0}\right)$$

The continuously compounded return over several periods is the sum of the single-period figures, because growth factors multiply and logs turn products into sums.

$$r_{0,T} = r_{0,1} + r_{1,2} + \cdots + r_{T-1,T}$$

The same relationship, rearranged, gives the price from a return or the starting price from an ending price.

Why do normal returns give lognormal prices?

If the continuously compounded return $r_{0,T}$ is normal, then $P_T = P_0\,e^{r_{0,T}}$ is the exponential of a normal variable, which is lognormal by definition. So a price cannot go below zero and its distribution leans right.

Two assumptions sit behind the story. Many investment models assume returns are independently and identically distributed (i.i.d.). Independent means past returns do not help predict future ones. Identically distributed is called stationarity : the mean and variance do not change from period to period.

Even when single-period returns are not normal, their sum over many periods is close to normal (the central limit theorem). That is why the lognormal model for prices holds up in practice.

Go deeper Mean and median of a lognormal price

The mean of a lognormal variable is larger than $e^{\mu}$ by a factor of $e^{\sigma^2/2}$. Here $\mu$ and $\sigma^2$ are the mean and variance of the associated normal variable, and the curriculum has you use the expression rather than learn it.

$$E(Y) = e^{\mu + \sigma^{2}/2}$$

With a one-year return of mean 5% and standard deviation 20%, a price of 100 has a median of $100 \times e^{0.05}$ = 105.13 and a mean of $100 \times e^{0.07}$ = 107.25. A larger $\sigma$ adds big upside outcomes while the downside stays capped at a price of zero, so the gap between mean and median widens.

Mean price on your calculator

TI BA II Plus HP 12C

1. 0.2x²÷2+0.05= 0.07, the exponent
2. 2NDe^x×100= 107.25

x² square · 2ND second function (the shifted label above a key) · e^x e to the power x

1. 0.2ENTER×2÷0.05+ 0.07, the exponent
2. ge^x100× 107.25

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Go deeper When the i.i.d. assumption fails

Stationarity is the first assumption to break in a crisis: volatility jumps and stays high, so the variance of returns is not the same period to period. Independence can also fail when big moves cluster. The square-root-of-time rule below is only as good as these two assumptions. For the lognormal price model itself, the Black–Scholes–Merton option pricing model assumes lognormal prices, which is why this lesson leads into option pricing.

How do you measure and scale volatility?

Volatility is the standard deviation of continuously compounded returns, quoted as an annual figure by convention. With i.i.d. returns, the T-period mean and variance are T times the one-period values, so the volatility is $\sqrt{T}$ times the one-period figure.

$$E(r_{0,T}) = \mu T, \qquad \sigma^{2}(r_{0,T}) = \sigma^{2}T, \qquad \sigma(r_{0,T}) = \sigma\sqrt{T}$$

Continuously compounded return over T periodsneeds independent and stationary returns

• Mean Memorize grows with T: $\mu T$
• Variance Memorize grows with T: $\sigma^2 T$
• Volatility Memorize grows with $\sqrt{T}$, not T

$$\sigma_{annual} = \sigma_{period}\sqrt{\text{periods per year}}, \quad \text{daily: } \sqrt{250}$$

Markets are open about 250 days a year, so daily volatility is scaled by $\sqrt{250}$ = 15.81.

In practice you estimate volatility from prices. Convert each price change into a continuously compounded return, take the sample standard deviation of those returns (divide by $n-1$), then annualize.

Four observations would be far too few for a real estimate of expected return, and the day-to-day noise swamps any signal about the mean. Volatility is estimated from long histories.

Exam tip Mean and variance scale with T. Volatility scales with $\sqrt{T}$. If you multiplied a volatility by T, redo it.