Alfredo Camacho Valle 1st february 2011
1. Introduction The credit risk problem is one of the most important problems discussed in mathematical ﬁnance and it consists of computing the default probability of a ﬁrm going into a debt. Two competing methodologies have emerged to model it: the structural approach that is concerned withmodeling and pricing credit risk that is speciﬁc to a paricular obligator and the reduce form aproach, in which the credit events are speciﬁed in terms of some exogenously speciﬁed jump process. We can distingish between two reduce form approach classes, the intensity - based models that are only concerned with the modeling of the default time and the migration models with migrations between creditrating grades in which we have identiﬁed elements of a ﬁnite set, referred to as the set of credit grades that most of the time are attributed by a commercial rating agency. Nowadyas, credit risk modeling by Markov process represents the most popular methodology used by the credit rating agencies; this approach working mainly on the generation of a transition matrix and there are copiousliterature about it (see Jarrow, Lando and Turnbull (1997); Nickell, Perraudin and Varotto (2000); Israel, Rosenthal and Wei (2001); Hu, Kiesel and Perraudin (2002)). However, there has been found some problems of the suitable of this model in the credit risk environmen that can be summarized in three points:
(a) There is dependence of transition probabilities on duration in a rating category orage; ie T must be consider as a random variable (Duﬃe and Singleton (2003), p. 87). (b) In general the rating evaluation depends on when it is carried out and, in particular, on the business cycle; ie a rating evaluation generally is diﬀerent each time (Nickell, Perraudin and Varoto (2000)). (c) There is dependence of the new rating on all the previous ones and no only on the last (see Nickell,Perraudin and Varoto (2000)). The ﬁrst problem can be solved by means of semi - Markov processes (HSMP) (DAmico, Janssen, Manca (2004)) since allows arbitrarily distributed sojourn times in any states and still have the markovian hypothesis, but in a more ﬂexible manner. The second problem can be dealt by means of a non - homogeneous enviroment ((DAmico, Janssen, Manca (2005)). Meanwhile, the thirdproblem exists in the case of a downward moving rating but not in the case of a upward moving rating, ie if a company gets a lower raing, there is a higher probbility that is subsequent rating will also be lower than the proceding one. It can be solved enlarging the state number (Christensen, Hansen,Landon (2004)), ie ﬁrms wich are downgraded into certain categories enter into an excited state inwhich are retained for a stochastic amount of time which a greater probability to downgrade that the normal state, if there is no movement in a certain time, it migrates to the normal state. To introduce the Semi Markov model in a credit risk enviroment lets consider the following reliability problem. let us consider a system having as state space the set I =1,...m with m ﬁnite. I is particionatedin two non-void subsets U and D, where U is the set of all functioning and D all the failed states such that a transition from a state of U to a state of D is seen as the end of the process. In order to perform a credit risk system in this enviroment, we can consider m possible states, where m-1 represents the good states and one default state, so we can see it as a reliability problem. 2
2.Non Homogeneous semi Markov process Let the two dimensional process in discrete time Jn , n ∈ ℵ with state space I =1, ...m be represent the state at the n - th transition, and Tn , n ∈ ℵ with state space equal to + be represent the time of the n-th transition, Jn : Ω → I, Tn : Ω → + We supposse that the process, (Jn , Tn ) is a non homogeneous markovian renewal process. Also, On complete...