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Dynamic Defaultable Term Structure Modeling Beyond The Intensity Paradigm

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  • Frank Gehmlich
  • Thorsten Schmidt

Abstract

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.

Suggested Citation

  • Frank Gehmlich & Thorsten Schmidt, 2018. "Dynamic Defaultable Term Structure Modeling Beyond The Intensity Paradigm," Mathematical Finance, Wiley Blackwell, vol. 28(1), pages 211-239, January.
  • Handle: RePEc:bla:mathfi:v:28:y:2018:i:1:p:211-239
    DOI: 10.1111/mafi.12138
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    Cited by:

    1. Claudio Fontana & Zorana Grbac & Sandrine Gümbel & Thorsten Schmidt, 2020. "Term structure modelling for multiple curves with stochastic discontinuities," Finance and Stochastics, Springer, vol. 24(2), pages 465-511, April.
    2. Tolulope Fadina & Thorsten Schmidt, 2019. "Default Ambiguity," Risks, MDPI, vol. 7(2), pages 1-17, June.
    3. Jan-Frederik Mai, 2019. "Pricing-Hedging Duality For Credit Default Swaps And The Negative Basis Arbitrage," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-17, September.
    4. Tahir Choulli & Catherine Daveloose & Michèle Vanmaele, 2020. "A martingale representation theorem and valuation of defaultable securities," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1527-1564, October.
    5. Claudio Fontana & Zorana Grbac & Thorsten Schmidt, 2022. "Term structure modelling with overnight rates beyond stochastic continuity," Papers 2202.00929, arXiv.org, revised Aug 2023.
    6. Sandrine Gumbel & Thorsten Schmidt, 2021. "Defaultable term structures driven by semimartingales," Papers 2103.01577, arXiv.org, revised Aug 2021.

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