Geometric Brownian Motion and the Lognormal Model for Share Prices

Understand Geometric Brownian Motion and Lognormal Model

Key rules to remember

GBM stochastic differential equation

dS(t) = μS(t) dt + σS(t) dZ(t)

μ is the drift and σ the volatility, both constant. Z(t) is standard Brownian motion.

Solution of the SDE

S(t) = S(0) exp{(μ − σ²/2)t + σZ(t)}

Obtained by applying Ito's lemma to ln S(t).

Distribution of log price

ln S(t) ~ N(ln S(0) + (μ − σ²/2)t, σ²t)

Normal, so S(t) is lognormal.

Distribution of log return over (t, t+s)

ln[S(t+s)/S(t)] ~ N((μ − σ²/2)s, σ²s)

Independent of the past and independent across non-overlapping intervals.

Mean of the share price

E[S(t)] = S(0) e^(μt)

Uses E[e^Y] = exp(m + v/2) for Y ~ N(m, v).

Variance of the share price

Var[S(t)] = S(0)² e^(2μt) (e^(σ²t) − 1)

Grows with σ² and t.

Lognormal moments (general)

If ln X ~ N(m, v): E[X] = e^(m + v/2), Var[X] = e^(2m + v)(e^v − 1)

Use for any lognormal variable.

Lognormal median

Median of S(t) = S(0) exp{(μ − σ²/2)t}

Lower than the mean.

Probability statement

P(S(t) > k) = 1 − Φ[(ln(k/S(0)) − (μ − σ²/2)t) ÷ (σ√t)]

Standardise ln S(t) and use normal tables.

How to solve Geometric Brownian Motion and Lognormal Model questions

1. 1Write down the model parameters: S(0), μ, σ and the time t in years. Check whether σ is given as σ or σ².
2. 2Define the log return: ln[S(t)/S(0)] ~ N((μ − σ²/2)t, σ²t). Compute the mean m and variance v numerically.
3. 3If the question asks for a probability, convert the price condition to a log condition, standardise using (x − m) ÷ √v, and read Φ from tables.
4. 4If it asks for the mean or variance of the price, use E[S] = S(0)e^(m + v/2) form, or the shortcut S(0)e^(μt) for the mean.
5. 5If it asks to derive the solution, set X = ln S, apply Ito's lemma, show the −σ²/2 term, then integrate.
6. 6For conditional questions, restart from the latest known price S(t) and use the time gap s only.
7. 7State the result with units and one line on assumptions, such as constant μ and σ and no dividends.

Quickest way: Log-first shortcut

Common mistakes in Geometric Brownian Motion and Lognormal Model

• Using μt as the mean of the log return.

  Students forget the −σ²/2 correction that comes from Ito's lemma.

  Fix: Always write (μ − σ²/2)t for the log return mean. Use μt only for E[S(t)]/S(0) in the exponent.

• Writing the variance of the log return as σt instead of σ²t.

  Confusing standard deviation with variance in the normal parameters.

  Fix: The N(m, v) notation uses variance. The standard deviation is σ√t.

• Applying ordinary calculus to ln S and missing the Ito term.

  Ordinary chain rule gives d(ln S) = dS/S, which looks natural.

  Fix: Use Ito's lemma: d(ln S) = dS/S − ½σ² dt. Remember (dZ)² = dt.

• Treating E[S(t)] as exp of the mean log price.

  Thinking expectation passes through the exponential.

  Fix: Use E[e^Y] = e^(m + v/2). The extra v/2 is the convexity adjustment.

• Mixing time units, such as using σ per year with t in months.

  Questions give t as 6 months or 90 days.

  Fix: Convert t to years before any calculation.

• Claiming the model is realistic because it fits the lognormal shape.

  Memorising the assumptions without the limitations.

  Fix: State the limits: constant volatility, no jumps, no dividends, thin tails, and no volatility clustering.

Worked examples