Geometric Brownian Motion and Lognormal Share Prices

Understand Geometric Brownian Motion and Lognormal Share Prices

Key rules to remember

GBM stochastic differential equation

dS(t) = μ S(t) dt + σ S(t) dB(t)

μ is the drift and σ the volatility, both per year. B(t) is standard Brownian motion.

Solution of GBM

S(t) = S(0) exp{(μ − σ²/2)t + σB(t)}

Follows from applying Ito's lemma to ln S(t).

Distribution of log return

ln(S(t)/S(0)) ~ N((μ − σ²/2)t, σ²t)

Normal, so S(t) is lognormal. Variance is σ²t, so the standard deviation is σ√t.

Lognormal parameters

If ln S(t) ~ N(m, v) then E[S(t)] = exp(m + v/2) and Var[S(t)] = exp(2m + v)(exp(v) − 1)

Here m = ln S(0) + (μ − σ²/2)t and v = σ²t.

Mean and variance of share price

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

Substitute m and v into the lognormal formulas.

Ratio over a period

S(t+s)/S(t) is lognormal and independent of the past up to time t

Same distribution as S(s)/S(0). Gives independent increments of log price.

Normal probability from log price

P(S(t) > K) = 1 − Φ[(ln K − m) ÷ √v]

Convert to the standard normal Z, then use tables.

How to solve Geometric Brownian Motion and Lognormal Share Prices questions

1. 1Write down the parameters: S(0), μ, σ and the time t in years. Convert months to years.
2. 2Write the log return distribution: ln(S(t)/S(0)) ~ N((μ − σ²/2)t, σ²t). Compute the mean m₀ = (μ − σ²/2)t and the variance v = σ²t.
3. 3If the question is about a probability, convert the price condition to a log condition, such as ln(K/S(0)). Standardise: Z = (x − m₀) ÷ √v.
4. 4Look up Φ(z) in the tables. Use symmetry for negative values: Φ(−z) = 1 − Φ(z).
5. 5If the question asks for the mean or variance of the price, use E[S(t)] = S(0)e^(μt) and Var[S(t)] = S(0)² e^(2μt)(e^(σ²t) − 1). Do not exponentiate the mean of the log return.
6. 6For a percentile or confidence interval, find the normal quantile for ln S, then exponentiate the end points.
7. 7State your assumptions (constant μ and σ, no dividends) and check the answer is sensible, for example a positive price and a mean above the median when σ > 0.

Quickest way: Log scale first, then lognormal formulas

Common mistakes in Geometric Brownian Motion and Lognormal Share Prices

• Using μt as the mean of the log return.

  Students forget that the log return has drift μ − σ²/2, not μ.

  Fix: Always write the log return as N((μ − σ²/2)t, σ²t). The μ alone appears only in E[S(t)] = S(0)e^(μt).

• Writing E[S(t)] = S(0)exp((μ − σ²/2)t).

  Students take the exponential of the mean of ln S and forget the lognormal correction.

  Fix: Use E[S] = exp(m + v/2). The v/2 = σ²t/2 cancels −σ²t/2, leaving S(0)e^(μt).

• Using σ instead of σ√t as the standard deviation of the log return.

  Students forget that variance grows linearly with time.

  Fix: The variance is σ²t. Take the square root only at the standardising step, giving σ√t.

• Mixing up volatility and variance, for example using σ = 0.04 when the question gives variance 0.04.

  The wording 'volatility 20%' and 'variance 0.04' are easy to confuse.

  Fix: Underline whether the question gives σ or σ². Volatility of 20% means σ = 0.2 and σ² = 0.04.

• Forgetting to convert the time period to years.

  Questions often give 6 months or 90 days, but μ and σ are annual.

  Fix: Convert first: 6 months is 0.5 years. Write t on the top line of your working.

• Applying the normal tables directly to the price, or using ln with the wrong base.

  Students forget that S is lognormal, not normal. Some use log base 10.

  Fix: Take the natural log of both sides of the inequality, then standardise. ln means base e throughout.

Worked examples