Chapter 1 · The Mathematical Architecture of Modern Finance
Part I · Uncertainty, Information, and Value
Chapter at a glance
Modern Finance Laboratory · Module 1 · Week 1
Module 1 implements the miniature market of Section 1.3 with every primitive live: the user sets (𝑆0 , 𝑢, 𝑑, 𝑅, 𝑝) and, optionally, the payoffs of a user-defined claim, and the module displays the replicating portfolio, state prices, risk-neutral probabilities, the SDF, and the claim’s price, recomputed continuously—with an arbitrage alarm that detects violations of 𝑑 < 𝑅 < 𝑢 and exhibits an explicit arbitrage portfolio when one exists.
Learning Outcome Statements
LOS 1.1 Describe modern finance as an integrated system linking uncertainty, information, valuation, dynamics, optimization, risk, and equilibrium, and identify these elements in a practical investment decision.
LOS 1.2 Explain the role and limits of mathematical modeling in finance, distinguishing between assumptions, mechanisms, and conclusions.
LOS 1.3 Construct a complete one-period, two-state market model and compute state prices, replicating portfolios, and risk-neutral probabilities within it.
LOS 1.4 Demonstrate, in the two-state model, why arbitrage-free valuation, replication, and risk-neutral expectation deliver identical prices, and why optimal portfolio demand is governed by the gap between physical and risk-neutral probabilities.
LOS 1.5 Map the fourteen chapters of this book onto the standard decision problems of an institutional investor.
Laboratory · Module 1 (book §1.10)
Module 1: Probability and State Space Explorer
Course Website · Week 1
Guided experiment (supports LOS 1.3–1.4). (i) Reproduce the running example and verify every number in Sections 1.3.2–1.3.5. (ii) Drag 𝑅 upward toward 𝑢 = 1.2 and watch 𝜓u → 0: dollars in the up state become worthless as claims as the bond comes to dominate the stock; at 𝑅 ≥ 𝑢 the alarm fires—read the arbitrage portfolio it reports and reconcile it with the proof of Theorem 1.6. (iii) Vary 𝑝 across (0, 1) with prices fixed and confirm that no pricing output moves while the optimal allocation 𝛼∗ of Section 1.4.1 swings from short to leveraged, crossing zero exactly at 𝑝 = 𝑞. (iv) Add a third state with no third asset and watch the unique price of a non-replicable claim dissolve into a bound—the door into Chapter 4.
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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 1.1. For each of the following, identify which of the book’s six primitives (uncertainty, information, value, time, decision, aggregation) are centrally involved, and name the chapter(s) of this book that address it: (a) a pension fund must post additional collateral when its derivatives lose value, forcing asset sales in a falling market; (b) a biotechnology firm staggers a drug program into phases, abandoning after bad trial results; (c) an index provider announces that a stock will join a major index, and its price jumps before any inclusion trading occurs; (d) a hedge fund’s backtest performs superbly but the live strategy does not.
Exercise 1.2. A colleague asserts: “The Black–Scholes formula was disproved by the 1987 crash.” Using the assumptions–mechanism–conclusions anatomy of Section 1.2, rewrite the assertion into a defensible statement. Which component of the model failed, and in what sense was the mathematics itself untouched?
Exercise 1.3. Explain, in at most five sentences and without formulas, why the price of the call option in Section 1.3.2 does not depend on the probability 𝑝, and why the Meridian committee’s decision problems (Sections 1.4.1–1.4.2) nevertheless do depend on 𝑝. Your answer should use the phrase “state prices” at least once.
Part B — Computations
Exercise 1.4. In the market of Example 1.3, price a put option with strike 𝐾 = 105 (payoff max(𝐾 − 𝑆1 , 0)) three ways: by replication, by state prices, and by risk-neutral expectation. Report the replicating portfolio and verify that all three prices agree.
Exercise 1.5. Recompute the state prices (1.7) for 𝑢 = 1.3, 𝑑 = 0.95, 𝑅 = 1.05, 𝑆0 = 100. Verify that 𝜓u ≠ 𝜓d , that 𝜓u + 𝜓d = 1/𝑅, and that the risk-neutral probability is 𝑞 = 2/7. Price the 𝐾 = 105 call in this market.
