Exact Smile Calibration in Path-Dependent Volatility Models

Definitions and mathematical statement

Write $\mathcal M(\sigma,\lambda)$ for the intended path-dependent volatility model, with volatility function $\sigma$ and mean-reversion parameter $\lambda$. The supplied record does not specify the model equations, the arguments or admissibility conditions of $\sigma$, the allowed values of $\lambda$, or the initial conditions and pricing conventions.

For a conditional formulation of the calibration question, let $D$ denote the intended set of strike--maturity pairs $(K,T)$, let $\Sigma_{\mathrm{mkt}}(K,T)$ denote the prescribed market implied volatility, and let $\Sigma_{\sigma,\lambda}(K,T)$ denote the implied volatility produced by $\mathcal M(\sigma,\lambda)$, wherever this quantity is well defined under the intended pricing convention. Let $\mathcal A$ denote the intended admissible class of parameter pairs. These objects are placeholders for information absent from the record, rather than additional hypotheses supplied by the source.

The exact-calibration question is whether \[ \exists (\sigma,\lambda)\in\mathcal A\quad \forall (K,T)\in D,\qquad \Sigma_{\sigma,\lambda}(K,T)=\Sigma_{\mathrm{mkt}}(K,T). \] The accompanying dynamics question asks whether the calibrated model reproduces the desired implied-volatility dynamics, particularly large negative short-term forward skews. The source does not define the forward implied-volatility quantity, its strike or moneyness coordinate, the relevant time limits, or a quantitative meaning of "large." It also does not specify whether this property is required of at least one exact calibration or of every exact calibration. Consequently, the second question cannot yet be expressed as a definite mathematical predicate.

The record likewise leaves open whether the calibration target is a single-maturity smile, a finite collection of market quotes, or an entire strike--maturity surface, and whether the question concerns a particular observed target or every target in a specified class. No universal calibration assertion or particular stochastic differential equation is inferred here.

Short English statement

Can a specified path-dependent volatility model be calibrated exactly to the intended market volatility smile by choosing $\sigma$ and $\lambda$, and can the calibrated model also exhibit large negative short-term forward skews? The model, calibration domain, admissibility conditions, and forward-skew criterion must be supplied before this becomes a precise existence problem.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003429$, titled "Exact Smile Calibration in Path-Dependent Volatility Models," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2017$ and no named proposer.
• Supplied report citation: Mathematics of Quantitative Finance (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/13, pp. 683--769. https://doi.org/10.4171/owr/2017/13. The background describes these reports as records of mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied; the citation and its year information have not been independently verified.
• Supplied literature reference: Foschi--Pascucci, path-dependent-volatility calibration. https://doi.org/10.1016/j.csda.2008.10.042.
• Supplied literature reference: Guyon--Lekeufack, four-factor path-dependent volatility. https://arxiv.org/abs/2406.02319.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and reports accurate numerical calibration in path-dependent and stochastic-volatility models, including recent joint smile calibration, and the ability to generate negative skews. It further reports that exact universal calibration with the requested dynamics is not proved. These are supplied dated curation claims, not independently verified mathematical facts or a new literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that notation, typography, and sentence structure were repaired while mathematical content was preserved. This supplied editorial label does not establish mathematical completeness or correctness of this draft.