AT1P can fit an entire time-zero survival curve exactly, provided its deterministic volatility is read as a clock. The usual calibration objection to structural credit models then loses much of its force. Instead of choosing a firm-value process and barrier, only to miss most of the CDS curve, Vrins and Brigo hold the barrier fixed and alter the speed of time.

The construction works by inversion. Begin with a latent first-passage model consisting of a process $X$, a constant barrier $k < X_0$, and a known running-infimum law, $G^X_k(t) = P(\underline{X}t > k)$. The implied clock is $\Theta(t) = I^X_k(G(t))$, with $I^X_k$ denoting the inverse of $G^X_k$. By construction, the first passage of $X{\Theta(t)}$ below $k$ has survival probability $G^X_k(\Theta(t)) = G(t)$ for every $t$. Lemma 3 establishes existence and uniqueness when $G$ and $G^X_k$ meet the regularity conditions in Definition 3.

Brownian latent dynamics make the inversion explicit. The reflection principle yields $G^W_k(t) = 2\Phi(-k/\sqrt{t}) - 1$, hence

$$\Theta^{G,W}_k(t) = \left(\frac{k}{\Phi^{-1}\left(\frac{1+G(t)}{2}\right)}\right)^2.$$

Dambis-Dubins-Schwarz returns the model to calendar time. The time-changed Brownian motion becomes the stochastic integral $S_t = \int_0^t \sigma(s)\,dB_s$, and its squared diffusion coefficient is the clock rate. The paper wants the model expressed on the original time scale because several processes may run on different clocks. Handling their multiple filtrations is, in its words, far from handy.

No CDS quotes are calibrated here. The targets are an exponential hazard of 10% and a piecewise-constant hazard curve drawn from Table 2 of Brigo, Morini and Tarenghi. The numerical exercise only checks Monte Carlo consistency, using $N = 1{,}000$ paths, step $0.01$, and horizon $T = 20$. Section 5.1 runs the Brownian and time-changed Brownian cases with $k = -1$ and $k = -5$. For AT1P, the inputs are $H = 0.4$, $S_0 = 1$, $\sigma = 0.15$, with $B = 0$ and $B = 0.5$. The simulated curves lie over $G$. The empirical evidence ends there, as the authors acknowledge.

AT1P reduced to its clock

The identification is the reason to read the paper. AT1P specifies a deterministic instantaneous volatility $\sigma(t)$ and lets the barrier drift with $r$, the payout ratio and $B\sigma^2$. Vrins and Brigo collapse those ingredients into a time change. Set $\mu = B - 1/2$ and $k = \log(K_0/S_0) < 0$. Then $G^{\text{AT1P}}(t) = G^{\tilde W}_k(\Sigma^2(t))$, where $\tilde W_t = \mu t + W_t$ and $\Sigma^2(t) = \int_0^t \sigma^2(u)\,du$. As the paper says, "the AT1P model is nothing else than the FPT of a drifted Brownian motion... time-changed using $\Theta$." Its deterministic volatility function is the clock.

This reading supplies a result the original calibration papers lacked. Brigo and co-authors calibrate AT1P by taking $\sigma$ piecewise constant over the CDS tenor grid, then solving iteratively for $\sigma_1, \ldots, \sigma_n$ against the quotes. Vrins and Brigo report that the literature contains no proof that a piecewise-constant $\sigma$ can always reproduce a specified term structure. Lemma 4 gives the corresponding grid-point result. Choose $\sigma_i = \sqrt{(\Theta(t_i) - \Theta(t_{i-1}))/(t_i - t_{i-1})}$, and the model matches $G$ exactly at $t_1, \ldots, t_n$.

Remark 4 draws the boundary carefully. Survival probabilities matched on a grid need not imply that quoted par spreads reprice under a particular pricer and date convention. The authors leave that question open.

The same identification accounts for a familiar calibration problem. According to the paper, the authors of [5] found that the first-bucket volatility $\sigma_1$ became "extremely large" when they fitted AT1P to actual market quotes with piecewise-constant volatility. This paper reports no value for $\sigma_1$. The clock interpretation makes the behavior unavoidable. A continuous-path process that begins strictly above its barrier has no immediate default. If the target hazard obeys $h(0^+) > 0$, the numerator satisfies $g(t) \to h(0^+) > 0$, while the latent first-passage density has $g^{\tilde W}_k(\Theta(t)) \to 0$ at operational time zero. The clock rate therefore diverges, as the authors state directly in Remark 1.

An integrable singularity

Theorem 2 provides the main technical result. For every regular $G$, without any growth condition on the hazard rate, $\Sigma^2(T) = \Theta(T) = k^2/z(T)^2 < \infty$ for all $T > 0$, where $z(T) = \Phi^{-1}((1+G(T))/2)$. Thus $\sigma \in L^2_{\text{loc}}$. Instantaneous variance may become unbounded near the origin, yet integrated variance remains finite over every horizon, leaving the stochastic integral well defined.

Remark 1 concedes the exploding clock rate, while Theorem 2 supports the paper's claim that the singularity is always harmless. The authors also argue that the implied clock resolves the vanishing short-spread problem of continuous-path structural models because CDS spreads depend only on $G$. Our reading is narrower. Their resolution requires an unbounded short-end variance rate, which the paper itself lists as one mitigation alongside jumps and a randomised barrier. A firm-value process whose variance rate diverges near the origin remains difficult to interpret economically.

In the pure Brownian model, $k$ is the single free parameter. The paper observes that $V(X_t) \propto k^2$, leaving the firm-value scale dependent on the modeller's declaration. It flags this restriction and addresses it in later extensions. The drifted latent process has both $\mu$ and $k$, which are nonredundant, while AT1P introduces $(K_0, B)$.

What does calibration determine?

Exact calibration covers one object, $P(\tau > t)$ viewed from time zero. The paper's footnote deserves repetition before anyone builds further on the result: "we stay silent about the future, conditional survival probabilities." It neither examines nor calibrates the conditional survival curves generated by the fitted diffusion. The conclusion does discuss dynamics. The clock changes the process's instantaneous variance and therefore its cross-variation with other risk sources, with CVA under wrong-way risk named as an application. No quantity is attached to either claim.

The construction does guarantee a unique solution to the calibration step. The original AT1P bootstrap applies an iterative root-find across $n$ tenors, without an existence guarantee stated in the original calibration papers. Here the task becomes one monotone inversion of a known continuous curve. The authors claim a substantial benefit when repeated recalibrations are needed, though they provide no runtime comparison and no repricing error.

Implementation also comes with a restriction. Regularity demands $\mu \le 0$, which gives $B \le 1/2$ in AT1P. When $\mu > 0$, the process has non-hitting probability $1 - e^{2\mu k} > 0$. Its latent curve consequently fails to decay to zero, and inversion becomes impossible beyond that level. The paper's numerical AT1P example uses $B = 0.5$, exactly at the boundary.

We could not test any of this. The paper makes no empirical performance claim, and its input is a market-implied survival curve bootstrapped from CDS quotes. We hold no issuer hazard curves and no credit instruments against which to price the calibrated model, so there is nothing to run.

We recently reviewed Brigo and Lucic on rough local stochastic volatility and raised a similar objection: an exact structural result stated and then not costed.

Same shape here.

Identifying AT1P's volatility function with a time change alters the interpretation of a model already in use. Repricing errors on a real CDS term structure, together with wall-clock time against the standard bootstrap, would turn the paper's comparative claim into something measurable.