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16
Credit Risk Models with Incomplete Information
, 2007
"... Incomplete information is at the heart of informationbased credit risk models. In this paper, we rigorously define incomplete information with the notion of “delayed filtrations”. We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time chang ..."
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Incomplete information is at the heart of informationbased credit risk models. In this paper, we rigorously define incomplete information with the notion of “delayed filtrations”. We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time change of filtrations and the second by finitely many marked point processes. This notion unifies the noisy information in Duffie and Lando (2001) and the partial information in CollinDufresne et al. (2004), under which structural models are translated into reducedform intensitybased models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an informationbased model. The authors are grateful to the Associate Editor and the two anonymous referees for their constructive suggestions and enlightening remarks.
Modeling the recovery rate in a reduced form model. Unpublished working paper
, 2005
"... This paper provides a model for the recovery rate process in a reduced form model. After default, a firm continues to operate, and the recovery rate is determined by the value of the firm’s assets relative to its liabilities. The debt recovers a different magnitude depending upon whether or not the ..."
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Cited by 10 (2 self)
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This paper provides a model for the recovery rate process in a reduced form model. After default, a firm continues to operate, and the recovery rate is determined by the value of the firm’s assets relative to its liabilities. The debt recovers a different magnitude depending upon whether or not the firm enters insolvency and bankruptcy. Although this recovery rate process is similar to that used in a structural model, the reduced form approach is maintained by utilizing information reduction in the sense of Guo, Jarrow and Zeng (2005). Our model is able to provide analytic expressions for a firm’s default intensity, bankruptcy intensity, and zerocoupon bond prices both before and after default. KEY WORDS: credit risk, recovery rates, reduced form model, filtration reduction ∗ Helpful comments from seminar participants at Cornell University, the Johannes
Absolutely Continuous Compensators
, 2010
"... We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener pro ..."
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We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Çinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and slightly of space, have such a representation. 1
Strict local martingales with jumps
, 2013
"... A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of “naturally occurring ” strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filt ..."
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A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of “naturally occurring ” strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a phenomenon we would contend is common in applications such as filtering, control, and especially in mathematical finance. We give a method for constructing such examples and analyze one particular method in detail. 1
ARBITRAGE OF THE FIRST KIND AND FILTRATION ENLARGEMENTS IN SEMIMARTINGALE FINANCIAL MODELS
"... Abstract. In a general semimartingale financial model, we study the stability of the No Arbitrage of the First Kind (NA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition under initial and under progressive filtration enlargements. In both cases, we provide a simple and general co ..."
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Abstract. In a general semimartingale financial model, we study the stability of the No Arbitrage of the First Kind (NA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition under initial and under progressive filtration enlargements. In both cases, we provide a simple and general condition which is sufficient to ensure this stability for any fixed semimartingale model. Furthermore, we give a characterisation of the NA1 stability for all semimartingale models.
On the stochastic behaviour of optional processes up to random times
 Forthcoming in the Annals of Applied Probability
, 2014
"... In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopp ..."
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In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopping time. This perspective has advantages in both the theoretical and practical study of optional processes up to random times. Applications are given to financial mathematics, as well as to the study of the stochastic behaviour of Brownian motion with drift up to its time of overall maximum as well as up to lastpassage times over finite intervals. Furthermore, a novel proof of the Jeulin–Yor decomposition formula via Girsanov’s theorem is provided. Introduction. Consider a filtered measurable space (,F), where F = (Ft)t∈R+ is a rightcontinuous filtration, as well as an underlying sigmaalgebra F over such that F ⊇ F ∞: = ∨t∈R+ Ft, where the last setinclusion may be strict. A random time is a [0,∞]valued, Fmeasurable random variable. The interplay between random times and the filtration F goes a long way back, with
Information asymmetry in pricing of credit derivatives
 International Journal of Theorical and Applied Finance
, 2010
"... We study the pricing of credit derivatives with asymmetric information. The managers have complete information on the value process of the firm and on the default threshold, while the investors on the market have only partial observations, especially about the default threshold. Different informatio ..."
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We study the pricing of credit derivatives with asymmetric information. The managers have complete information on the value process of the firm and on the default threshold, while the investors on the market have only partial observations, especially about the default threshold. Different information structures are distinguished using the framework of enlargement of filtrations. We specify risk neutral probabilities and we evaluate default sensitive contingent claims in these cases.
Random Times and Enlargements of Filtrations
, 2012
"... Although the path of research is largely a lonesome one, it would not have been possible without the generous support of numerous individuals to whom I am intellectually and personally indebted. However, before I take this opportunity to thank the many people who have made this thesis possible. I m ..."
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Although the path of research is largely a lonesome one, it would not have been possible without the generous support of numerous individuals to whom I am intellectually and personally indebted. However, before I take this opportunity to thank the many people who have made this thesis possible. I must first apologize to my grandparents to whom I bear the guilt of not been able to fulfill my duties as a grandson. I must gratefully acknowledge my parents for providing me with the opportunity to pursue my selfish endeavours and to my mother for her quiet, everyday support that enabled me to focus on writing my thesis. I must reserve the bulk of my gratitude for my friends Elise, Alister, Ken, Laurence, Jon, Jaq, Bao and Huiming whose friendship, empathy and understanding continues to transcend both time and geography. I must thank my friends in Sydney for keeping me alive and sane. Thank you Belina, Sheryl and Emi for feeding me when I am hungry, for looking after me when I am sick and for comforting me when I am down. Thank you Etienne, John and Jay for entertaining me when I