Chapter 5 · Martingales, Change of Measure, and Risk-Neutral Valuation
Part I · Uncertainty, Information, and Value
Chapter at a glance
Modern Finance Laboratory · Module 5 · Week 5
126 5 Martingales, Change of Measure, and Risk-Neutral Valuation Module 5 has three panels. The fair-game sandbox lets the reader design predictable betting rules in a visual editor—bet sizes as arbitrary functions of the visible history— and runs thousands of Monte Carlo bettors, displaying average gains with error bands; a prominently labeled peek toggle grants the rule one step of foresight and regenerates Figure 5.1 live. The measure panel shows the working tree with P- and Q-weights side by side, a slider tilting the one-step 𝑞, and the density process 𝑍𝑡 recomputed along every path; a claim-pricing readout evaluates EQ [𝐻]/𝑅𝑇 and E[𝑍 𝐻]
Learning Outcome Statements
LOS 5.1 Define martingales, supermartingales, and submartingales, and verify the property for the standard financial examples: forecast processes, likelihood ratios, and discounted prices. 109 110 5 Martingales, Change of Measure, and Risk-Neutral Valuation.
LOS 5.2 Prove and apply the martingale transform theorem: predictable trading in a fair game has zero expected profit.
LOS 5.3 State and prove the optional stopping theorem on a finite horizon, compute first-passage probabilities with it, and use it to rule out timing schemes.
LOS 5.4 Change probability measures with Radon–Nikodym derivatives and density processes, apply the abstract Bayes rule, and relate 𝑑Q/𝑑P to the stochastic discount factor.
LOS 5.5 Prove the multiperiod fundamental theorem on finite trees and price and hedge attainable claims by backward induction and risk-neutral expectation.
LOS 5.6 Audit a structured hedge by computing its risk-neutral value, its mispricing, and its replicating strategy.
Laboratory · Module 5 (book §5.11)
Module 5: Fair Games, Measure Change, and Dynamic Hedging
Course Website · Week 5
Guided experiment (supports LOS 5.2–5.6). (i) Build three predictable rules of increasing deviousness in the sandbox and confirm each averages to zero within error bands; engage the peek and record the slope it buys. (ii) Reproduce the collar audit end to end: put 1.95, call 4.81, collar −2.85, zero-cost strike 118.3. (iii) Hedge the dealer’s side manually along five simulated paths, then let the autopilot hedge; verify the autopilot’s terminal error is zero to machine precision and diagnose, node by node, where your manual hedge leaked. (iv) Set transaction costs to ten basis points and re-run the autopilot fifty times; describe the distribution of hedging shortfall and explain why the fair quote should move.
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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 5.1. Classify each process as a martingale, submartingale, supermartingale, or none, with a one-line justification: (a) the stock 𝑆𝑡 of Section 5.6 under P with 𝑝 = 0.60; (b) the discounted stock under Q; (c) 𝑀𝑡2 for the fair walk 𝑀; (d) the density process 𝑍𝑡 ; (e) the running maximum max𝑠≤𝑡 𝑀𝑠 .
Exercise 5.2. A fund letter reports: “Our monthly rebalancing rule outperformed buyand-hold by 2% in a fifty-year backtest; this demonstrates that disciplined rebalancing 5.12 Exercises 127 adds alpha.” Using Theorem 5.6, write the four-sentence referee report: what the result would have to mean about prices under P, what alternative explanations (risk premia loaded differently, look-ahead in the data, luck) must be excluded, and what single additional statistic you would request.
Exercise 5.3. The doubling strategy on a fair coin—double the stake after every loss, stop after the first win—ends ahead with probability one. Identify precisely which hypothesis of Theorem 5.6 it violates on an infinite horizon, and verify that on the finite horizon 𝑇 its expected gain is exactly zero by computing the two-outcome distribution of its terminal wealth.
Part B — Computations
Exercise 5.4. (The collar, end to end.) On the quarterly tree of Section 5.6: (a) verify 𝑞 = 0.5294 and the terminal Q-weights; (b) price the 90-put and 110-call by (5.7) and confirm the collar value −2.85; (c) reproduce the (𝑉, Δ) lattice of Figure 5.3 by backward induction; (d) verify the zero-cost strike 𝐾 ∗ = 118.3 and compute the collar’s value if implied volatility rises so that 𝑢 = 1.11, 𝑑 = 0.90—which way does the fair zero-cost strike move, and why?
Exercise 5.5. On the two-period tree of Figure 5.2, take the claim 𝐻 = (𝑆2 − 100) + with 𝑆0 = 100, 𝑢 = 1.09, 𝑑 = 0.92, 𝑅 = 1.01. Price it two ways—EQ [𝐻]/𝑅 2 directly, and E[𝑍 𝐻]/𝑅 2 using the path values of 𝑍 in the figure—and confirm agreement. Then verify E[𝑍1 |F0 ] = 1 and that 𝑍1 𝑆e1 has P-expectation 𝑆0 , illustrating Proposition 5.10.
