Exercise: The Itô Correction — Why $\mathbb{E}[\ln S_t/S_0] \ne \ln \mathbb{E}[S_t/S_0]$

Prerequisites: Geometric Brownian Motion, Itô's Lemma, Expectation and Variance

Problem

Let $S_t$ follow geometric Brownian motion with drift $\mu$ and volatility $\sigma$.

1. Compute both $\mathbb{E}\!\left[\ln(S_t / S_0)\right]$ and $\ln\mathbb{E}\!\left[S_t / S_0\right]$ in closed form. Express the difference as a single clean quantity.
2. Using the closed-form answers, argue via Jensen's inequality that the two quantities cannot be equal (unless $\sigma = 0$). Identify the convex function and the direction of the inequality.
3. Explain what goes wrong if you "derive" GBM by naively applying the ordinary chain rule to $\ln S_t$ (i.e. writing $d(\ln S_t) = dS_t/S_t$ without the Itô correction). Specifically, what would you conclude about $\mathbb{E}[\ln(S_t/S_0)]$?
4. A portfolio manager claims: "My fund has averaged 8% per year in log-returns over 20 years. By exponentiating, that translates to an arithmetic compound return of 8% per year for my investors." Is this claim correct? If not, what is the correct statement relating log-returns to arithmetic returns?

Hint

For parts 1 and 2: you already know $\ln(S_t/S_0) \sim \mathcal{N}((\mu - \tfrac{1}{2}\sigma^2)t, \sigma^2 t)$ and $\mathbb{E}[S_t/S_0] = e^{\mu t}$.