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Mathematics and Financial Economics
Published in: Mathematics and Financial Economics 2/2020

01-01-2020

On the probability of default in a market with price clustering and jump risk

Authors: Shiyu Song, Yongjin Wang, Guangli Xu

Published in: Mathematics and Financial Economics | Issue 2/2020

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Abstract

In this paper, we consider the probability of default for financial variables under a tractable stochastic model which can capture both the price clustering phenomenon and the jump risk. We assume that the logarithm of the price dynamics is driven by a so-called sticky double exponential jump diffusion process. In particular, the price clustering is incorporated by time changing the jump diffusion process, and the jump risk is described through a Poisson stochastic integral related to the compound Poisson process with double exponential jumps. Under such a model, we study the default events in the structural credit risk framework. The explicit solutions of Laplace transforms associated with default times are obtained. Based on the derived results, we numerically analyze the effects of the price clustering and the jump risk on the distributions of default times and on the bond’s credit spreads.
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Appendix
Available only for authorised users
Footnotes
1
Because of the jump part, when \(Y_{t}\) crosses \(a^{*}\), sometimes it hits \(a^{*}\) exactly, and sometimes it incurs an “overshoot” (if \(Y_{0}<a^{*}\)) or “undershoot” (if \(Y_{0}>a^{*}\)) over \(a^{*}\). See, for example, [17, 22] in which the overshoot and undershoot problems for jump diffusions were considered respectively.
 
2
As shown in the proof of the proposition, \({\mathbf {P}}(\varrho _{a^{*}}<+\infty |Y_{0}=x)>0\) for any \(x\in \mathbb {R}\).
 
3
Mathematically, this is the first passage time of \(Y_{t}\) to z in the case when \(Y_{0}\ge z\). So we adopt the notation (A.1) appearing in the proof of Proposition 1 with \(a^{*}\) replaced by z.
 
4
In the case when \(a^{*}\le l\), \(\tau _{l}\) has the same distribution as that of the first passage time of the double exponential jump diffusion process which corresponds exactly to Corollary 2 in the current paper.
 
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Metadata
Title
On the probability of default in a market with price clustering and jump risk
Authors
Shiyu Song
Yongjin Wang
Guangli Xu
Publication date
01-01-2020
Publisher
Springer Berlin Heidelberg
Published in
Mathematics and Financial Economics / Issue 2/2020
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-019-00253-x

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