Explain why stock prices are lognormal, how rates of return are distributed, and how expected return is computed.

FRM® Part I · Valuation and Risk Models · The Black-Scholes-Merton Model · FRM1.VR.15.a

The idea

Black-Scholes-Merton assumes a stock price follows geometric Brownian motion. Over T years the continuously compounded return ln(S_T ÷ S_0) is then normally distributed. Its mean is (μ − σ²÷2) × T and its variance is σ² × T. Here μ is the expected return and σ the volatility. The price equals S_0 times e raised to that normal return, so the price itself is lognormal. It can never be negative and has a right-skewed distribution. Expected price = S_0 × e^(μ × T). So μ is the expected arithmetic rate of return, while the expected continuously compounded return is lower by σ²÷2. The usual mistakes are to treat μ as the mean of the log return, or to add the half variance instead of subtracting it. Exam questions test which of the two means applies.

Formulas

Expected stock price under geometric Brownian motion

E(ST) = S0 × e^(μ × T)

S0: the stock price today
μ: the expected return per year
T: the horizon in years

No σ² adjustment here. Using μ − σ² ÷ 2 in the exponent gives the median price, which is lower.

Mean of the continuously compounded return over T years

Mean = (μ − σ² ÷ 2) × T

μ: the expected return per year
σ: the annual volatility
T: the horizon in years

The variance of the same return is σ² × T. Adding the half variance instead of subtracting it is the classic error.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why is the price of a stock assumed to be lognormally distributed in the Black-Scholes-Merton model?

Its continuously compounded return over any period is normally distributed, and the price is the starting price multiplied by the exponential of that return. - correct
Its simple return in each period is normal, so the price, which is a sum of those returns, is normal as well.Why not: Treats price as a sum of normal returns; with continuous compounding, returns add in logs, so price is lognormal, not normal.
Volatility is assumed to rise as the price rises, which makes large upward moves more likely than large downward moves.Why not: Confuses lognormal shape with price-dependent volatility; BSM holds volatility constant.
In a risk-neutral world the expected return equals the risk-free rate, and that drift is what makes the price lognormal.Why not: Confuses the drift used for valuation with the shape of the distribution, which comes from normal log returns.

In geometric Brownian motion ln(S_T/S_0) is normal, and S_T = S_0 x exp(that return). A price that is the exponential of a normal variable is lognormal, which is also why it cannot be negative.

A harder one

A stock has an expected return of 12% per year and a volatility of 30% per year. The expected continuously compounded annual return is closest to:

12.0%Why not: Treats the expected return mu as the expected log return; the log return has mean mu - sigma²/2.
16.5%Why not: Adds sigma²/2 = 4.5% to mu instead of subtracting it.
7.5% - correct
3.0%Why not: Subtracts the full variance sigma² = 9% rather than half of it.

Expected continuously compounded return = mu - sigma²/2 = 12% - 0.5 x 0.30² = 12% - 4.5% = 7.5%.

The hardest tier

A stock has an expected return of 9% per year and a volatility of 20% per year. The average continuously compounded return per year over a four-year period is normally distributed. Using 1.96 standard deviations, its 95% confidence interval is closest to:

-10.6% to 28.6%Why not: Centres the interval on mu = 9% rather than mu - sigma²/2 = 7%.
-12.6% to 26.6% - correct
-32.2% to 46.2%Why not: Uses a standard deviation of sigma = 20% instead of sigma/sqrt(T) = 10% for the average annual return.
-8.6% to 30.6%Why not: Centres on mu + sigma²/2 = 11%; the adjustment to the mean log return is negative.

The average log return per year is N(mu - sigma²/2, sigma²/T) = N(7%, 0.10²). The 95% interval is 7% +/- 1.96 x 10% = 7% +/- 19.6%, from -12.6% to 26.6%.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →