E[S(T)] = —
Median S(T) = —
Mode S(T) = —
GBM: S(t) = S₀·exp((μ−σ²/2)t + σW(t))
Log-normal S(T): log S(T) ~ N(log S₀ + (μ−σ²/2)T, σ²T)
Itô correction: μ−σ²/2 (not μ) is the growth rate of log S — Jensen's inequality gap.
Black-Scholes: risk-neutral pricing replaces μ with r; call price = S·N(d₁)−Ke^{−rT}·N(d₂).
Middle chart: distribution of S(T) at maturity. Bottom: log-returns are Gaussian.
Log-normal S(T): log S(T) ~ N(log S₀ + (μ−σ²/2)T, σ²T)
Itô correction: μ−σ²/2 (not μ) is the growth rate of log S — Jensen's inequality gap.
Black-Scholes: risk-neutral pricing replaces μ with r; call price = S·N(d₁)−Ke^{−rT}·N(d₂).
Middle chart: distribution of S(T) at maturity. Bottom: log-returns are Gaussian.