Chapter 7 · Nonlinear Valuation: Indifference Prices, Good-Deal Bounds, and Convex Risk Measures

Chapter at a glance

When a claim is unspanned, linear pricing fails and holder-relevant valuation must become nonlinear. The chapter develops indifference pricing under exponential utility, good-deal bounds that cap the attainable Sharpe ratio, convex risk measures with their dual representations, and the dynamic BSDE formulation. Its organizing image is the operator ladder: Chapter 3’s no-arbitrage ambiguity, narrowed by declared assumptions into ever-tighter nested intervals.

Learning Outcome Statements

LOS 7.1 — Explain why holder-relevant valuation of unspanned claims must violate additivity, and state the axioms a defensible nonlinear valuation operator should retain.

LOS 7.2 — Compute certainty equivalents under exponential utility and derive the exponential–Gaussian indifference price in closed form, isolating the residual-risk.

LOS 7.3 — Prove the structural properties of indifference prices: buyer–seller ordering, containment in the no-arbitrage interval, volume monotonicity, and span consistency.

LOS 7.4 — Construct good-deal bounds as coherent, measure-set-based operators and position them between no-arbitrage bounds and preference-based prices.

LOS 7.5 — State the axioms of convex and coherent risk measures, prove the robust representation on finite state spaces, and recognize the entropic measure as the riskmeasure form of exponential indifference.

LOS 7.6 — Formulate dynamically consistent nonlinear valuation through BSDEs and 𝑔-expectations, and identify the drivers corresponding to linear, ambiguity-penalized,.

LOS 7.7 — Size an unspanned position by equating marginal indifference value to price, and defend the resulting quantity-dependent bid to an investment committee.

Laboratory (book §7.9)

Module: Indifference Pricing Engine — open in the Laboratory

Price unspanned claims with the full nonlinear toolkit side by side. The organizing display is the operator ladder: for one claim, the no-arbitrage interval, the good-deal interval inside it, the [p_b, p_s] indifference pair inside that, and the Davis point they straddle. E1 hedgeable-fraction sweep (watch the indifference band collapse onto replication); the buyer–seller gap is a variance bill γσ_e²q²/R; only the residual variance is charged.

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

7.1 Write the one-paragraph memo reconciling the CIO and the risk head, using only the concepts marginal price, demand curve, and residual variance—no formulas—and state what single number each officer was implicitly quoting.

7.2 Audit these practice rules against the chapter’s three axioms (monotonicity, span consistency, cash translation): “NAV minus 20%”; “discount rate plus 300bp for 118 7 Nonlinear Valuation: Indifference Prices, Good-Deal Bounds, and Convex Risk Measures illiquidity”; “lower of cost and NAV”; “expected value under our house measure.” Identify each failure and its consequence with a two-state example.

7.3 Explain why the buyer–seller gap of Proposition 7.4 is a preference-side bid–ask, distinct from the search-side discount of Chapter 5, and describe a transaction in which both operate simultaneously with opposite comparative statics in market depth.

7.4 The good-deal ceiling ℎ and the risk aversion 𝛾 are both “declared worldviews.” Contrast what evidence could falsify each declaration, and explain why a mark defensible under both disciplines (Section 7.5) is more audit-robust than one defensible under either alone.

Mathematical Problems

7.5 Complete the proof of Proposition 7.4: (a) the buyer’s lower bound in (ii) via subreplication; (b) volume monotonicity (iii) for general concave utility via the certainty-equivalent gain’s concavity, including the implicit-function step.

7.6 Extend Proposition 7.4 to a residual correlated with one traded factor (𝑒 = 𝛽𝑒 𝑅 m + 𝑒): ˜ show the hedge absorbs 𝛽𝑒 and the price formula holds with 𝑒’s ˜ moments—“only the orthogonal residual is charged.”

7.7 Prove that the entropic risk measure is convex and cash invariant but not positively homogeneous, and verify its robust representation on a two-point Ω by direct maximization over with the entropy penalty.

7.8 Prove the coherent-case necessity argument of Theorem 7.6 in full detail on Ω = {𝜔1 , 𝜔2 , 𝜔3 }, exhibiting the representing set for 0.5 explicitly.

7.9 For the one-period binomial analogue of the driver 𝑔 = ℎ|𝑧| (define the discrete 𝑔-expectation by backward recursion), prove the comparison theorem and show the induced static operator at time 0 coincides with the good-deal bound over the corresponding one-step Girsanov interval.

Computational Problems

7.10 Exercises 119

7.11 Solve the quadratic-driver BSDE for a two-factor simulated claim by backward Euler with regression; validate against the closed-form exponential indifference price on the Gaussian sub-case, reporting the grid-refinement convergence rate.

7.12 (Calibrating 𝛾.) Take three stylized accepted deals (size, residual variance, price paid); back out the implied 𝛾 from each via Proposition 7.4; test internal consistency and compute the demand curve the median 𝛾 implies for the co-investment.

7.13 In secondary markets, observed block discounts reflect the marginal buyer’s indifference price. Combine this chapter’s exponential–Gaussian pricing with Chapter 5’s search model into an equilibrium in which heterogeneous-𝛾 buyers arrive at rate 𝜈: characterize the stationary distribution of accepted discounts, and design an estimation strategy that separates preference dispersion from search frictions using data on time-on-market and realized discounts jointly.