Solution: The Itô Correction

Exercise: The Itô Correction

Part 1 — Closed-form values and their difference

From the lesson, $\ln(S_t / S_0) \sim \mathcal{N}((\mu - \tfrac{1}{2}\sigma^2)t, \sigma^2 t)$ , so:

\mathbb{E}\!\left[\ln\tfrac{S_t}{S_0}\right] = \left(\mu - \tfrac{1}{2}\sigma^2\right)t

Also $\mathbb{E}[S_t/S_0] = e^{\mu t}$ , so:

\ln\mathbb{E}\!\left[\tfrac{S_t}{S_0}\right] = \mu t

The gap is:

\ln\mathbb{E}\!\left[\tfrac{S_t}{S_0}\right] - \mathbb{E}\!\left[\ln\tfrac{S_t}{S_0}\right] = \mu t - \left(\mu - \tfrac{1}{2}\sigma^2\right)t = \tfrac{1}{2}\sigma^2 t

The gap is exactly the Itô correction times

t

. This is not a coincidence — the same

\sigma^2/2

appears in three places (Itô correction, Jensen gap for log-returns, and the drift shift from

\mu

\mu - \sigma^2/2

) because they are all the same mathematical phenomenon.

Part 2 — Jensen's inequality

\ln

is a concave function. Jensen's inequality for concave

g

states:

g(\mathbb{E}[X]) \ge \mathbb{E}[g(X)]

with equality iff $X$ is almost surely constant. Applying to $X = S_t / S_0$ and $g = \ln$ :

\ln\mathbb{E}\!\left[\tfrac{S_t}{S_0}\right] \ge \mathbb{E}\!\left[\ln\tfrac{S_t}{S_0}\right]

with equality iff $S_t / S_0$ is almost surely constant — which happens only when $\sigma = 0$ . The gap $\tfrac{1}{2}\sigma^2 t$ is strictly positive for any non-degenerate GBM, and it quantifies how much concavity of $\ln$ pulls down the expected log-return relative to the log of the expected return.

Part 3 — The naive "derivation" without Itô

Applying the ordinary chain rule to $f(S) = \ln S$ would give:

d(\ln S_t) \stackrel{\text{wrong}}{=} \frac{dS_t}{S_t} = \mu\,dt + \sigma\,dW_t

Integrating: $\ln(S_t/S_0) = \mu t + \sigma W_t$ , from which $\mathbb{E}[\ln(S_t/S_0)] = \mu t$ .

This contradicts the correct value

(\mu - \tfrac{1}{2}\sigma^2)t

. The discrepancy is precisely the

-\tfrac{1}{2}\sigma^2 t

correction that Itô's lemma introduces because

(dW_t)^2 = dt

does not vanish. So the practical consequence of skipping the Itô correction is overstating the expected log-return by $\tfrac{1}{2}\sigma^2$ per unit time. For a stock with

\sigma = 30\%

, that is a 4.5 percentage-point overstatement per year — easily the difference between a winning and losing pricing model.

Part 4 — The fund manager's claim

The claim is wrong in general. If the fund's log-returns averaged 8% per year, then the annualised geometric return (median terminal wealth per year) is

e^{0.08} - 1 \approx 8.33\%

, but the arithmetic return is higher by the Jensen/Itô bump. If the fund's log-return volatility is

\sigma

, then:

\text{expected arithmetic return} = e^{\mu + \sigma^2/2} - 1

where $\mu = 0.08$ is the average log-return. For $\sigma = 0.15$ (a typical equity-fund volatility): arithmetic return $\approx e^{0.08 + 0.01125} - 1 \approx 9.56\%$ . So the manager's verbal translation ("8% log = 8% arithmetic") understates investor-facing arithmetic returns by roughly 1.5 percentage points per year — every year, compounded.

Conversely, a prospectus that quotes an arithmetic mean return of 10% hides the fact that the corresponding log-return (which determines typical long-run wealth) is only

10 - \sigma^2/2

. The two conventions disagree in different directions depending on what is quoted, and the disagreement is

\sigma^2/2

— the same Itô term, again.

Takeaways

$\tfrac{1}{2}\sigma^2$ is the single quantity connecting three apparently different things: the Itô correction in stochastic calculus, the Jensen gap for log vs. log-of-expectation, and the spread between arithmetic and geometric returns. They are all identical.
Concavity of $\ln$ is the reason. Any statement "the log of the mean equals the mean of the log" is quietly assuming the distribution is degenerate.
Performance reporting is ambiguous without specifying log vs. arithmetic. "My fund returned 10% per year" can mean several things; the gap is $\sigma^2/2$ , which matters most for high-volatility strategies.
If you forget the Itô correction in a pricing model, you will get systematic biases on the order of $\sigma^2/2$ per unit time. For equity options at 25% vol, that is a 3% per-year pricing error — not a rounding issue.