Questions: Geometric Brownian Motion

Question 1

In the GBM solution S(t) = S(0)exp((μ - σ²/2)t + σW(t)), the drift of ln(S) is μ - σ²/2, not μ. Where does the -σ²/2 come from?

• A. It is a discretization error from converting continuous to discrete time
• B. It is the Itô correction: applying Itô's formula to ln(S) with dS = μS dt + σS dW produces d(ln S) = (μ - σ²/2)dt + σ dW
• C. It compensates for the fact that geometric means are smaller than arithmetic means
• D. It ensures the process remains positive

Answer: (B)

Apply Itô's formula to f(S) = ln(S) with f'(S) = 1/S and f''(S) = -1/S². Then d(ln S) = (1/S)(μS dt + σS dW) + (1/2)(-1/S²)(σS)² dt = (μ - σ²/2)dt + σ dW. The -σ²/2 is the Itô correction term (1/2)f''(S)(σS)² dt = -σ²/2 dt. While option C is heuristically related (Jensen's inequality for the concave log function), the precise source is Itô's formula.

Question 2

S(t) follows GBM with μ = 0.10 and σ = 0.30. The expected value E[S(t)] grows at rate:

• A. μ - σ²/2 = 0.055, the drift of ln(S)
• B. μ = 0.10, because E[S(t)] = S(0)e^{μt} regardless of σ
• C. μ + σ²/2 = 0.145, because volatility increases the expected level
• D. σ² = 0.09, because in GBM the variance dominates the growth

Answer: (B)

E[S(t)] = S(0)E[exp((μ-σ²/2)t + σW(t))] = S(0)exp(μt), using the moment generating function of the normal distribution: E[e^{aW(t)}] = e^{a²t/2}. The (μ-σ²/2) in the exponent is the growth rate of the median (and the geometric mean), but the expected value grows at rate μ because the lognormal distribution's right tail contributes extra. This distinction between median growth (μ-σ²/2) and mean growth (μ) is a consequence of Jensen's inequality applied to the convex exponential function.

Question 3

Geometric Brownian motion can take negative values if the initial condition S(0) > 0, provided σ is large enough and enough time passes.

• T. True
• F. False

Answer: False

S(t) = S(0)exp((μ-σ²/2)t + σW(t)) is the exponential of a real number, which is always strictly positive. If S(0) > 0, then S(t) > 0 for all t almost surely, regardless of μ or σ. This positivity is a key property that makes GBM suitable for modeling quantities like stock prices or populations that cannot go negative. The process can get arbitrarily close to zero but never reaches or crosses it.

Question 4

Explain why geometric Brownian motion, despite its widespread use, is a flawed model for real stock prices.

Model answer: GBM assumes constant volatility σ and drift μ, producing returns that are i.i.d. normal. Real stock returns exhibit volatility clustering (periods of high and low volatility), heavy tails (extreme moves more frequent than the normal distribution predicts), mean reversion in volatility, and leverage effects (negative correlation between returns and volatility changes). GBM also cannot produce jumps — sudden large price moves observed in real markets. More realistic models include stochastic volatility (Heston), jump-diffusion (Merton), and local volatility models.

GBM is the 'spherical cow' of finance — an idealization that captures the essential features (positivity, multiplicative growth, randomness) while missing second-order effects. Its value is as a tractable baseline that admits closed-form solutions (Black-Scholes), not as a precise description of reality.