  In this article, we consider a 2 factors-model for pricing defaultable bond
with discrete default intensity and barrier where the 2 factors are stochastic
risk free short rate process and firm value process. We assume that the default
event occurs in an expected manner when the firm value reaches a given default
barrier at predetermined discrete announcing dates or in an unexpected manner
at the first jump time of a Poisson process with given default intensity given
by a step function of time variable. Then our pricing model is given by a
solving problem of several linear PDEs with variable coefficients and terminal
value of binary type in every subinterval between the two adjacent announcing
dates. Our main approach is to use higher order binaries. We first provide the
pricing formulae of higher order binaries with time dependent coefficients and
consider their integrals on the last expiry date variable. Then using the
pricing formulae of higher binary options and their integrals, we give the
pricing formulae of defaultable bonds in both cases of exogenous and endogenous
default recoveries and credit spread analysis.
