arXiv subjects

Frédéric Vrins

Publications and source records attributed to Frédéric Vrins.

14 recordsLinked to original sources

arXiv2026

Exact calibration of structural models via time-change

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.

q-fin.MF↗

arXiv2026

On the boundaries, asymptotic law and Bernoulli-Doob representation of homogeneous bounded martingales

We study homogeneous diffusion martingales evolving in a bounded state space $D=[a,b]$, where $a$ and $b$ are zeros of the diffusion coefficient. We call a process of the form $Z_t=\mathbb{E}[B\mid\mathcal{F}_t]$, with $B$ a Bernoulli random variable, a Bernoulli-Doob martingale. Our main results establish a complete equivalence: every such diffusion martingale is a Bernoulli-Doob martingale (Theorem 2) and, conversely, every continuous time-homogeneous Markov Bernoulli-Doob martingale on a Brownian filtration arises from such a diffusion (Theorem 3). The intuitive reason is that a bounded martingale has constant expectation while accumulating variance, so it converges to the maximum-variance distribution with given mean and range, namely the Bernoulli. We further show that this Bernoulli limit is truly asymptotic: for any fixed finite horizon $T$, the probability of not yet having reached the boundary is strictly positive (Theorem 4), even when the individual boundaries are accessible. We clarify the relationship between Feller's boundary classification, the pathwise SDE framework, and the martingale constraint, showing that the martingale property forces absorption at any attainable boundary. The theory is illustrated with the $Φ$-martingale, the Jacobi martingale, and applications to credit-risk modelling.

math.PR↗

arXiv2023

Electrode Selection for Noninvasive Fetal Electrocardiogram Extraction using Mutual Information Criteria

Blind source separation (BSS) techniques have revealed to be promising approaches for, among other, biomedical signal processing applications. Specifically, for the noninvasive extraction of fetal cardiac signals from maternal abdominal recordings, where conventional filtering schemes have failed to extract the complete fetal ECG components. From previous studies, it is now believed that a carefully selected array of electrodes well-placed over the abdomen of a pregnant woman contains the required `information' for BSS, to extract the complete fetal components. Based on this idea, in previous works array recording systems and sensor selection strategies based on the Mutual Information (MI) criterion have been developed. In this paper the previous works have been extended, by considering the 3-dimensional aspects of the cardiac electrical activity. The proposed method has been tested on simulated and real maternal abdominal recordings. The results show that the new sensor selection strategy together with the MI criterion, can be effectively used to select the channels containing the most `information' concerning the fetal ECG components from an array of 72 recordings. The method is hence believed to be useful for the selection of the most informative channels in online applications, considering the different fetal positions and movements.

eess.SP↗

arXiv2020

Affine term structure models : a time-changed approach with perfect fit to market curves

We address the so-called calibration problem which consists of fitting in a tractable way a given model to a specified term structure like, e.g., yield or default probability curves. Time-homogeneous jump-diffusions like Vasicek or Cox-Ingersoll-Ross (possibly coupled with compounded Poisson jumps, JCIR), are tractable processes but have limited flexibility; they fail to replicate actual market curves. The deterministic shift extension of the latter (Hull-White or JCIR++) is a simple but yet efficient solution that is widely used by both academics and practitioners. However, the shift approach is often not appropriate when positivity is required, which is a common constraint when dealing with credit spreads or default intensities. In this paper, we tackle this problem by adopting a time change approach. On the top of providing an elegant solution to the calibration problem under positivity constraint, our model features additional interesting properties in terms of implied volatilities. It is compared to the shift extension on various credit risk applications such as credit default swap, credit default swaption and credit valuation adjustment under wrong-way risk. The time change approach is able to generate much larger volatility and covariance effects under the positivity constraint. Our model offers an appealing alternative to the shift in such cases.

q-fin.MF↗

arXiv2019

An arbitrage-free conic martingale model with application to credit risk

Conic martingales refer to Brownian martingales evolving between bounds. Among other potential applications, they have been suggested for the sake of modeling conditional survival probabilities under partial information, as usual in reduced-form models. Yet, conic martingale default models have a special feature; in contrast to the class of Cox models, they fail to satisfy the so-called \emph{immersion property}. Hence, it is not clear whether this setup is arbitrage-free or not. In this paper, we study the relevance of conic martingales-driven default models for practical applications in credit risk modeling. We first introduce an arbitrage-free conic martingale, namely the $Φ$-martingale, by showing that it fits in the class of Dynamized Gaussian copula model of Crépey et al., thereby providing an explicit construction scheme for the default time. In particular, the $Φ$-martingale features interesting properties inherent on its construction easing the practical implementation. Eventually, we apply this model to CVA pricing under wrong-way risk and CDS options, and compare our results with the JCIR++ (a.k.a. SSRJD) and TC-JCIR recently introduced as an alternative.

