Chapter 4 · Valuation, No-Arbitrage, and State Prices
Part I · Uncertainty, Information, and Value
Chapter at a glance
Modern Finance Laboratory · Module 4 · Week 4
Module 4 operationalizes the chapter on user-supplied markets. The extraction panel accepts a table of assets (price, payoff by state), reports the rank of 𝐷 and hence completeness, and displays the set Ψ: a point when complete, a segment or polytope— rendered on the state-price simplex—when not. The dictionary panel converts any selected 𝜓 into 𝑞 and 𝑚 and prices arbitrary payoffs three ways in parallel, watching (4.2) agree to the cent. The bounds panel solves the sub- and super-replication programs of Proposition 4.11 for a user payoff and animates the interval tightening as assets are added one at a time. The HJ panel plots (E[𝑚], 𝜎(𝑚)) f
Learning Outcome Statements
LOS 4.1 Formulate general one-period markets—payoff matrices, portfolios, marketed subspaces—and state and verify the law of one price.
LOS 4.2 Define arbitrage precisely and prove the first fundamental theorem of asset pricing by a separation argument, extracting state prices from traded prices.
LOS 4.3 Translate fluently among state prices, risk-neutral probabilities, and stochastic discount factors, and compute all three from market data.
LOS 4.4 Prove the second fundamental theorem and determine market completeness from the rank of the payoff matrix.
LOS 4.5 Compute sub- and super-replication bounds for non-marketed payoffs in incomplete markets, and assess valuation claims against them.
LOS 4.6 Derive the Hansen–Jagannathan bound and use it to discipline proposed models of the stochastic discount factor.
Laboratory · Module 4 (book §4.11)
Module 4: State Prices, Completeness, and Bounds
Course Website · Week 4
Guided experiment (supports LOS 4.2–4.6). (i) Enter the running example’s bond and equity; confirm the segment of Figure 4.2(a) and read off the stake’s bounds. (ii) Add the recession claim at $23.52; watch Ψ collapse to 𝜓 ∗ and the interval to 147.0; then reprice the claim at $25 and observe the extractor report an arbitrage, exhibiting the portfolio. (iii) Price the stake three ways in the dictionary panel and verify Example 4.10’s 𝑞 and 𝑚. (iv) In the HJ panel, compute the maximal traded Sharpe ratio and confirm it equals 𝜎(𝑚)/E[𝑚] in the completed market; then delete the equity and watch the admissible wedge widen—fewer assets, weaker discipline.
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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 4.1. A market has three states and two assets with payoffs (1, 1, 1) and (0, 1, 2). Which of the following payoffs are marketed: (2, 3, 4); (1, 0, 0); (3, 3, 3); (0, 0, 1)? For one non-marketed payoff, exhibit the two-dimensional plane 𝑀 explicitly and describe geometrically why the payoff lies off it. 4.12 Exercises 103
Exercise 4.2. A colleague announces: “Our risk-neutral model says the probability of recession is 29%, but the economics desk says 25%; one of them must be wrong.” Write the three-sentence reply, using the dictionary of Proposition 4.7 and Example 4.10’s numbers, that dissolves the disagreement.
Exercise 4.3. For each price system on states {𝜔1 , 𝜔2 }, either exhibit a strictly positive state-price vector or exhibit an arbitrage portfolio: (a) bond (1.05, 1.05) at 1, stock (120, 90) at 100; (b) bond (1.05, 1.05) at 1, stock (120, 90) at 115; (c) stock (120, 90) at 100, call with strike 110 (payoff (10, 0)) at 12.
Part B — Computations
Exercise 4.4. (The running example, end to end.) With the bond and equity of Section 4.1: (a) derive the state-price family and the positivity interval for 𝑡; (b) derive the stake’s value function 26.25𝑡 + 139.66 and the bounds (142.69, 152.56); (c) verify that adding the recession claim at 23.52 selects 𝑡 ∗ = 0.28 and reproduce 𝜓 ∗ , 𝑞, and 𝑚; (d) confirm by direct computation that the claim’s expected return is −4.3% and reconcile the sign with (4.3).
Exercise 4.5. A four-state market trades three assets. Their prices are (1, 100, 20); their payoff vectors, in the same order, are (1.02, 1.02, 1.02, 1.02), then (80, 100, 110, 130), and then (40, 20, 10, 0). Determine whether the market is complete; find all strictly positive state-price vectors; and give the no-arbitrage price interval for the payoff (0, 0, 0, 100).
