Pricing credit-risky bonds using recovery rate uncertainty and macro-regime switching

A Multi-Credit-Rating Migration Model with Asymmetric Migration Boundaries

“…In this section, we prove the existence of solutions to the PDE problem (7). It is not a general semi-unbounded problem or a Dirichlet problem, since the value of the solution on the boundaries is not given.…”

“…Lemma 2. Φ H (x, t), Φ M (x, t) and Φ L (x, t) defined by (10) form a solution for the PDE problem (7).…”

“…We use the maximum principle to prove the uniqueness of the solution of problem (7). Since the asymmetric boundary of each region is inside another region, the solution of each region cannot obtain the maximum or minimum value on the boundary.…”

“…In detail, Jarrow, Lando and Turnbull [1] were the first to apply a Markov chain, and they adopted a transfer intensity matrix to capture the credit migration process. This reduced-form approach is developed naturally (see Arvanitis et al [2], Hurd and Kuznetsov [3], Lando [4], Thomas et al [5], Nguyen [6], Chen et al [7] and so forth). The transfer intensity matrix is usually derived from general statistical data and mainly reflects the impact of macroeconomic factors on credit rating.…”