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In this work, a pricing model for a defaultable corporate bond with creditrating migration risk is established. The model turns out to be a free boundaryproblem with two free boundaries. The latter are the level sets of the solutionbut of different kinds. One is from the discontinuous second order term, theother from the obstacle. Existence, uniqueness, and regularity of the solutionare obtained. We also prove that two free boundaries are $C^\infty$. Theasymptotic behavior of the solution is also considered: we show that itconverges to a traveling wave solution when time goes to infinity. Moreover,numerical results are presented.

Double free boundary problem for defaultable corporate bond with credit rating migration risks and their asymptotic behaviors