Chapter 11 · Optimal Stopping and Real Options
Part III · Optimization, Control, and Learning
Chapter at a glance
Modern Finance Laboratory · Module 11 · Week 11
Module 11 has four panels. The Snell panel runs the recursion (11.2) on a visible lattice, coloring the stopping region as it forms and displaying the stopped envelope’s martingale property as a live flat-mean readout—Theorem 11.1 as animation. The American panel prices puts and calls with a dividend slider: the call’s exercise region is empty at 𝛿 = 0 (Proposition 11.2 on screen) and materializes as 𝛿 rises. The pasting panel lets the user drag the boundary of a candidate perpetual-put solution and watch the value-matched curve kink, the supermartingale test fail, and the attained value fall— maximized exactly at tangency. The platform panel
Learning Outcome Statements
LOS 11.1 Formulate when-decisions as optimal stopping problems over stopping times.
LOS 11.2 Construct the Snell envelope by backward recursion, prove its characterization as the smallest dominating supermartingale, and prove the optimality of the first-contact stopping time.
LOS 11.3 Price American options on lattices, extract exercise boundaries, and prove that American calls on non-dividend assets are never exercised early.
LOS 11.4 Solve perpetual stopping problems in closed form using value matching and smooth pasting, and explain why the tangency condition locates the boundary.
LOS 11.5 Apply the McDonald–Siegel model to irreversible investment: compute hurdle multiples, option premia over NPV, and the value of exclusivity windows.
LOS 11.6 Write a commit-or-wait recommendation that states its spanning assumptions and prices the alternatives.
Laboratory · Module 11 (book §11.10)
Module 11: The Timing Desk
Course Website · Week 11
Guided experiment (supports LOS 11.2–11.6). (i) On the Snell panel, price the American 90-put and verify 1.826 and the boundary of Figure 11.1(b). (ii) Set 𝛿 = 0 and then 3% on the American panel; record when the call’s region appears. (iii) On the pasting panel, report the attained values at boundaries 60, 66, and 75, confirming the
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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 11.1. Classify as stopping problems or not, with one line of justification: (a) when to refinance a mortgage; (b) how much to hedge each day; (c) when to fire an underperforming manager; (d) whether to buy insurance today. For the stopping problems, name the reward process.
Exercise 11.2. “While waiting is optimal, the problem is a fair game.” Locate this sentence in Theorem 11.1’s clauses, and explain in two sentences why a strictly positive drift of the envelope in the continuation region would contradict optimality.
Exercise 11.3. The platform memo says volatility is “partly an asset.” Reconcile this with Chapter 9, where volatility drag made variance a pure cost to growth: identify precisely what the option holder owns that the portfolio holder does not, and which formula in each chapter carries the sign.
Part B — Computations
Exercise 11.4. By hand, on a three-date binomial tree (𝑆0 = 100, 𝑢 = 1.2, 𝑑 = 0.9, 𝑅 = 1.05—Chapter 1’s market): compute the Snell envelope of the American 100-put, exhibit the stopping region, and compare with the European value. Verify the stopped envelope’s martingale property on one path. 254 11 Optimal Stopping and Real Options
Exercise 11.5. Reproduce the chapter’s lattice numbers: American 90-put = 1.826 at 1,000 steps; the boundary values at 𝑡 = 0.25, 0.5, 0.75; and the convergence of the American price across 𝑛 = 50, 200, 1,000 steps.
Exercise 11.6. Verify the perpetual put arithmetic: 𝑝 = −2.754, 𝑆 ∗ = 66.0, 𝑉 (100) = 7.64; check both value matching and smooth pasting numerically; and recompute (𝑆 ∗ , 𝑉 (100)) at 𝜎 = 25%, explaining the direction of both moves.
Exercise 11.7. Verify the platform numbers: 𝛽1 = 1.739, hurdle 2.35, 𝑉 ∗ = 176.5, 𝐹 (82) = 26.8; then tabulate the hurdle multiple over 𝜎 ∈ {15%, 25%, 35%} and 𝛿 ∈ {3%, 4%, 6%}, reproducing Figure 11.3(b)’s ordering.
Part C — Proofs and extensions
Exercise 11.8. Prove that for a submartingale reward the horizon is always an optimal stopping time, and for a supermartingale reward, stopping immediately is. Use the first to re-derive Proposition 11.2 in one line, and exhibit the dividend-paying counterexample to the submartingale hypothesis.
Exercise 11.9. (The certificate in action.) Using Theorem 11.1(i), prove that the function 𝑈𝑛 := max(𝑍 𝑛 , E[𝑍 𝑁 |F𝑛 ]) is in general not the Snell envelope by constructing a three-date counterexample where intermediate stopping strictly beats both “now” and “at the end”—and verify your counterexample by running (11.2).
Exercise 11.10. Complete Proposition 11.3: solve the value-matching/smooth-pasting pair to obtain (11.3); verify dominance 𝑉 ≥ (𝐾 − 𝑆) + everywhere; and confirm by direct differentiation that 𝑉 is 𝐶 1 at 𝑆 ∗ but that 𝑉 ′′ jumps, computing both one-sided second derivatives.
Exercise 11.11. ∗ Derive the first-passage machinery behind the perpetual solutions: for GBM and a level 𝑏 < 𝑆0 , use the exponential martingale (optional stopping with the care of Section 7.6∗ ) to prove EQ [𝑒 −𝑟 𝜏𝑏 ] = (𝑆0 /𝑏) 𝑝 with 𝑝 = −2𝑟/𝜎 2 , and hence re-derive 𝑉 (𝑆) = (𝐾 − 𝑆 ∗ ) (𝑆/𝑆 ∗ ) 𝑝 as “payoff at the boundary, discounted by the hitting-time transform”—the probabilistic reading of (11.3).
Part D — Modeling and application
Exercise 11.12. Meridian’s risk officer proposes a drawdown rule: “de-risk the portfolio at first touch of −15%.” Cast it as a stopping time; explain, using the Snell logic, why a rule fixed in advance is generically suboptimal against a rule derived from a reward process; and specify what reward process would have to be believed for the −15% rule to be exactly optimal.
Exercise 11.13. Write the two-page memo for the item-four vote: the recommendation (wait; revisit at the window’s edge); the four numbers that carry it (𝐹 = $26.8M 11.11 Exercises 255 perpetual, 𝐹0.5 = $9.6M in-window, NPV = $7M, hurdle 𝑉 ∗ = $176.5M); the value of extending exclusivity to one and five years as negotiation currency; the sensitivity table over 𝛿 ∈ {3%, 4%, 6%} with the 𝛿 at which the recommendation flips; and the spanning caveat stated in Chapter 4’s language of admissible pricing measures.
Part E — Laboratory
Exercise 11.14. (Laboratory Module 11; supports LOS 11.2–11.6.) Perform the fourpart guided experiment of Section 11.10. Submit: (a) the Snell screen with the boundary; (b) the dividend threshold at which the call’s region appears; (c) the three attained values on the pasting panel; (d) the platform readouts and your flip-𝛿 sentence.
Exercise 11.15. (Course Website, Week 11, Notebook 11.) Extend the notebook: implement Longstaff–Schwartz least-squares Monte Carlo for the American 90-put; report the price against the lattice’s 1.826 with standard errors at 104 and 105 paths; and write three sentences on why regression-based stopping matters once the state has more dimensions than a lattice can hold.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 11; they are not posted here. Problem-set files are on the Course Website, Week 11.