Exercise 4.6. In the completed running example, price a call option on the listed equity with strike 105—payoff max(𝑆 − 105, 0) = (0, 5, 45)—three ways: by state prices, by discounted risk-neutral expectation, and by E[𝑚𝑋]. Then compute its expected return under P and explain, via (4.3), why it exceeds the equity’s.
Exercise 4.7. A market’s best available Sharpe ratio is 0.45 and the riskless rate is 3%. (a) What is the minimum volatility of any SDF pricing this market? (b) A strategist proposes 𝑚 = 𝑎 − 𝑏 𝑟 mkt with 𝜎(𝑟 mkt ) = 0.18; how large must 𝑏 𝜎(𝑟 mkt ) be, and what does that imply about 𝑏? (c) If the consumption-CRRA model is imposed with 𝜎𝑐 = 1.8%, what 𝛾 is required?
Part C — Proofs and extensions
Exercise 4.8. Prove that no arbitrage implies the law of one price, and give a two-state, two-asset example satisfying the LOP that admits an arbitrage—so the implication is strict. Conclude which of Proposition 4.3 and Theorem 4.6 does the heavier lifting, and why linear pricing alone cannot guarantee positive prices for positive payoffs. 104 4 Valuation, No-Arbitrage, and State Prices
Exercise 4.9. Prove the converse bijections of Proposition 4.7 in detail: starting from any probability vector 𝑞 ≫ 0 together with 𝑅, and separately from any strictly positive random variable 𝑚 with E[𝑚] = 1/𝑅, construct the corresponding state-price vector and verify (4.2). Where does each construction use strict positivity?
Exercise 4.10. On the state space of Example 4.8, exhibit explicitly a nonzero vector 𝜂 orthogonal to the marketed subspace, verify that 𝜓(𝑡) = 𝜓(0.28) + 𝜀𝜂 reparametrizes the family, and determine the exact interval of 𝜀 preserving strict positivity. Relate your 𝜂 to the direction in which Figure 4.2(b)’s value function increases.
Exercise 4.11. ∗ (Duality and attainment.) For the super-replication problem of Proposition 4.11: (a) show that its linear-programming dual is max{𝜙 · 𝑋 : 𝜙 ≥ 0, 𝐷𝜙 = 𝑝}, whose feasible set is the closure of Ψ; (b) invoke (or prove, using Lemma 4.5) strong duality for feasible finite LPs to conclude 𝑉 (𝑋) = max 𝜙∈Ψ 𝜙 · 𝑋; (c) show by example that the maximizer can sit on the boundary of Ψ—some state price zero—so the supremum over Ψ itself need not be attained.
Part D — Modeling and application
Exercise 4.12. Write the two-page memorandum resolving agenda item two. Structure it as Example 4.12 does—consistency check against the bounds, position within the interval, sensitivity to the completing instrument’s price, and an explicit list of the judgment calls that remain—and add a governance recommendation: what standing evidence should the private-markets team submit with each future valuation so that the committee can rerun this chapter’s analysis in ten minutes?
Exercise 4.13. Meridian’s draft capital-market assumptions project a global-equity Sharpe ratio of 0.55 for the coming decade. Write the one-page challenge memo based on Theorem 4.13: compute the implied lower bound on 𝜎(𝑚)/E[𝑚]; state what the consumption-CRRA benchmark would require; list two economic mechanisms (from the menu previewed at the end of Section 4.6) that could justify the number and one observable implication of each; and recommend a Sharpe assumption you would defend.
Part E — Laboratory
Exercise 4.14. (Laboratory Module 4; supports LOS 4.2–4.6.) Perform the four-part guided experiment of Section 4.11. Submit: (a) the extraction panel’s rendering of Ψ before and after the recession claim trades; (b) the arbitrage portfolio the extractor exhibits when the claim is repriced at $25, with a one-sentence explanation of the trade; (c) the three-way pricing screen for the stake; (d) the HJ panel’s maximal Sharpe ratio and its comparison with 𝜎(𝑚)/E[𝑚]. 4.13 Notes and Sources 105
Exercise 4.15. (Course Website, Week 4, Notebook 4.) Extend the notebook to a fivestate market: begin with a bond and one risky asset, then add three more assets one at a time (the notebook supplies candidates). After each addition, solve the two linear programs for a fixed target payoff and plot the value interval’s width against the number of assets. Report the rank of 𝐷 at each step, identify the addition that produces the largest tightening, and explain its payoff pattern in one sentence.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 4; they are not posted here. Problem-set files are on the Course Website, Week 4.