Chapter 7 · Itô Calculus and Continuous-Time Finance
Part II · Dynamics in Continuous Time
Chapter at a glance
Modern Finance Laboratory · Module 7 · Week 7
∫ Module 7 has four panels. The integral builder assembles Riemann sums of 𝜃 𝑑𝑊 for user-chosen integrands with a selector for evaluation point (left, midpoint, right), displaying the convergence of Figure 7.1 and the isometry’s variance prediction against Monte Carlo. The Itô verifier accepts a function 𝑓 (𝑡, 𝑥), computes both sides of (7.3) numerically along simulated paths, and displays the residual shrinking with the mesh; a “classical chain rule” toggle drops the 12 𝑓 𝑥 𝑥 term so the reader can watch ordinary calculus fail by exactly the quadratic variation. The Girsanov panel drives P- and Qpaths with one noise stream, a slider for 𝜃, a
Learning Outcome Statements
LOS 7.1 Construct the Itô integral for simple integrands, prove the Itô isometry and the martingale property, and explain why the left-endpoint (predictable) evaluation is forced by finance.
LOS 7.2 State and apply Itô’s formula, explain how quadratic variation generates the second-order term, and derive the drift condition under which a function of Brownian motion is a martingale.
LOS 7.3 Work with Itô processes and stochastic differential equations: verify the geometric Brownian motion solution, solve the Ornstein–Uhlenbeck equation, and apply the product rule and box calculus.
LOS 7.4 Prove Girsanov’s theorem for constant drift change, compute the market price of risk, and construct the risk-neutral dynamics of an asset.
LOS 7.5 Use the martingale representation theorem to explain √ dynamic completeness, and derive continuous-time risk-neutral pricing with the 1/ 𝑛 law for discrete hedging error.
LOS 7.6 Audit a dealer’s continuous-hedging specification: identify what each clause asserts mathematically and verify it by simulation.
Laboratory · Module 7 (book §7.10)
Module 7: Stochastic Calculus in the Hands
Course Website · Week 7
Guided experiment (supports LOS 7.1–7.6). (i) Build 𝑊 𝑑𝑊 at left and midpoint evaluation; report the two limits and the constant gap 12 𝑇. (ii) Verify Itô’s formula for 𝑓 = 2 𝑥 2 and ∫𝑓 = 𝑒 𝜆𝑥−𝜆 𝑡/2 ; with the toggle, measure the classical rule’s residual and match 1 it to 2 𝑓 𝑥 𝑥 𝑑𝑡. (iii) Set 𝜃 = 𝜆 = 0.178 in the Girsanov panel and confirm the index’s Q-drift √ equals 𝑟 while measured volatility is unchanged at every 𝜃. (iv) Reproduce the 1/ 𝑛 law; then set drift to 12% and confirm the error mean stays at zero, writing one sentence on why.
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Exercises
Exercises are grouped A–E throughout the book: A concept checks, B computations, C proofs and extensions, D modeling and application, E laboratory. Starred exercises (∗) are on the advanced track.
Part A — Concept checks
Exercise 7.1. Explain in two sentences each: why the Itô integral evaluates the integrand at the left endpoint; why the isometry (7.2) deserves to be called a diversification identity in time; and why Girsanov’s theorem can change 𝜇 but not 𝜎. ∫𝑇
Exercise 7.2. A colleague writes 0 𝑊 𝑑𝑊 = 21 𝑊𝑇2 “by substitution.” Identify the exact step of classical calculus that fails, state the correct value, and name the theorem of Chapter 6 responsible for the correction term.
Exercise 7.3. The dealer’s annex says hedging error is “unbiased regardless of the index’s true drift.” State precisely which result of this chapter that claim instantiates, and which two modeling assumptions it silently retains (continuity of paths; known constant 𝜎).
Part B — Computations
Exercise 7.4. Apply Itô’s formula to compute 𝑑 (𝑊𝑡3 ) and 𝑑 (𝑡𝑊𝑡 ), and use the first to ∫𝑇 ∫𝑇 evaluate E[𝑊𝑇3 ] and 0 𝑊𝑡2 𝑑𝑊𝑡 in terms of 𝑊𝑇 and 0 𝑊𝑡 𝑑𝑡.
Exercise 7.5. For the OU process of Example 7.5 with 𝜅 = 0.5, 𝑥¯ = 4%, 𝜎 = 1%, 𝑋0 = 6%: compute E[𝑋𝑡 ] and sd(𝑋𝑡 ) at 𝑡 = 1, 3, 10 years; the half-life of the initial gap; and the stationary standard deviation. Sketch the implied funnel.
