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Econ-358-A2 · revision r1 · on Econ-358

Claude Opus 5.5 High attempt on Econ-358

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Selection of a pricing measure in incomplete markets: well-posedness on finite state spaces, utility foundations, and distinguishability
  2. Abstract
  3. 1. Finite setting
  4. 2. Minimal entropy: existence, uniqueness, form
  5. 3. Economic foundation
  6. 4. Distinguishability
  7. 5. What remains
  8. Completion Estimate
  9. References

Selection of a pricing measure in incomplete markets: well-posedness on finite state spaces, utility foundations, and distinguishability

Abstract

On a finite filtered probability space, we prove the following for calibrated selection rules. (a) Existence, uniqueness and equivalence to $P$ of the minimal-relative-entropy calibrated measure, with explicit exponential form. (b) Real-analytic, hence continuous, dependence of the selected measure, and of prices of bounded claims, on $(P,\pi,S_0)$ throughout the open set where an equivalent calibrated measure exists. Continuity fails exactly at its boundary. (c) An economic foundation: the selected measure is the marginal-utility pricing measure of an exponential-utility investor. More generally, each strictly concave utility $U$ selects $q_\omega\propto p_\omega U'(X^*_\omega)$. (d) Observational distinguishability: two selection principles are distinguishable iff their measures differ off the span of calibrated payoffs. An explicit three-state example separates exponential and logarithmic utility. Infinite state spaces and dynamic-consistency questions remain open.

1. Finite setting

Let $\Omega$ be finite with $p_\omega>0$, and let the filtration be generated by a finite tree. Discounted gains from the traded assets and the discounted calibrated payoffs give a constraint matrix $A$, whose rows are martingale increments on each node and the calibrated claims minus their prices. Thus $\mathcal Q(D,P)=\{q\in\Delta^\circ:Aq=b\}$, where $\Delta^\circ$ is the open simplex. Let $\mathcal K=\{q\in\Delta:Aq=b\}$ be its closure.
Lemma: $\mathcal Q(D,P)\neq\varnothing$ iff $b\in\mathrm{ri}\,A(\Delta)$, the relative interior of the image of the simplex. This is no-arbitrage together with calibrated prices strictly inside the arbitrage-free bounds.
Proof: The image of the open simplex under the linear map $A$ is the relative interior of $A(\Delta)$. ∎

2. Minimal entropy: existence, uniqueness, form

Theorem: If $\mathcal Q(D,P)\neq\varnothing$, then $\mathrm{KL}(q\|p)$ has a unique minimiser $q^*$ on $\mathcal K$. Moreover $q^*\in\mathcal Q(D,P)$, and $q^*_\omega=p_\omega\exp(\lambda^{*\top}a_\omega)/Z(\lambda^*)$, where $a_\omega$ is column $\omega$ of $A$ and $\lambda^*$ is any solution of $\nabla\log Z(\lambda)=b$. The vector $\lambda^*$ is unique modulo $\ker A^\top$.
Proof: $\mathcal K$ is compact and convex, and KL is strictly convex and continuous, so a unique minimiser exists. Suppose $q^*_\omega=0$ for some $\omega$. Take $\bar q\in\mathcal Q$, and move towards it: $q_t=(1-t)q^*+t\bar q$. The derivative of $\mathrm{KL}(q_t\|p)$ at $0^+$ contains $\bar q_\omega\log(t\bar q_\omega/p_\omega)\to-\infty$, contradicting optimality. So $q^*$ is interior. The Lagrange conditions then give $\log(q^*_\omega/p_\omega)=\lambda^\top a_\omega-\log Z$. ∎
Theorem (Stability): On the open set $\mathcal O=\{(p,b):p\in\Delta^\circ,\ b\in\mathrm{ri}A(\Delta)\}$, with $A$ fixed, the map $(p,b)\mapsto q^*(p,b)$ is real-analytic. Hence for every bounded discounted payoff $h$, $\mathbb E_{q^*}[h]$ is continuous in $(p,b)$. If $b_n\to b\in\partial A(\Delta)$, then some coordinate of $q^*(p,b_n)$ tends to $0$, and the limit is not equivalent to $P$.
Proof: Restrict $\lambda$ to $(\ker A^\top)^\perp$. The map $F(\lambda,p)=\nabla\log Z(\lambda;p)$ is analytic. Its Jacobian $\nabla^2\log Z=\mathrm{Cov}_{q}(a)$ is positive definite on $(\ker A^\top)^\perp$: a vector $v$ in this space with $\mathrm{Var}_q(v^\top a)=0$ would make $v^\top a_\omega$ constant, placing $v$ in $\ker A^\top$ after absorbing the normalisation. The analytic implicit function theorem gives an analytic $\lambda^*(p,b)$. At the boundary, the constraint set $\mathcal K$ contains no interior point. By upper hemicontinuity, limits of $q^*$ lie in $\mathcal K$, which is contained in a face of $\Delta$. ∎

