Chapter 3 · Information, Conditional Expectation, and Filtrations
Part I · Uncertainty, Information, and Value
Chapter at a glance
Modern Finance Laboratory · Module 3 · Week 3
Module 3 makes information structures manipulable. The tree panel implements the three-period market of Figure 3.2: the user selects any random variable on the path space (terminal price, running maximum, a digital payoff) and any date, and the module displays the conditional expectation atom by atom, recomputing live as the user coarsens or refines the filtration. The strategy panel runs trading rules through a clairvoyance detector: submitted rules are checked, node by node, for F𝑡 −1 -measurability, and rejected rules are returned with the exact node and the exact future information they consumed. The two-observer panel runs the same marke
Learning Outcome Statements
LOS 3.1 Model information as partitions and sigma-algebras on a state space, and translate between “what is knowable” and the formal structure.
LOS 3.2 Compute conditional expectations with respect to partitions, and characterize conditional expectation by its defining property.
LOS 3.3 Prove and apply the tower property, linearity, taking-out-what-is-known, and the conditional Jensen inequality in finite settings.
LOS 3.4 Construct filtrations for multi-period financial models, and determine whether given processes and trading strategies are adapted.
LOS 3.5 Define stopping times, and distinguish exercise and trading rules that are implementable from rules that are not.
LOS 3.6 Explain how differential information—insider versus public, observed versus latent—is formalized by nested filtrations, and derive the variance-smoothing effect of coarse observation.
Laboratory · Module 3 (book §3.10)
Module 3: Information and Conditional Expectation Simulator
Course Website · Week 3
Guided experiment (supports LOS 3.2–3.6). (i) Reproduce every number of Example 3.9, then verify the tower property numerically by averaging date-2 forecasts to date-1 forecasts. (ii) Submit the rule “hold the stock only in periods the stock rises” to the clairvoyance detector; read the rejection, then repair the rule into its adapted cousin (“hold after a rise”) and compare the two backtests—the gap is look-ahead bias made visible. (iii) In the two-observer panel, grant the insider one period of foresight on earnings and record the difference in mean squared forecast error and in trading profit; relate both to Proposition 3.7. (iv) Drag the smoothing slider from monthly to annual observation and watch measured volatility fall while true volatility is unchanged; read off the two terms of (3.2) and reproduce Wednesday’s 18%-versus-6% pattern.
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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 3.1. For each observer below, describe the partition of the relevant state space they hold, and give one financially material question each cannot answer: (a) a rating agency that observes annual audited financials of a private borrower; (b) an index-futures 80 3 Information, Conditional Expectation, and Filtrations trader who observes the index level tick by tick but not its constituents; (c) Meridian’s board, which receives the private-infrastructure appraisal quarterly.
Exercise 3.2. Restate each form of the efficient-markets hypothesis—weak, semistrong, strong—as a statement of the form “E[ excess return𝑡+1 | G𝑡 ] = 0 for G𝑡 = . . .,” identifying each G𝑡 precisely. Which form does Friday’s compliance episode bear on, and why?
Exercise 3.3. Classify each rule as a stopping time or not, with a one-sentence justification grounded in Definition 3.10: (a) sell when the drawdown from the running peak first exceeds 12%; (b) sell at the price path’s maximum over the year; (c) exercise the option the first date it is in the money; (d) sell one day before the next Federal Reserve rate cut; (e) rebalance on the first date that realized year-to-date volatility exceeds 20%.
Part B — Computations
Exercise 3.4. On the tree of Figure 3.2 (𝑆0 = 100, 𝑢 = 1.2, 𝑑 = 0.9, 𝑝 = 0.6, independent periods): (a) compute E[𝑆3 |F𝑡 ] at every node for 𝑡 = 0, 1, 2 and verify the pattern E[𝑆3 |F𝑡 ] = 1.08 3−𝑡 𝑆𝑡 ; (b) verify the tower property numerically along the down branch; (c) let 𝑀3 = max𝑡 ≤3 𝑆𝑡 and compute E[𝑀3 |F1 ] on both atoms, explaining why no formula as simple as (a) exists.
Exercise 3.5. A payoff 𝑋 on the four-state macro space takes values (12, 4, −2, −8) on (expansion, low), (expansion, high), (recession, low), (recession, high), with probabilities (0.4, 0.2, 0.15, 0.25). Compute E[𝑋 |G] for (a) G generated by the business-cycle partition; (b) G generated by the inflation partition; and verify E[E[𝑋 |G]] = E[𝑋] in both cases.