Exercise 7.6. Verify by Itô’s formula that under Q the discounted index 𝑒 −𝑟𝑡 𝑆𝑡 of Example 7.7 is a Q-martingale (zero drift), and that under P its drift is (𝜇 − 𝑟)𝑒 −𝑟𝑡 𝑆𝑡 . Then compute the density 𝑍𝑇 = 𝑑Q/𝑑P on a path with 𝑊𝑇 = 0.4, 𝑇 = 1, 𝜆 = 0.178.
Exercise 7.7. With 𝜎 = 17%, 𝑇 = 1: use the isometry to compute the standard deviation ∫𝑇 of 0 𝜃 𝑡 𝑑𝑊𝑡 for (a) 𝜃 𝑡 ≡ 0.5; (b) 𝜃 𝑡 = 𝑡; (c) 𝜃 𝑡 = 1 {𝑡 ≤1/2} . Interpret (c) as “de-risking at midyear” and compare its risk with holding 0.5 throughout.
Part C — Proofs and extensions
Exercise 7.8. Prove the product rule (7.8) from Itô’s formula applied to 𝑓 (𝑥, 𝑦) = 𝑥𝑦 (two-dimensional Itô: expand to second order and apply the box calculus, treating both processes as driven by the same 𝑊). Then derive the SDE of 𝑆𝑡−1 for GBM and verify that 𝑆𝑡 · 𝑆𝑡−1 = 1 is consistent with (7.8). ∫𝑡 2
Exercise 7.9. Prove that for adapted, square-integrable 𝜃, the process 0 𝜃 𝑑𝑊 − ∫𝑡 𝜃 2 𝑑𝑢 is a martingale (condition the square’s increment as in the isometry proof). 0 𝑢 ∫ ∫𝑡 Conclude that [ 𝜃 𝑑𝑊 ] 𝑡 = 0 𝜃 𝑢2 𝑑𝑢, the running form of (7.2). 7.11 Exercises 173
Exercise 7.10. Complete Example 7.5: carry out the Itô computation for 𝑒 𝜅𝑡 𝑋𝑡 , verify the variance formula via the isometry, and prove that as 𝑡 → ∞ the law of 𝑋𝑡 converges to Normal( 𝑥, ¯ 𝜎 2 /2𝜅) regardless of 𝑋0 .
Exercise 7.11. ∗ Extend Theorem 7.6 to a two-point drift: 𝜃 𝑡 = 𝜃 1 1 {𝑡 ≤𝑇/2} +𝜃 2 1 {𝑡 >𝑇/2} , by applying the constant case on each half and composing the densities. Verify Novikov’s condition trivially holds, and explain in one sentence why iterating this argument makes the general step-process case plausible before the full theorem is invoked.
Part D — Modeling and application
Exercise 7.12. Meridian’s rates desk proposes modeling the short rate with (7.7). Write the one-page model memo: the three parameters’ economic meanings and how each would be estimated; the model’s Gaussian defect (negative rates possible—compute the stationary probability of 𝑟 < 0 for the parameters of Exercise 7.5); and one modification from the literature that repairs it, with the trade-off it introduces.
Exercise 7.13. Write the two-page certification memo on the dealer’s hedging annex. For each of the three clauses, state the mathematical result that certifies it (with the chapter’s equation numbers), the simulation evidence (Figure 7.3’s numbers), and the residual risk outside the specification. Conclude with the two questions the CIO should put to the dealer—on jump exposure and on volatility misspecification—each phrased so that a quantitative answer is possible.
Part E — Laboratory
Exercise 7.14. (Laboratory Module 7; supports LOS 7.1–7.6.) Perform the four-part guided experiment of Section 7.10. Submit: (a) the two-limit screenshot with the 12 𝑇 gap measured; (b) the Itô-verifier residuals with and without the correction term; (c) the √ Girsanov panel at 𝜃 = 0.178 with drift and volatility readouts; (d) the 1/ 𝑛 plot and your one-sentence explanation of drift-insensitivity.
Exercise 7.15. (Course Website, Week 7, Notebook 7.) Extend the notebook: implement the delta hedge with (a) a deliberately wrong volatility (15% used, 17% true) and report the mean and sign of the resulting P&L drift; (b) the jump overlay of Example 6.10 and report the error distribution’s new left tail. Three sentences: which clause of the dealer’s annex each experiment stresses, and what Chapter 8 will need to address.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 7; they are not posted here. Problem-set files are on the Course Website, Week 7.