Exact calibration of structural models via time-change
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 K̃ and process S̃, and time-change those using a deterministic clock Θ to get Kt=K̃Θ(t) and St:=S̃Θ(t), leading to the final FTP model (K,S,Θ). As the market curve G and the latent model (K̃,S̃) 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: Ṽt=μt+Wt and K̃t=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.
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BibTeX
@misc{vrins2026exact,
title = {{Exact calibration of structural models via time-change}},
author = {Frédéric Vrins and Damiano Brigo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12666},
eprint = {2609.12666},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12666},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact calibration of structural models via time-change
AU - Frédéric Vrins
AU - Damiano Brigo
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.12666
UR - https://arxiv.org/abs/2609.12666
AB - 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 K̃ and process S̃, and time-change those using a deterministic clock Θ to get K_t=K̃_Θ(t) and S_t:=S̃_Θ(t), leading to the final FTP model (K,S,Θ). As the market curve G and the latent model (K̃,S̃) 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: Ṽ_t=μt+W_t and K̃_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.
ER -