How valuation rules can reveal the uncertainty behind prices

A valuation rule translates a future payoff into a value today. In conventional modeling, that calculation starts with assumptions about how an economic state evolves and how future payments are discounted. But if valuations are observable while those assumptions are hidden, can the calculation be run backward?

In Valuation Reveals Uncertainty, Jongjin Park, David Criens and Hyungbin Park establish a mathematical recovery result under specified conditions. Their preprint constructs an uncertainty structure whose robust valuations reproduce the original dynamic rule. The recovered structure describes uncertainty jointly in the evolution of the economic state and in discounting.

This is a theoretical representation result, not an empirical demonstration that ordinary market quotes reveal investors’ actual beliefs.

What the valuation rule contains

The starting point is more extensive than the price of one security. The authors consider a family of valuation operators covering different time horizons, initial states and bounded, continuous payoff functions.

These operators must satisfy several requirements. A larger payoff cannot receive a smaller value. Multiplying a payoff by a nonnegative amount scales its valuation accordingly, while the valuation of a combined payoff cannot exceed the sum of the separate valuations. Together, these last two properties define sublinearity.

The rule must also satisfy continuity conditions, a bound preventing valuations from exceeding the payoff’s maximum absolute size, and recursive consistency: valuing over two successive periods must agree with valuing over their combined duration.

The robust representation takes the supremum of expected discounted payoffs across the permitted models. It therefore produces an upper valuation; applying the rule to the negative payoff and reversing the sign produces a corresponding lower valuation. Although the paper describes robust valuation in worst-case terms, its displayed formula uses this upper-envelope convention.

Reconstructing uncertainty in four steps

First, the authors extract the valuation rule’s infinitesimal generator. This records how the value of a payoff changes over an increasingly short time interval.

Their locality assumption restricts that instantaneous behavior to information near the current state. Under this assumption, the generator depends on the payoff’s value, its first derivatives and its second derivatives at that state. These describe its level, slope and curvature.

Second, convex duality turns this generating function into a support set at each state. Each element specifies a possible combination of a discounting coefficient, drift and covariance. Drift describes directional movement; covariance describes the local variability of the state process. The support sets preserve the permitted combinations rather than treating these sources of uncertainty separately.

Third, the authors use those sets to construct possible state-process laws and cumulative discounting processes. The discounting coefficient determines accumulated discounting, while drift and covariance constrain the state dynamics through a martingale problem—a probabilistic characterization of their evolution.

Finally, they prove that taking robust valuations across the constructed family reproduces the starting valuation rule.

The proof builds substantially on an earlier framework involving processes that can be sent to an absorbing terminal state, a device called killing. The present paper recovers discounting and underlying state-process laws from those technical objects, giving them an economic interpretation.

Why the assumptions matter

The recovery theorem requires more than sublinearity. Its local-generator assumption places the analysis in a continuous-path, diffusion-type setting. The authors explicitly exclude genuinely nonlocal effects such as jumps and identify their treatment as future work.

A comparison principle for the associated nonlinear partial differential equation ensures that the local generator determines the global valuation rule uniquely. A further Lyapunov-type condition controls behavior toward the boundary of the state space and supports the probabilistic construction.

These conditions are essential to the stated result. The theorem constructs a representing uncertainty structure; it is not an unrestricted claim that any observed collection of prices identifies hidden economic uncertainty.

What option prices could reveal

The paper illustrates identification using put options. In its general argument, valuations of every finite signed put portfolio, across all positive initial states and sufficiently short horizons, determine the generator. Crucially, portfolio valuations cannot generally be reconstructed from individual options’ bid and ask prices under a nonlinear rule.

A second example reduces the information required by assuming a particular uncertain-volatility structure. Within that class, sufficiently smooth, compatible put bid and ask surfaces can identify lower and upper volatility functions through Dupire’s formula. Matching put prices alone does not establish that the structural assumption is correct.

Both examples use idealized valuation information. The authors explicitly leave model selection and calibration to finitely many, potentially noisy quotations outside the study. Their contribution is the mathematical bridge: under stated assumptions, a dynamic valuation rule contains enough information to construct models that reproduce it.