In this article, we study the problem of pricing defaultable bond with discrete default intensity , discrete default barrier and exogenous default recovery. The motivation is the fact that the investor outside of the firm can know the firm information like default barrier only in some discrete dates such as announcing dates of firm management information and the credit rating of the firm that reflects the default intensity is generally not changed between the two adjacent announcing dates. In our model, the risk free short rate follows a generalized Hull-White model. The default event occurs in an expected or unexpected manner when the firm value reaches a certain lower threshold - the default barrier at predetermined discrete announcing dates or  at the first jump time of a Poisson process with given default intensity , respectively . Then our pricing problem is derived to a solving problem of PDE with constant default intensity and terminal value of binary type in every subinterval between the two adjacent announcing dates. Our main method to give the pricing of defaultable bonds is to transform several PDEs for pricing in every subinterval between the two adjacent announcing dates into a pricing problem of a higher order binary option using several changes of variable and unknown function including change of numeraire and then to use the pricing  formulae of higher binary options  .
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 .
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study the problem of
consider a 2 factors-model for
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, discrete default barrier and exogenous default recovery. The motivation is the fact that the investor outside of the firm can know the firm information like default barrier only in some discrete dates such as announcing dates of firm management information and the credit rating of the firm that reflects the default intensity is generally not changed between the two adjacent announcing dates. In our model, the
and barrier where the 2 factors are stochastic
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follows a generalized Hull-White model. The
process and firm value process. We assume that the
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or unexpected
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certain lower threshold - the
given
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in an unexpected manner
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, respectively
given by a step function of time variable
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problem is derived to
model is given by
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PDE with constant default intensity
several linear PDEs with variable coefficients
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method to give the pricing of defaultable bonds is to transform several PDEs for pricing in every subinterval between the two adjacent announcing dates into a pricing problem of a higher order binary option using several changes of variable and unknown function including change of numeraire and then to use the
approach is to use higher order binaries. We first provide the
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formulae of higher order binaries with time dependent coefficients and consider their integrals on the last expiry date variable. Then using the pricing
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and their integrals, we give the pricing formulae of defaultable bonds in both cases of exogenous and endogenous default recoveries and credit spread analysis