## Valuation of default sensitive claims under imperfect information. Working paper (2006)

Citations: | 22 - 0 self |

### BibTeX

@MISC{Coculescu06valuationof,

author = {Delia Coculescu and Hélyette Geman and Monique Jeanblanc},

title = {Valuation of default sensitive claims under imperfect information. Working paper},

year = {2006}

}

### OpenURL

### Abstract

We propose an evaluation method for financial assets subject to default risk, when investors face imperfect information about the state variable triggering the default. The model we propose generalizes the one by Duffie and Lando (2001) in the following way: (i) it incorporates informational noise in continuous time, (ii) it respects the (H) hypothesis, (iii) it precludes arbitrage from insiders. The model is sufficiently general to encompass a large class of structural models. In this setting we show that the default time is totally inaccessible in the market’s filtration and derive the martingale hazard process. Finally, we provide pricing formulas for default-sensitive claims and illustrate with particular examples the shapes of the credit spreads and the conditional default probabilities. An important feature of the conditional default probabilities is they are non Markovian. This might shed some light on observed phenomena such as the ”rating momentum”. 1

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Citation Context ... default probabilities documented in the literature, such as the “rating momentum,” which is the dependence of the new rating on the previous ones instead of the last observed one (see, for instance, =-=[18]-=- or [20]).Valuation of default-sensitive claims under imperfect information 213 Fig. 4 Impact of the noise volatility s on the credit spreads Fig. 5 Impact of the assets’ volatility σ on the credit s... |

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Citation Context ...w. Proof We begin by recalling an important result involving the first passage time of a Brownian motion through a continuous barrier (for developments on the subject, see [4, 10] and, more recently, =-=[22]-=-). Let (Wt)t≥0 be a P-Brownian motion and h a continuous function with h(0) < 0. We introduce the hitting time Th = inf { t : Wt ≤ h(t) } and its distribution function π P,h (t) := P(Th ≤ t), t ≥ 0.2... |

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Citation Context ...erty also known as the (H)-hypothesis. This200 D. Coculescu et al. property is essential for the valuation of securities and also has important mathematical consequences, which were first studied in =-=[3]-=-. Let us introduce U as the optional projection of the process 1(τ≤t) onto the filtration (Ft). Then Ut = P(τ ≤ t|Ft) a.s., (3.1) and an equivalent formulation of the (H)-hypothesis is that P(τ ≤ t|Ft... |

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Citation Context ...�t. This kind of approximation of the first passage time density was already employed in default models (see [6]) as well as insurance models √ 〈N〉i �t and bi,j ci �t −cj �t = for j < i and bi,i = ∞. =-=[1]-=-. 1 We define ai = ci �t √ 〈N〉i �t −〈N〉j �t We approximate (3.9) by considering that default may only occur at the ends of the subperiods. Thus, we write Φ(an) = n∑ i=1 Φ(bn,i)P ( τ ∈ ( (i − 1)�t, i�t... |

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Citation Context ...iesecke and Goldberg [12], where the asset’s default point is not observed; Çetin et al. [5], where only the sign of the fundamental process is supposed to be publicly known. In Jeanblanc and Valchev =-=[15]-=-, a study of the credit spreads for different types of discrete information is proposed, while Guo et al. [13] model a delayed information arrival, which can be either discrete or continuous. See also... |

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Citation Context ...√ 〈N〉i �t −〈N〉j �t We approximate (3.9) by considering that default may only occur at the ends of the subperiods. Thus, we write Φ(an) = n∑ i=1 Φ(bn,i)P ( τ ∈ ( (i − 1)�t, i�t ]∣ ) ∣F∞ . (3.11) 1 See =-=[2]-=- and the references therein for a study of the convergence of this type of approximations.Valuation of default-sensitive claims under imperfect information 203 From the above equality we derive a rec... |

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Citation Context ...on of the standard normal law. Proof We begin by recalling an important result involving the first passage time of a Brownian motion through a continuous barrier (for developments on the subject, see =-=[4, 10]-=- and, more recently, [22]). Let (Wt)t≥0 be a P-Brownian motion and h a continuous function with h(0) < 0. We introduce the hitting time Th = inf { t : Wt ≤ h(t) } and its distribution function π P,h (... |

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