Exercise 1.6. In the market of Example 1.3: (a) compute the SDF values 𝑚(𝜔u ), 𝑚(𝜔d ) for physical probabilities 𝑝 = 0.5, 𝑝 = 0.6, and 𝑝 = 0.8; (b) verify E[𝑚] = 1/𝑅 in each 30 1 The Mathematical Architecture of Modern Finance case; (c) for 𝑝 = 0.6, verify the covariance decomposition (1.12) numerically for the stock and for the call option of Section 1.3.2.
Exercise 1.7. A forward contract on the stock, struck at 𝐹, pays 𝑆1 − 𝐹 at 𝑡 = 1 (in both states; no optionality). (a) Using state prices, find the strike 𝐹 ∗ making the contract worth zero today, and show that 𝐹 ∗ = 𝑆0 𝑅 regardless of (𝑢, 𝑑, 𝑝). (b) Replicate the forward and interpret the portfolio. (c) A dealer quotes 𝐹 = 106 in the market of Example 1.3; exhibit the arbitrage.
Part C — Proofs and extensions
Exercise 1.8. Prove that in the two-state market with 𝑑 < 𝑅 < 𝑢, the state prices satisfying (1.6) are unique, and prove that the replicating portfolio of any claim is unique. (Both are statements about a 2 × 2 linear system; identify precisely which hypothesis rules out degeneracy.)
Exercise 1.9. (The 𝑝 = 𝑞 theorem.) Consider the allocation problem (1.13) with a general strictly increasing, strictly concave, differentiable utility 𝑢(·) in place of the logarithm. Prove that 𝛼∗ = 0 if and only if 𝑝 = 𝑞, where 𝑞 is the risk-neutral probability (1.8). Interpret: risk-taking is compensated exactly by the divergence between physical and risk-neutral measures.
Exercise 1.10. (Put–call parity.) In the general two-state market, let 𝐶0 and 𝑃0 be the arbitrage-free prices of a call and a put with common strike 𝐾. Prove directly from state pricing that 𝐶0 − 𝑃0 = 𝑆0 − 𝐾/𝑅, and explain why the argument will survive, verbatim, in every arbitrage-free model of this book, including those of Chapter 8.
Exercise 1.11. ∗ (Three states.) Let Ω = {𝜔1 , 𝜔2 , 𝜔3 } with the same two assets (bond returning 𝑅; stock returning 𝑢 > 𝑚 > 𝑑 in the three states). (a) Show that a claim is replicable if and only if its payoff vector lies in a two-dimensional subspace of R3 , and exhibit a non-replicable claim. (b) Show that state prices consistent with the two asset prices form a one-parameter family, and characterize the set of arbitrage-free prices of your claim from (a) as an interval. (c) Explain in one paragraph why this interval is the correct formalization of the valuation dispute in Meridian’s second agenda item. (This exercise is the trailhead of Chapter 4.)
Part D — Modeling and application
Exercise 1.12. Write the one-page memorandum that Meridian’s chief investment officer should circulate before the committee votes on agenda item four (commit now versus wait), using the analysis of Section 1.4.2. Your memo must: state the assumptions under which the $8.48 waiting premium is valid (spanning, exclusivity, unchanged cost); identify which assumption is most fragile in the actual renewable-energy setting; and recommend a decision rule rather than a decision. 1.11 Exercises 31
Exercise 1.13. Construct a two-state model of a decision from your own professional or personal experience (a job offer with an uncertain bonus, a property purchase, a currency exposure). Specify the states, the traded instruments available to you, and the claim being valued; compute the state prices if the market you face is complete, or the pricing bounds if it is not; and state honestly which feature of your situation the two-state model most seriously misrepresents.
Part E — Laboratory
Exercise 1.14. (Laboratory Module 1; supports LOS 1.3–1.4.) Perform the four-part guided experiment of Section 1.10. Submit: (a) a screenshot or table of the reproduced running example; (b) the arbitrage portfolio reported by the alarm at 𝑅 = 1.25, together with a verification by hand that it satisfies Definition 1.4; (c) a plot of 𝛼∗ against 𝑝 ∈ (0.05, 0.95) with the crossing at 𝑝 = 𝑞 marked; (d) for the three-state extension, the price bounds of the claim (10, 0, 0) and one sentence on why they are bounds.
Exercise 1.15. (Course Website, Week 1, Notebook 1.) The notebook implements the module’s computations in Python. Extend it to price a claim with payoffs (𝑋u , 𝑋d ) = (25, −10) (note the negative payoff), verify the price against a hand computation by state prices, and explain why negative payoffs require no modification of any formula in this chapter.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 1; they are not posted here. Problem-set files are on the Course Website, Week 1.