Chapter 3 · No-Arbitrage, State Prices, and Incomplete Markets

Chapter at a glance

The chapter turns three theorems into manipulable objects: the fundamental theorem of asset pricing, the state-price/SDF/risk-neutral trinity, and superhedging duality. It makes incompleteness visible — the pricing set as a geometric object, the valuation interval as its shadow, and completeness as the razor’s-edge special case it actually is.

Learning Outcome Statements

LOS 3.1 — Define arbitrage in one-period, multi-period, and continuous-time markets, and state the corresponding fundamental theorems with their exact hypotheses.

LOS 3.2 — Prove the one-period fundamental theorem via the separating hyperplane argument and extract state prices from the separation.

LOS 3.3 — Translate freely among state prices, stochastic discount factors, and riskneutral measures, and normalize prices correctly under each representation.

LOS 3.4 — Characterize the set of equivalent martingale measures of an incomplete market, compute it explicitly in finite-state examples, and relate its dimension to unspanned risk.

LOS 3.5 — Decompose a private-market claim into spanned and residual components and show that valuation ambiguity is carried entirely by the residual.

LOS 3.6 — State superhedging duality in discrete and continuous time and explain why superhedging bounds, while correct, are too wide to serve as prices for private-market.

LOS 3.7 — Compute no-arbitrage price bounds by linear programming and interpret the width of the valuation interval as a measure of market incompleteness.

Laboratory (book §3.9)

Module: State Prices and Pricing Bounds — open in the Laboratory

Build a finite-state traded market; watch the risk-neutral set appear, shrink, and vanish as assets are edited; compute no-arbitrage bounds with hedging portfolios by linear programming. E1 completeness as a knife edge; E2 manufacture an arbitrage and read the Farkas certificate; E3 the spanning curve (interval width vs spanned fraction); E4 P versus Q (reconcile two desks as an SDF disagreement).

Downloads: Python notebook · Excel workbook · Slides

Exercises

Solutions are distributed to instructors with the Instructor’s Solutions Manual; they are not posted here.

Conceptual Problems

3.1 The two desks of Section 3.1 are asked by a regulator to converge to a single number. List the assumptions each desk must add to no-arbitrage to defend its mark, classify each assumption as a restriction on 𝐷, on 𝑀, or on the valuation operator (Remark 3.3), and identify which differences are empirically adjudicable in principle.

3.2 Explain why the non-traded claim’s existence does not enlarge the arbitrage opportunities of the traded market, and why, nevertheless, adding the same claim as a traded asset at Desk B’s price could create an arbitrage against Desk A’s clients. What institutional feature of private markets prevents this?

3.3 A pitch deck asserts: “our fund’s returns are market-neutral, so the risk-neutral and physical expected returns of our fund coincide.” Diagnose the error using Definition 3.3 and the ratio 𝑞 𝑗 /𝑝 𝑗 , and construct a two-state counterexample.

3.4 For each market, state whether completeness is plausible and name the unspanned factors if not: (a) a binomial-tree stock market with a bond; (b) listed equities plus index options across strikes and maturities; (c) the market “visible” to the holder of a bilateral middle-market loan; (d) a farmland portfolio hedged with grain futures.

Mathematical Problems

3.5 Complete the measurable-selection-free part of Theorem 3.4.1: in a two-period model with finite Ω and a filtration generated by a partition, run the one-period separation of Theorem 3.2 conditionally on each time-1 atom and glue the resulting one-step densities into an EMM via the tower property, verifying equivalence and the martingale property explicitly.

3.6 Prove Theorem 3.5 in the finite one-period case: no arbitrage plus completeness iff (3.1) has exactly one solution, which is strictly positive; exhibit, for 𝐽 = 3 and a market consisting of a bond only, the full two-dimensional pricing set.

3.7 (Boundary marks.) In the trinomial market of Figure 3.2, show that 𝜋(𝑋) is approached only as 𝑞 𝑚 → 0+ along . Interpret a mark at 0.083 as an assertion about the middle state’s price, and exhibit the non-equivalent limiting “measure” explicitly.

3.8 Prove that adding a column to 𝐴 (a new traded asset, priced without creating arbitrage) weakly shrinks [𝜋(𝑋), 𝜋(𝑋)] for every claim 𝑋, with strict shrinkage at the upper end iff the maximizer set of the dual LP contains no 𝜓 pricing the new asset correctly. Conclude the monotone convergence of bounds along any nested sequence of market expansions.

Computational Problems

3.9 Implement Algorithm 3.8 (any LP library). Reproduce the interval of Figure 3.2 exactly; then verify Exercise 3.8 numerically by adding an Arrow security on the middle state and reporting the collapsed interval.

3.10 Build a 𝐽 = 8 market from a discretized two-factor model (market factor, rates factor) with a bond, an index, and a rates instrument traded. Import (or simulate, per Chapter 2) a private-credit claim with a default jump loading 60% on an idiosyncratic state variable. Compute the interval, the spanning fraction, and the width attribution of Proposition 3.6; then trace experiment E3’s width-vs-spanning curve.

3.11 (Width scaling.) In a one-period Gaussian-factor model discretized on 𝐽 states, vary the unspanned volatility 𝜎⊥ of a claim and fit the scaling of interval width against 𝜎⊥ ; compare with the heuristic of Section 3.8 and report where discretization distorts the relationship. Research Extensions

3.12 Short-sale and position-limit constraints replace the subspace by a convex cone, and pricing systems by superlinear functionals. Formulate the constrained analogue of Theorem 3.2 and of the duality (3.4), and design a study of how borrowing constraints faced by secondary buyers widen the practical valuation interval for LP interests, connecting to the discount evidence discussed in Chapter 5.

3.13 The spanning fraction ∥ 𝑋 ∥ ∥/∥ 𝑋 ∥ of private-equity cash flows by public factors is an empirical quantity with a contested literature (see the replication and listedproxy debates, e.g. 1, 15). Propose an estimation design using fund-level cash flows, stating explicitly how appraisal smoothing (Chapters 1 and 6) biases naive estimates of the spanned fraction and how your design corrects for it.