Questions: Itô's Formula (Itô's Lemma)

Question 1

Apply Itô's formula to f(x) = x² with X(t) = W(t) (standard Brownian motion). What is d(W²)?

• A. 2W(t) dW(t), exactly as the classical chain rule gives
• B. 2W(t) dW(t) + dt, because the Itô correction term (1/2)f''(W)(dW)² = (1/2)(2)(dt) = dt
• C. W²(t) dt + 2W(t) dW(t)
• D. 2W(t) dW(t) − dt

Answer: (B)

With f(x) = x², we have f'(x) = 2x and f''(x) = 2. Itô's formula gives df(W) = f'(W)dW + (1/2)f''(W)(dW)² = 2W dW + (1/2)(2)(dt) = 2W dW + dt. Integrating both sides recovers the earlier result: W(T)² = 2∫₀ᵀ W dW + T, or equivalently ∫₀ᵀ W dW = (W(T)² - T)/2. The +dt correction is the hallmark of Itô calculus — it is absent from the classical chain rule because smooth functions have zero quadratic variation.

Question 2

Itô's formula applied to f(x) = eˣ with dX = μ dt + σ dW gives d(eˣ) = eˣ(μ + σ²/2)dt + eˣσ dW. The μ + σ²/2 drift — larger than the naive μ — is sometimes called:

• A. The Girsanov correction
• B. The convexity adjustment (or Jensen's inequality effect), reflecting that the exponential's curvature amplifies volatility into drift
• C. The risk-neutral drift
• D. The martingale compensation

Answer: (B)

For a convex function like eˣ, the average of the function exceeds the function of the average (Jensen's inequality). In continuous time, this manifests as the (1/2)f''σ² = (1/2)eˣσ² term in Itô's formula. The exponential curves upward, so random fluctuations (volatility) systematically push the expected value of eˣ above what you'd predict from the drift μ alone. This σ²/2 'convexity adjustment' appears throughout mathematical finance — it is why the drift of geometric Brownian motion differs from what naive application of the chain rule would suggest.

Question 3

A student claims that Itô's formula is just the ordinary chain rule with an error term added. Explain why this framing is misleading.

Model answer: Itô's formula is not an approximation to the classical chain rule with a correction — it is the exact chain rule for processes driven by Brownian motion. The 'extra' (1/2)f''σ²dt term arises naturally and necessarily from the Taylor expansion when (dW)² = dt rather than 0. In classical calculus, (dx)² = 0 because smooth paths have zero quadratic variation, so the second-order Taylor term vanishes. For Brownian motion, (dW)² = dt is a first-order quantity that contributes to the expansion at the same order as dW and dt. The formula is exact and complete — there is no error.

Calling the (1/2)f''σ²dt an 'error term' suggests it is small or negligible. It is neither — it is often the dominant effect. In geometric Brownian motion, the convexity adjustment σ²/2 determines whether the stock price grows faster or slower than the deterministic case. The correct framing is that the classical chain rule is the special case of Itô's formula when σ = 0 (no noise), not that Itô's formula is the classical rule plus noise.

Question 4

Itô's formula requires the function f to be twice continuously differentiable (C²). This regularity condition cannot be relaxed.

• T. True
• F. False

Answer: False

The C² condition is sufficient but not necessary. The Tanaka formula extends Itô's formula to the non-smooth function f(x) = |x|, which has a kink at x = 0 and no second derivative there. The result involves a local time term that captures the singular behavior at the non-differentiable point: d|W| = sgn(W)dW + dL₀(t), where L₀ is the local time at zero. More generally, Itô-Tanaka-Meyer formulas extend to convex functions and functions of bounded variation. The C² assumption is a clean starting point, but the theory extends well beyond it.