Exercise 3.6. For the payoff of the previous exercise with the business-cycle sigmaalgebra: compute both terms of the total-variance decomposition (3.2), report the fraction of total variance visible to the cycle-only observer, and interpret the fraction as a smoothing ratio in the sense of Section 3.5.2.
Exercise 3.7. On the two-period tree (𝑢 = 1.2, 𝑑 = 0.9, 𝑝 = 0.6, gross riskless return 1 for simplicity), compare two rules for holding 𝜃 ∈ {0, 1} shares each period: (adapted) 𝜃 𝑡 = 1 if the previous period’s move was up, else 0; (anticipating) 𝜃 𝑡 = 1 if the current period’s move will be up, else 0. Compute each rule’s expected total trading gain and identify exactly where the anticipating rule’s “profit” violates F𝑡 −1 -measurability.
Part C — Proofs and extensions
Exercise 3.8. Prove the two rules of Proposition 3.6 deferred in the text: (i) linearity, and (iv) that if 𝑋 is independent of G (meaning P({𝑋 = 𝑥} ∩ 𝐴) = P(𝑋 = 𝑥)P( 𝐴) for 3.11 Exercises 81 all 𝑥 and 𝐴 ∈ G), then E[𝑋 |G] = E[𝑋]. Use the characterizing property (3.1), not atom averages.
Exercise 3.9. Let H ⊆ G be sigma-algebras on a finite probability space. Prove that the tower property fails in general when the nesting is dropped: construct sigma-algebras G, H on a four-state containing the other, and a random variable 𝑋 with space, neither E E[𝑋 |G] H ≠ E E[𝑋 |H ] G . Conclude in one sentence why “iterated estimates” are order-dependent between observers who hold different information.
Exercise 3.10. Prove that for a stopping time 𝜏 the family F𝜏 := { 𝐴 : 𝐴 ∩ {𝜏 = 𝑡} ∈ F𝑡 for all 𝑡} is a sigma-algebra (the “information at the stopping time”), that 𝜏 is F𝜏 measurable, and that for an adapted process 𝑋 the stopped value 𝑋 𝜏 is F𝜏 -measurable. (This object returns in Chapters 5 and 11.)
Exercise 3.11. ∗ Prove the converse half of Definition 3.3 in sigma-algebra language: on a finite space, 𝑋 is G-measurable in the general sense ({𝑋 ≤ 𝑥} ∈ G for all 𝑥) if and only if 𝑋 is constant on the atoms of G. Then show by example that on Ω = [0, 1] with the Borel sigma-algebra, “constant on atoms” is vacuous while measurability is not—the precise reason Section 3.6∗ defines conditional expectation through (3.1).
Part D — Modeling and application
Exercise 3.12. Write the two-page audit memorandum Meridian’s CIO should commission on Monday’s backtest. Your memo must: define look-ahead bias via adaptedness in one committee-readable paragraph; list at least five concrete filtration checks (publication lags of each data source; index-membership as of the trade date; restatement and revision history; timestamp conventions of the database; execution and price-availability assumptions); specify the point-in-time protocol that repairs each; and state which single check you would run first and why.
Exercise 3.13. Write the one-page reply to Wednesday’s trustee. Your reply must: state Proposition 3.12 in words; explain why the 6% appraisal volatility is a property of the observation filtration rather than of the asset; propose two remedies (an unsmoothing adjustment, and model-based filtering per Chapter 12) with one limitation each; and recommend what number, or pair of numbers, the risk report should carry instead, citing the specialist treatment in [1] for the committee’s further reading.
Part E — Laboratory
Exercise 3.14. (Laboratory Module 3; supports LOS 3.2–3.6.) Perform the four-part guided experiment of Section 3.10. Submit: (a) the completed forecast tree of experiment (i) with the tower verification; (b) the clairvoyance detector’s rejection report and the adapted-versus-anticipating backtest gap from experiment (ii); (c) the insider’s forecast-error and profit advantages from experiment (iii); (d) the measured-volatility 82 3 Information, Conditional Expectation, and Filtrations curve against observation frequency from experiment (iv), annotated with the two terms of (3.2).
Exercise 3.15. (Course Website, Week 3, Notebook 3.) Extend the notebook’s lookahead study: simulate 10,000 paths of a market with i.i.d. returns (zero true predictability), run (i) an adapted momentum rule and (ii) the same rule granted a one-period peek, and report both Sharpe ratios with Monte Carlo standard errors. Then repeat with the peek replaced by a 45-day-equivalent publication lag error on a simulated fundamental signal, and write three sentences translating the result into the language of
Exercise 3.12.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 3; they are not posted here. Problem-set files are on the Course Website, Week 3.