What happens after a default: The conditional density approach

By Nicole El Karoui, Monique Jeanblanc, Ying Jiao. 2009

130 citations recorded by the source corpus.

We present a general model for default times, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow us to compute the price of defaultable claims, except in the case where the immersion property is satisfied. We propose in this paper a density approach for default times. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and on the immersion property.

Subfields: Accounting, Finance

View Paper: https://www.semanticscholar.org/paper/abc236af7881bc07f85dc2ca1dcd21e0162ace2c

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Citations in the corpus

Optimal investment with counterparty risk: a default-density modeling approach (2009, background): “Under (DH) Hypothesis, a (P, F)-brownian motion W is a G-semimartingale and admits an explicit decomposition in terms of the density α given by (see [9], [6], [3]): Wt = Ŵ G t + ∫ t∧τ 0 d 〈Ws, Gs〉 Gs + ∫ t τ d 〈Ws, αs(τ)〉 αs(τ) =: Ŵ G t + At, 0 ≤ t ≤ T, where Ŵ G is a (P, G)-brownian motion, and A i”
Counterparty Risk and Funding: A Tale of Two Puzzles (2014, methodology): “By application of the results of Section 4 in El Karoui, Jeanblanc, and Jiao (2009), the G-compensated martingale of the default indicator process Ht = 1{τ≤t} is given by Mt = Ht − ∫ t∧τ 0 λvdv, (7.14) for the pre-default intensity of τ given by λt = at(t) At .”
Martingale representation property in progressively enlarged filtrations (2012, unknown): “[7, 9, 16, 18, 19]) show that 1(τ,∞) Γ(X) can have various different forms.”