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Econ-258 · economics · Economics

Stable valuation bounds for incompletely hedgeable claims

LLM status Open unverified

Statement

FI-04 |  Robust asset valuation

Research agenda formalization

Definitions and assumptions

Use the finite-horizon market of FI-03 and let $\mathcal Q_{\rm adm}(D)\ne\varnothing$ be a specified subset of calibrated martingale measures, incorporating declared restrictions on beliefs, preferences, robustness or equilibrium. Let a terminal claim $H$ satisfy $H/B_T\in L^\infty$. For admissible linear valuation rules \[ V_Q(H)=B_0\E_Q[H/B_T],\qquad Q\in\mathcal Q_{\rm adm}(D), \] define the sharp envelopes \[ \underline V_D(H)=\inf_{Q\in\mathcal Q_{\rm adm}(D)}V_Q(H),\qquad \overline V_D(H)=\sup_{Q\in\mathcal Q_{\rm adm}(D)}V_Q(H). \] Sharp means that no smaller interval contains every admissible valuation; endpoint attainment is a separate issue. In the special normalization $B_0=B_T=1$ these reduce to expectations of $H$.

Formal question

Find conditions ensuring informative widths $\overline V_D(H)-\underline V_D(H)$, dual characterizations by feasible hedging or penalized hedging problems, and stability under perturbations of prices and model restrictions. Characterize when additional observed claims or equilibrium restrictions strictly tighten the interval and when the endpoints are attained. Construct methods that remain valid when calibration data are estimated with uncertainty, with guarantees in a declared metric on discounted payoff laws.

Scope and status

The envelope inequalities hold by definition for the stated valuation class; proving them is not the open task. The research issue is their economic content, duality, stability and empirical recovery in a specified incomplete-market family. Discounting is necessary unless the numeraire is normalized as above. Arbitrary subjective laws need not be martingale measures, and arbitrary nonlinear utility-indifference prices are not automatically covered by this linear valuation class. The source motivates the combined-restriction agenda without stating this exact optimization problem as a conjecture.

References for the source of the problem

Zhang Chen and Chen Kay, Historical Developments in Probability Measures for Asset Pricing: From State Prices to Modern Pricing Kernels, arXiv:2605.27658 (2026), Sections 18.6--18.7 for incomplete-market and robust valuation, and Section 20, pp. 23--24, for partially hedgeable risks and combined restrictions. https://arxiv.org/pdf/2605.27658.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-258-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-258

Stable incomplete-market bounds from interior calibration

Problem ID: G13-254. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and finite-state market

The problem asks for stable valuation intervals for claims that cannot be fully hedged, using economically justified restrictions on pricing measures. The source review [CK] distinguishes physical probabilities from pricing measures. Here we prove a finite-state stability theorem, clarify the role of equivalent measures, and identify exactly when the interval collapses. This does not cover the full continuous-time path-space problem.

Work in discounted units in a one-period market with $N$ states, each of positive physical probability. Cash pays one. The vector of payoffs of $d$ other traded assets in state $s$ is $a_s\in{\mathbb{R}}^d$, and their observed price vector is $p$. Remove redundant affine coordinates so that $D=\operatorname{conv}\{a_1,\ldots,a_N\}$ has nonempty interior in its ambient space. Let \[ Q(p)=\{q\in{\mathbb{R}}^N:q_s\ge0,\ \sum_s q_s=1,\ \sum_s q_sa_s=p\}, \qquad Q^+(p)=\{q\in Q(p):q_s>0\ \forall s\}. \] For a claim $h=(h_s)$ with $|h_s|\le M$, define $U_h(p)=\max_{q\in Q(p)}\sum_s q_sh_s$ and $L_h(p)=\min_{q\in Q(p)}\sum_s q_sh_s$.

Sharp bounds, equivalence, and replication

Theorem (Finite-state valuation interval): For $p\in\operatorname{int}D$, the following hold.
  1. $Q^+(p)$ is nonempty and dense in $Q(p)$. Hence the supremum and infimum over equivalent pricing measures are $U_h(p)$ and $L_h(p)$, although those endpoints need not be attained by equivalent measures.
  2. The upper endpoint has the attained superhedging representation \[ U_h(p)=\min_{a\in{\mathbb{R}},\lambda\in{\mathbb{R}}^d} \left\{a+\lambda\cdot p:a+\lambda\cdot a_s\ge h_s\ \forall s\right\}. \] The lower endpoint is the analogous maximum over subhedges.
  3. $U_h(p)=L_h(p)$ if and only if there are $a,\lambda$ such that $h_s=a+\lambda\cdot a_s$ for every state.
Proof: Let $\bar a=N^{-1}\sum_s a_s$. For sufficiently small $\epsilon>0$, the point $p'=(p-\epsilon\bar a)/(1-\epsilon)$ lies in $D$, since it tends to the interior point $p$. Represent $p'$ by a convex combination with weights $q'$. Then $q^0=(1-\epsilon)q'+\epsilon(1/N,\ldots,1/N)$ belongs to $Q^+(p)$. For any $q\in Q(p)$, $(1-t)q+tq^0$ is strictly positive for $t>0$ and tends to $q$. Compactness of $Q(p)$ and continuity of $q\mapsto q\cdot h$ prove part 1.