Exercise 5.6. (a) Verify in one line that 𝑀𝑡2 − 𝑡 is a martingale for the fair coin walk. (b) A desk starts at 5 with boundaries 0 and 12: compute the probability of reaching the target and the expected time to a boundary. (c) Now tilt the coin to 𝑝 = 0.55 and show that ( 1−𝑝𝑝 ) 𝑀𝑡 is a martingale; use it to compute the ruin probability from 5 with the same boundaries.
Exercise 5.7. Using (5.7) on the quarterly tree, price (a) a digital option paying 10 if 𝑆4 ≥ 120; (b) an at-the-money straddle |𝑆4 − 100|; (c) verify put–call parity at strike 100 numerically, and state which theorem of this chapter parity instantiates.
Part C — Proofs and extensions
Exercise 5.8. Prove the equivalence claimed in Definition 5.11: a strategy is selffinancing if and only if its discounted value satisfies (5.3). (Divide the self-financing identity by 𝐵𝑡 and telescope.) Conclude that the account holdings 𝛽 are determined by (𝑉0 , 𝜃), so a self-financing strategy is exactly “an initial capital and a predictable exposure.” 128 5 Martingales, Change of Measure, and Risk-Neutral Valuation
Exercise 5.9. Let (𝑀𝑡 ) and (𝑁𝑡 ) be martingales for the same filtration. Prove or disprove with a counterexample: (a) 𝑀𝑡 + 𝑁𝑡 is a martingale; (b) 𝑀𝑡 𝑁𝑡 is a martingale; (c) max(𝑀𝑡 , 𝑁𝑡 ) is a submartingale. For (b), exhibit the correction term for the fair coin 2 |F ] − 𝑀 2 —and connect it to Proposition 5.5. walk—compute E[𝑀𝑡+1 𝑡 𝑡
Exercise 5.10. Prove the following converse to Example 5.3 on a finite horizon: every martingale (𝑀𝑡 )𝑡 ≤𝑇 is the Doob martingale of its terminal value, 𝑀𝑡 = E[𝑀𝑇 |F𝑡 ]. Conclude that on finite horizons “martingale” and “honest forecast of something” are the same concept, and explain in one sentence why this fails on infinite horizons (consider the doubling wealth).
Exercise 5.11. ∗ Prove the optional stopping inequality for submartingales (E[𝑀 𝜏 ] ≤ E[𝑀𝑇 ] for 𝜏 ≤ 𝑇) via the transform 𝜃 𝑡 = 1 { 𝜏<𝑡 } , and use it to prove Doob’s maximal inequality on a finite space: 𝜆 P max𝑡 ≤𝑇 𝑀𝑡 ≥ 𝜆 ≤ E[𝑀𝑇+ ] for a submartingale 𝑀. (Stop at the first crossing of 𝜆.) This inequality controls path maxima throughout Part II.
Part D — Modeling and application
Exercise 5.12. Write the two-page memorandum resolving agenda item one. Include: the audit table (put, call, collar values; dollar impact at $800M); the two acceptable executions (upfront payment, or restrike to 118.3); the expected-return cost of collaring under the committee’s P and its covariance-pricing interpretation; a sensitivity table of the collar value and 𝐾 ∗ to the volatility input; and a recommendation with the negotiation script’s first sentence written out.
Exercise 5.13. Write the one-page policy note on agenda item three. State what Theorem 5.6 guarantees rebalancing cannot do; list the three legitimate channels through which a rebalancing policy adds value (exposure control against drift, harvesting of genuinely non-martingale dynamics if evidenced, behavioral discipline) with the evidentiary standard each requires; and propose the martingale benchmark against which the CIO should measure any future “rebalancing alpha” claim.
Part E — Laboratory
Exercise 5.14. (Laboratory Module 5; supports LOS 5.2–5.6.) Perform the four-part guided experiment of Section 5.11. Submit: (a) your three predictable rules with their zero-consistent averages and the peek slope; (b) the collar audit screen; (c) the manualversus-autopilot hedging comparison with your leak diagnosis; (d) the transaction-cost shortfall histogram with a three-sentence pricing implication.
Exercise 5.15. (Course Website, Week 5, Notebook 5.) Extend the notebook: simulate 10,000 paths of the quarterly tree under P and confirm by Monte Carlo (with standard errors) that (a) the autopilot hedge of the collar reproduces the payoff with zero variance; (b) the collar’s average discounted payoff under P differs from its price, 5.13 Notes and Sources 129 while under importance weights 𝑍 it matches; (c) a momentum rule’s average gain on the discounted stock under Q-simulated paths is zero. Report all three as tests of, respectively, Theorem 5.13, Proposition 5.10, and Theorem 5.6.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 5; they are not posted here. Problem-set files are on the Course Website, Week 5.