q-fin.MF↗

arXiv2019

Conditional survival probabilities under partial information: a recursive quantization approach with applications

We consider a structural model where the survival/default state is observed together with a noisy version of the firm value process. This assumption makes the model more realistic than most of the existing alternatives, but triggers important challenges related to the computation of conditional default probabilities. In order to deal with general diffusions as firm value process, we derive a numerical procedure based on the recursive quantization method to approximate it. Then, we investigate the error approximation induced by our procedure. Eventually, numerical tests are performed to evaluate the performance of the method, and an application is proposed to the pricing of CDS options.

q-fin.MF↗

arXiv2018

Minimum Rényi Entropy Portfolios

Accounting for the non-normality of asset returns remains challenging in robust portfolio optimization. In this article, we tackle this problem by assessing the risk of the portfolio through the "amount of randomness" conveyed by its returns. We achieve this by using an objective function that relies on the exponential of Rényi entropy, an information-theoretic criterion that precisely quantifies the uncertainty embedded in a distribution, accounting for higher-order moments. Compared to Shannon entropy, Rényi entropy features a parameter that can be tuned to play around the notion of uncertainty. A Gram-Charlier expansion shows that it controls the relative contributions of the central (variance) and tail (kurtosis) parts of the distribution in the measure. We further rely on a non-parametric estimator of the exponential Rényi entropy that extends a robust sample-spacings estimator initially designed for Shannon entropy. A portfolio selection application illustrates that minimizing Rényi entropy yields portfolios that outperform state-of-the-art minimum variance portfolios in terms of risk-return-turnover trade-off.

q-fin.PM↗

arXiv2018

Closed form expression of the multivariate standard Normal distribution under a weighted sum constraint

In this letter we derive the $(n-1)$-dimensional distribution corresponding to a $n$-dimensional i.i.d. Normal standard vector $Z=(Z_1,Z_2,\ldots,Z_n)$ subjected to the weighted sum constraint $\sum_{i=1}^n w_i Z_i=c$, $w_i\neq 0$. We first address the $n=2$ case before proceeding with the general $n\geq 2$ case. The resulting distribution is a Normal distribution whose mean vector $μ$ and covariance matrix $Σ$ are explicitly derived as a function of $w_1,\ldots,w_n,c$. The derivation of the density relies on a very specific positive definite matrix for which the determinant and inverse can be computed analytically.

math.PR↗

arXiv2018

A subordinated CIR intensity model with application to Wrong-Way risk CVA

Credit Valuation Adjustment (CVA) pricing models need to be both flexible and tractable. The survival probability has to be known in closed form (for calibration purposes), the model should be able to fit any valid Credit Default Swap (CDS) curve, should lead to large volatilities (in line with CDS options) and finally should be able to feature significant Wrong-Way Risk (WWR) impact. The Cox-Ingersoll-Ross model (CIR) combined with independent positive jumps and deterministic shift (JCIR++) is a very good candidate : the variance (and thus covariance with exposure, i.e. WWR) can be increased with the jumps, whereas the calibration constraint is achieved via the shift. In practice however, there is a strong limit on the model parameters that can be chosen, and thus on the resulting WWR impact. This is because only non-negative shifts are allowed for consistency reasons, whereas the upwards jumps of the JCIR++ need to be compensated by a downward shift. To limit this problem, we consider the two-side jump model recently introduced by Mendoza-Arriaga \& Linetsky, built by time-changing CIR intensities. In a multivariate setup like CVA, time-changing the intensity partly kills the potential correlation with the exposure process and destroys WWR impact. Moreover, it can introduce a forward looking effect that can lead to arbitrage opportunities. In this paper, we use the time-changed CIR process in a way that the above issues are avoided. We show that the resulting process allows to introduce a large WWR effect compared to the JCIR++ model. The computation cost of the resulting Monte Carlo framework is reduced by using an adaptive control variate procedure.

