General Dynamic Term Structures Under Default Risk

Fontana, Claudio; Schmidt, Thorsten

doi:10.2139/ssrn.3154361

Cited by 8 publications

(13 citation statements)

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“…To conclude the proof, we finally turn to the discontinuous part. Note that Lemma 3.5 already provides us with parameters γ, a set J ν and the validity of (12c) and (16). Taking the continuous increasing function A c from the first part of the proof and inserting jumps of strictly positive hight at each time t P J ν we obtain an increasing function A with continuous part A c and jump set J A " J ν .…”

Section: The Characterization Of Affine Semimartingalesmentioning

confidence: 92%

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Affine processes beyond stochastic continuity

Keller‐Ressel¹,

Schmidt²,

Wardenga³

2019

Ann. Appl. Probab.

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In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale characteristics have affine form and that the conditional characteristic function can be represented with solutions to measure differential equations of Riccati type. We prove existence of affine Markov processes and affine semimartingales under mild conditions and elaborate on examples and applications including affine processes in discrete time. arXiv:1804.07556v2 [math.PR] E " e xu,Xty |F s ‰ " exp`φ s pt, uq`xψ s pt, uq, X s y˘(3)for all 0 ď s ď t and u P U. Moreover, X is called time-homogeneous, if φ s pt, uq " φ 0 pt´s, uq and ψ s pt, uq " ψ 0 pt´s, uq, again for all 0 ď s ď t and u P U.Note that the left-hand side of (3) is always well-defined and bounded in absolute value by 1, due to the definition of U.Remark 2.2. Comparing Definition 2.1 with the definition of an affine process in [10] (which treats the time-homogeneous case) and [14] (which treats the time-inhomogeneous case), we have replaced the Markov assumption of [10,14] with a semimartingale assumption. In view of [10, Thm. 2.12] this seems to slightly restrict the scope of the definition, since it excludes non-conservative processes. On the other hand, and this is the central point of our paper, we do not impose a stochastic continuity assumption on X, as has been done in [10,14]. It turns out that omitting this assumption leads to a significantly larger class of stochastic processes and to a substantial extension of the results in [10,14].To continue, we introduce an important condition on the support of the process X. Recall that the support of a generic random variable X, is the smallest closed set C such that P pX P Cq " 1; we denote this set by supppXq. For a set A we write convpAq for its convex hull, i.e. the smallest convex set containing A.Condition 2.3. We say that an affine semimartingale X has support of full convex span, if convpsupppX t qq " D for all t ą 0.Under Condition 2.3, φ and ψ are uniquely specified:Lemma 2.4. Let X be an affine semimartingale satisfying the support condition 2.3. Then φ s pt, uq and ψ s pt, uq are uniquely specified by (3) for all 0 ă s ď t and u P U.Proof. Fix 0 ă s ď t and suppose that r φ s pt, uq and r ψ s pt, uq are also continuous in u P U and satisfy (3). Write p s pt, uq :" r φ s pt, uq´φ s pt, uq and q s pt, uq :" r φ s pt, uq´φ s pt, uq. Due to (3) it must hold that p s pt, uq`xq s pt, uq, X s y takes values in t2πik : k P Nu a.s. @ u P U.However, the set U is simply connected, and hence its image under a continuous function must also be simply connected. It follows that u Þ Ñ p s pt, uq`xq s pt, uq, X s y is constant on U and therefore equal to p s pt, 0q`xq s pt, 0q, X s y " 0. Hence, p s pt, uq`xq s pt, uq, xy " 0, for all x P supppX s q and u P U. T...

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Section: The Characterization Of Affine Semimartingalesmentioning

confidence: 92%

“…For (iii) note that γ has been defined in such a way that (18) becomes (12d). The jump equations (16) are a direct consequence of (9). If J ν Ă J A , then γpt, uq " 0 whenever t R J A and it follows that γ is a good parameter.…”

Section: The Characterization Of Affine Semimartingalesmentioning

confidence: 99%

Affine processes beyond stochastic continuity

Keller‐Ressel¹,

Schmidt²,

Wardenga³

2019

Ann. Appl. Probab.

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“…with a strictly positive random variable , independent of G. As long as Assumption (2.7) holds, the setup of Bélanger et al (2004) is included in our framework. Beyond Assumption (2.7), numerous technical problems arise, which are covered in full generality in Fontana and Schmidt (2016).…”

Section: The Relation Tomentioning

confidence: 99%

“…() is included in our framework. Beyond Assumption , numerous technical problems arise, which are covered in full generality in Fontana and Schmidt ().…”

Section: A General Account On Bond Markets With Credit Riskmentioning

confidence: 99%

Dynamic Defaultable Term Structure Modeling Beyond the Intensity Paradigm

Gehmlich

Schmidt

2016

Mathematical Finance

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The two main approaches in credit risk are the structural approach pioneered by Merton and the reduced‐form framework proposed by Jarrow and Turnbull and by Artzner and Delbaen. The goal of this paper is to provide a unified view on both approaches. This is achieved by studying reduced‐form approaches under weak assumptions. In particular, we do not assume the global existence of a default intensity and allow default at fixed or predictable times, such as coupon payment dates, with positive probability. In this generalized framework, we study dynamic term structures prone to default risk following the forward‐rate approach proposed by Heath, Jarrow, and Morton. It turns out that previously considered models lead to arbitrage possibilities when default can happen at a predictable time. A suitable generalization of the forward‐rate approach contains an additional stochastic integral with atoms at predictable times and necessary and sufficient conditions for an appropriate no‐arbitrage condition are given. For efficient implementations, we develop a new class of affine models that do not satisfy the standard assumption of stochastic continuity. The chosen approach is intimately related to the theory of enlargement of filtrations, for which we provide an example by means of filtering theory where the Azéma supermartingale contains upward and downward jumps, both at predictable and totally inaccessible stopping times.

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“…For more details on the reduced form approach, we refer to Bielecki and Rutkowski (2002), and the references therein. Structural models can be embedded into (generalized) reduced form models as pointed out in Bélanger et al (2004) and Fontana and Schmidt (2018).…”

Section: Introductionmentioning

confidence: 99%

Advances in Credit Risk Modeling and Management

Vrins¹

2020

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Correctly assessing credit risk still represents an important challenge for both practitioners and scholars. On the one hand, credit risk measures play a central role in the banking sector's regulations, governing the profitability of financial institutions which remain at the heart of our economic system. On the other hand, effectively computing such measures in a sound and rigorous way triggers important challenges because of the lack of relevant information and/or models. It is therefore important that academics pursue efforts to improve their models. This book presents some recent advances which methodologically and/or computationally contribute to the more rigorous and reliable management of credit risk of firms. The book covers default and recovery rate models, trade credit, counterparty credit risk, and hybrid product pricing.

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General Dynamic Term Structures Under Default Risk

Cited by 8 publications

References 51 publications

Affine processes beyond stochastic continuity

Affine processes beyond stochastic continuity

Dynamic Defaultable Term Structure Modeling Beyond the Intensity Paradigm

Advances in Credit Risk Modeling and Management

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