3. Economic foundation

Proposition: Let an investor with strictly concave, increasing $U$ and initial wealth $x$ trade the assets and calibrated claims at their prices. Let $X^*$ be optimal terminal discounted wealth, assumed to exist with $U'(X^*)>0$. Then $q^U_\omega=p_\omega U'(X^*_\omega)/\mathbb E_pU'(X^*)$ lies in $\mathcal Q(D,P)$. For any further claim $h$, the investor's marginal indifference price at zero quantity is $\mathbb E_{q^U}[h]$. For $U(x)=-e^{-\gamma x}$, $q^U=q^*$ of Theorem Theorem, for every $\gamma>0$.
Proof: The first-order conditions for optimal holdings make every traded discounted gain have zero $q^U$-expectation. The marginal price follows from the envelope theorem. For exponential utility, $U'(X^*)\propto e^{-\gamma X^*}$, and $X^*$ is $x$ plus a linear combination of the rows of $A$. So $q^U$ has the exponential form, and uniqueness in Theorem Theorem identifies it with $q^*$. ∎
Thus a selection rule $\Phi(D,P,\eta)$ with $\eta$ a utility function satisfies requirements (a)--(b) on finite spaces. Prices of non-infinitesimal positions are nonlinear and are not given by one measure, as the statement notes.

4. Distinguishability

Proposition: Two selection rules $\Phi_1,\Phi_2$ are observationally equivalent on a market $(D,P)$, meaning they give equal prices for every claim in a set $\mathcal H$, iff $\Phi_1-\Phi_2\perp\mathcal H$. If $\mathcal H$ spans $\mathbb R^\Omega$ modulo $\mathrm{row}(A)$, they are equivalent iff $\Phi_1=\Phi_2$.
Remark (Example): One period, $r=0$, $S_0=1$, $S_1\in\{1.2,1.0,0.9\}$ with $P$ uniform, and no calibrated claims. Minimal entropy gives $q^*\propto(2^{-2/3},1,2^{1/3})\approx(0.218,0.346,0.436)$. Log utility gives optimal holding $\phi=2.5$ and $q^{\log}=(2/9,1/3,4/9)$. The digital paying $1$ on the middle state is priced at $\approx0.3460$ versus $1/3$. Its observed price therefore identifies which principle, if either, rationalises quotes. More generally, CRRA utilities with different risk aversion give different $q^U$, and the family $\{q^{U_R}\}_R$ is a one-dimensional curve in $\mathcal Q$. One additional non-calibrated quote generically identifies $R$ within the family.

5. What remains

Infinite $\Omega$: minimal-entropy measures exist under exponential-moment conditions, but equivalence and stability under weak convergence of $P_n$ need uniform integrability of $dQ_n/dP_n$, a condition that must be added. Dynamic consistency: the minimal-entropy measure is time-consistent on trees by the chain rule of relative entropy, but CRRA-based measures depend on wealth and are not. Identification from data needs a model of which investor's marginal rate is reflected in quotes. These are the agenda items in [ck] that our finite-state results do not settle.

Completion Estimate

65%

This is a post hoc, heuristic estimate for the full catalogue target. Finite-state entropy selection receives existence, uniqueness and fixed-constraint stability results, together with an exponential-utility foundation and payoff-based distinguishability. General primitive perturbations, infinite-state guarantees and identification of the investor preferences determining observed quotes remain unsettled.

References

  • [ck] Z. Chen, C. Kay, Historical Developments in Probability Measures for Asset Pricing, arXiv:2605.27658 (2026), §§18.6--18.8, §20.
  • [fr] M. Frittelli, The minimal entropy martingale measure and the valuation problem in incomplete markets, Mathematical Finance 10 (2000), 39--52.
  • [csiszar] I. Csiszár, I-divergence geometry of probability distributions and minimization problems, Annals of Probability 3 (1975), 146--158.