Every superhedge gives $q\cdot h\le a+\lambda\cdot p$, so the displayed minimum is at least $U_h(p)$. For the converse, $U_h$ is concave: mixtures of feasible probability vectors at two prices are feasible at their mixture price and have the corresponding mixture value. Its hypograph is a closed convex polyhedron, explicitly the downward closure of the convex hull of $(a_s,h_s)$. It has a supporting hyperplane at $(p,U_h(p))$. This follows directly by separating points $(p,U_h(p)+\epsilon)$ from that closed convex set and taking a normalized convergent subsequence of separating normals. The vertical coefficient of a supporting normal is nonnegative by downward closure and cannot be zero: a nonzero horizontal normal would support $D$ at the interior point $p$. Normalize the vertical coefficient to one. We obtain a vector $\lambda$ with \[ U_h(x)\le U_h(p)+\lambda\cdot(x-p),\qquad x\in D. \] Since $U_h(a_s)\ge h_s$, the intercept $a=U_h(p)-\lambda\cdot p$ gives the required superhedge, with equality in its price. Applying this argument to $-h$ gives the lower representation.

For part 3, a replicating affine payoff has the same expectation under all feasible measures. Conversely, suppose all feasible expectations coincide. If $v$ satisfies $\sum_s v_s=0$ and $\sum_s v_sa_s=0$, strict positivity of $q^0$ makes $q^0\pm tv$ feasible for small enough $t>0$. Equality of their expectations implies $h\cdot v=0$. In finite-dimensional Euclidean space, the orthogonal complement of the kernel of the matrix with columns $(1,a_s)$ is its row space: this follows by decomposing a vector orthogonally into that row space and its orthogonal complement. Thus $h$ is an affine combination of the traded payoffs. ∎

A two-state cash-only market with $h=(0,1)$ already shows nonattainment: equivalent measures have claim values strictly between zero and one, while the sharp closed interval is $[0,1]$. Writing "minimum over equivalent measures" without additional compactness would be incorrect.

A quantitative interior stability theorem

Theorem (Calibration stability): Suppose the closed Euclidean balls of radius $r>0$ about $p,p'\in D$ are contained in $D$. Then \[ |U_h(p)-U_h(p')|\le\frac{2M}{r}\|p-p'\|, \qquad |L_h(p)-L_h(p')|\le\frac{2M}{r}\|p-p'\|. \] Consequently the width $W_h=U_h-L_h$ changes by at most $4M\|p-p'\|/r$.
Proof: Take an optimal superhedge at $p$, with slope $\lambda$, as constructed above. Its supporting inequality at $x=p-r\lambda/\|\lambda\|$, if $\lambda\ne0$, gives $-M\le U_h(x)\le U_h(p)-r\|\lambda\|\le M-r\|\lambda\|$. Hence $\|\lambda\|\le2M/r$. Evaluate the same affine majorant at $p'$ to obtain $U_h(p')-U_h(p)\le2M\|p-p'\|/r$. Reverse the roles of the two points for the opposite inequality. Apply the argument to $-h$ and add to obtain the remaining claims. ∎

Suppose an estimator $\widehat p$ has a confidence region $C$ containing the true calibration $p_*$ with probability at least $1-\alpha$. The robust bounds \[ \underline v_C=\inf_{p\in C\cap D}L_h(p),\qquad \overline v_C=\sup_{p\in C\cap D}U_h(p) \] then contain every price consistent with the true calibration on that event. If $C$ is a radius-$\delta$ ball about $\widehat p$, and every point of $C\cap D$ and $p_*$ has an interior ball of radius $r$, the preceding theorem gives \[ 0\le (\overline v_C-\underline v_C)-W_h(p_*)\le\frac{8M\delta}{r} \] on the coverage event. Indeed, each such price lies at distance at most $2\delta$ from $p_*$, giving $4M\delta/r$ on each endpoint. This separates statistical calibration error from the irreducible incomplete-market width. It is a conditional stability guarantee: a confidence set reaching the boundary need not have the required common $r$.

Proposition (When an additional quote strictly helps): Let a further payoff $g$ have observed price $c$, and assume the augmented feasible set $Q' =\{q\in Q(p):q\cdot g=c\}$ is nonempty. The upper bound strictly decreases precisely when no maximizer of $q\cdot h$ over $Q(p)$ belongs to $Q'$. The lower bound strictly increases precisely when no old minimizer belongs to $Q'$.
Proof: Both sets are compact. If an old optimizer is feasible, its old value is retained and restriction cannot improve it. If the value is retained, compactness supplies an optimizer in the restricted set attaining the old value, contradicting absence of an old optimizer. Apply the same reasoning to $-h$. ∎

Remaining gap

These results do not choose a unique pricing measure by imposing an arbitrary physical-probability preference. They determine a sharp interval from replication and observed quotes. Extending the stability statement to continuous-time martingale measures requires tightness, closure, admissible trading, and payoff-continuity assumptions that finite states suppress. In particular, small Wasserstein distance alone does not control all bounded claims: point masses at $-\epsilon$ and $\epsilon$ have distance $2\epsilon$, but the digital claim ${\mathbf 1}\{x\ge0\}$ changes expectation by one. Payoff regularity or restrictions near its discontinuity are necessary for that kind of extension.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt characterizes sharp finite-state valuation bounds, attained hedge duals, replication, interior calibration stability and when an additional quote strictly tightens the interval. Continuous-time markets and economically restricted pricing classes still require closure, tightness and payoff regularity conditions supporting comparable empirical stability guarantees.

References

  • [CK] Zhang Chen and Chen Kay (2026), "Historical Developments in Probability Measures for Asset Pricing: From State Prices to Modern Pricing Kernels," arXiv:2605.27658, Sections 18.6--18.7 and 20. This is the catalogue's review source; the finite-state arguments above are supplied in full.

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