q-fin.MF↗

arXiv2017

Piecewise Constant Martingales and Lazy Clocks

This paper discusses the possibility to find and construct \textit{piecewise constant martingales}, that is, martingales with piecewise constant sample paths evolving in a connected subset of $\mathbb{R}$. After a brief review of standard possible techniques, we propose a construction based on the sampling of latent martingales $\tilde{Z}$ with \textit{lazy clocks} $θ$. These $θ$ are time-change processes staying in arrears of the true time but that can synchronize at random times to the real clock. This specific choice makes the resulting time-changed process $Z_t=\tilde{Z}_{θ_t}$ a martingale (called a \textit{lazy martingale}) without any assumptions on $\tilde{Z}$, and in most cases, the lazy clock $θ$ is adapted to the filtration of the lazy martingale $Z$. This would not be the case if the stochastic clock $θ$ could be ahead of the real clock, as typically the case using standard time-change processes. The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on (intervals of) $\mathbb{R}$.

math.PR↗

arXiv2016

Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment

A key driver of Credit Value Adjustment (CVA) is the possible dependency between exposure and counterparty credit risk, known as Wrong-Way Risk (WWR). At this time, addressing WWR in a both sound and tractable way remains challenging: arbitrage-free setups have been proposed by academic research through dynamic models but are computationally intensive and hard to use in practice. Tractable alternatives based on resampling techniques have been proposed by the industry, but they lack mathematical foundations. This probably explains why WWR is not explicitly handled in the Basel III regulatory framework in spite of its acknowledged importance. The purpose of this paper is to propose a new method consisting of an appealing compromise: we start from a stochastic intensity approach and end up with a pricing problem where WWR does not enter the picture explicitly. This result is achieved thanks to a set of changes of measure: the WWR effect is now embedded in the drift of the exposure, and this adjustment can be approximated by a deterministic function without affecting the level of accuracy typically required for CVA figures. The performances of our approach are illustrated through an extensive comparison of Expected Positive Exposure (EPE) profiles and CVA figures produced either by (i) the standard method relying on a full bivariate Monte Carlo framework and (ii) our drift-adjustment approximation. Given the uncertainty inherent to CVA, the proposed method is believed to provide a promising way to handle WWR in a sound and tractable way.

q-fin.MF↗

arXiv2016

Wrong-Way Risk Models: A Comparison of Analytical Exposures

In this paper, we compare static and dynamic (reduced form) approaches for modeling wrong-way risk in the context of CVA. Although all these approaches potentially suffer from arbitrage problems, they are popular (respectively) in industry and academia, mainly due to analytical tractability reasons. We complete the stochastic intensity models with another dynamic approach, consisting in the straight modeling of the survival (Azéma supermartingale) process using the $Φ$-martingale. Just like the other approaches, this method allows for automatic calibration to a given default probability curve. We derive analytically the positive exposures $V^+_t$ "conditional upon default" associated to prototypical market price processes of FRA and IRS in all cases. We further discuss the link between the "default" condition and change-of-measure techniques. The expectation of $V^+_t$ conditional upon $τ=t$ is equal to the unconditional expectation of $V^+_tζ_t$. The process $ζ$ is explicitly derived in the dynamic approaches: it is proven to be positive and to have unit expectation. Unfortunately however, it fails to be a martingale, so that Girsanov machinery cannot be used. Nevertheless, the expectation of $V^+_tζ_t$ can be computed explicitly, leading to analytical expected positive exposure profiles in the considered examples.

q-fin.MF↗

arXiv2016

Characteristic Function of Time-Inhomogeneous Lévy-Driven Ornstein-Uhlenbeck Processes

Distributional properties -including Laplace transforms- of integrals of Markov processes received a lot of attention in the literature. In this paper, we complete existing results in several ways. First, we provide the analytical solution to the most general form of Gaussian processes (with non-stationary increments) solving a stochastic differential equation. We further derive the characteristic function of integrals of Lévy-processes and Lévy driven Ornstein-Uhlenbeck processes with time-inhomogeneous coefficients based on the characteristic exponent of the corresponding stochastic integral. This yields a two-dimensional integral which can be solved explicitly in a lot of cases. This applies to integrals of compound Poisson processes, whose characteristic function can then be obtained in a much easier way than using joint conditioning on jump times. Closed form expressions are given for gamma-distributed jump sizes as an example.

math.PR↗

arXiv2016

Conic Martingales from Stochastic Integrals

In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in $[0,1]$. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion and Geometric Brownian motion) having a separable coefficient $σ(t,y)=g(t)h(y)$ and that can be obtained via a time-homogeneous mapping of Gaussian diffusions. The approach is exemplified to the modeling of stochastic conditional survival probabilities in the univariate (both conditional and unconditional to survival) and bivariate cases.

math.PR↗