{
  "abstract": "In this note, we propose a general structural approach to model a default time $τ$ as the first-passage time (FPT) of a (``firm-value'') process $S$ below a (``debt'') barrier $K$ that comply with a pre-specified survival probability curve $G(t)=\\Pr(τ>t)$. Following an idea of Mbaye and Vrins (Mathematical Finance, 2022) applied to reduced-form models, our approach consists in two steps: choose a latent FPT model driven by a barrier $\\tilde{K}$ and process $\\tilde{S}$, and time-change those using a deterministic clock $Θ$ to get $K_t=\\tilde{K}_{Θ(t)}$ and $S_t:=\\tilde{S}_{Θ(t)}$, leading to the final FTP model $(K,S,Θ)$. As the market curve $G$ and the latent model $(\\tilde{K},\\tilde{S})$ are assumed to be given, the calibration step simply consists in finding the clock $Θ$ such that the distribution of the FPT of $S$ below $K$ coincides with the survival curve $G$. We show that this is achievable for a broad class of specified curves $G$ and latent FTP models. The calibration amounts to a simple inversion of a function, which is almost immediate provided that the latent model is tractable enough. In particular, we show that the AT1P model of Brigo, Morini and Tarenghi - which is able to reproduce a broad range of CDS term-structures - can be regarded as the FPT of a time-changed drifted Brownian motion to a constant barrier: $\\tilde{V}_t=μt+W_t$ and $\\tilde{K}_t=k<0$. This connection offers an elegant interpretation for the instantaneous volatility function featured in AT1P and yields an immediate calibration of the latter to perfectly match a target survival curve.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12666",
  "author_total": 2,
  "authors": [
    {
      "name": "Frédéric Vrins"
    },
    {
      "name": "Damiano Brigo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4dgb28trk6nsb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12666",
  "fulltext_url": null,
  "id": "4dgb28trk6nsb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12666",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "q-fin.MF"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7212782525",
  "pdf_url": "https://arxiv.org/pdf/2609.12666",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4dgb28trk6nsb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.12666",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12666"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Exact calibration of structural models via time-change",
  "title_text": "Exact calibration of structural models via time-change",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/4dgb28trk6nsb",
  "venue": "arXiv",
  "versions": []
}