No Arbitrage Theory for Bond Markets

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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“…Theorem 5.2 in Klein et al (2016) shows that NAFL is equivalent to the existence of an ELMM, which we recall here for convenience.…”

Section: Absence Of Arbitragementioning

confidence: 99%

“…The market of defaultable bonds contains an infinite number of assets and is treated here in the spirit of large financial markets following Klein, Schmidt, and Teichmann (2016). It is the first time that this concept is applied to a market with credit risk.…”

Section: Absence Of Arbitragementioning

confidence: 99%

Dynamic Defaultable Term Structure Modeling Beyond the Intensity Paradigm

Gehmlich

Schmidt

2016

Mathematical Finance

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“…Remark 3.1 It is possible to generalize the notion of trading strategies and the concept of arbitrage. The notion of trading strategies can be generalized by allowing for measure-valued trading strategies [4] or by using methods from large financial markets [21]. There are also other no-arbitrage concepts related to bond markets, which are based on the theory of large financial markets [8].…”

Section: Arbitrage-free Forward Rate Dynamicsmentioning

confidence: 99%

Term structure modeling under volatility uncertainty

Hölzermann

2021

Math Finan Econ

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In this paper, we study term structure movements in the spirit of Heath et al. (Econometrica 60(1):77–105, 1992) under volatility uncertainty. We model the instantaneous forward rate as a diffusion process driven by a G-Brownian motion. The G-Brownian motion represents the uncertainty about the volatility. Within this framework, we derive a sufficient condition for the absence of arbitrage, known as the drift condition. In contrast to the traditional model, the drift condition consists of several equations and several market prices, termed market price of risk and market prices of uncertainty, respectively. The drift condition is still consistent with the classical one if there is no volatility uncertainty. Similar to the traditional model, the risk-neutral dynamics of the forward rate are completely determined by its diffusion term. The drift condition allows to construct arbitrage-free term structure models that are completely robust with respect to the volatility. In particular, we obtain robust versions of classical term structure models.

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“…(ii) Under the additional hypothesis of locally bounded bond prices, [41,Theorem 5] shows that the existence of an equivalent local martingale measure is equivalent to the no asymptotic free lunch (NAFL) condition. In the present setting, NAFL is equivalent to NAFLVR.…”

Section: Hjm-type Conditions For General Defaultable Term Structuresmentioning

confidence: 99%

General Dynamic Term Structures Under Default Risk

Fontana

Schmidt

2017

SSRN Journal

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We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath, Jarrow and Morton (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes.

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No Arbitrage Theory for Bond Markets

Cited by 15 publications

References 33 publications

Dynamic Defaultable Term Structure Modeling Beyond the Intensity Paradigm

Dynamic Defaultable Term Structure Modeling Beyond the Intensity Paradigm

Term structure modeling under volatility uncertainty

General Dynamic Term Structures Under Default Risk

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