In this paper, we explore a pricing model for corporate bond accompanied with multiple credit rating migration risk and stochastic interest rate. The bond price volatility strongly depends on potentially multiple credit rating migration and stochastic change of interest rate. A free boundary problem of partial differential equation is presented, which is the equivalent transformation of the pricing model. The existence, uniqueness, and regularity for the free boundary problem are established to guarantee the rationality of the pricing model. Due to the stochastic change of interest rate, the discontinuous coefficient in the free boundary problem depends explicitly on the time variable but is convergent as time tends to infinity. Accordingly, an auxiliary free boundary problem is constructed, whose coefficient is the convergent limit of the coefficient in the original free boundary problem. With some constraint on the risk discount rate satisfied, we prove that a unique traveling wave exists in the auxiliary free boundary problem. The inductive method is adopted to fit the multiplicity of credit rating. Then we show that the solution of the original free boundary problem converges to the traveling wave in the auxiliary free boundary problem. Returning to the pricing model with multiple credit rating migration and stochastic interest rate, we conclude that the bond price profile can be captured by a traveling wave pattern coupling with a guaranteed bond price with face value equal to one at the maturity.

中文翻译：

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具有随机利率的多重信用评级迁移模型

在本文中，我们探索了具有多重信用评级迁移风险和随机利率的企业债券定价模型。债券价格的波动性在很大程度上取决于潜在的多重信用评级迁移和利率的随机变化。提出了偏微分方程的自由边界问题，即定价模型的等效变换。建立了自由边界问题的存在，唯一性和规律性，以保证定价模型的合理性。由于利率的随机变化，自由边界问题中的不连续系数明显取决于时间变量，但随着时间趋于无穷而收敛。因此，构造了一个辅助自由边界问题，其系数是原始自由边界问题中系数的收敛极限。在满足一定的风险折现率约束的情况下，我们证明了辅助自由边界问题中存在唯一的行波。采用归纳法来拟合信用评级的多样性。然后我们证明了原始自由边界问题的解收敛到辅助自由边界问题中的行波。回到具有多个信用评级迁移和随机利率的定价模型，我们得出结论，可以通过行权波动模型结合债券的价格曲线来获取债券价格特征，该债券的担保债券价格在到期时的票面价值等于1。我们证明辅助自由边界问题中存在唯一的行波。采用归纳法来拟合信用评级的多样性。然后我们证明了原始自由边界问题的解收敛到辅助自由边界问题中的行波。回到具有多个信用评级迁移和随机利率的定价模型，我们得出结论，可以通过行权波动模型结合债券的价格曲线来获取债券价格特征，该债券的担保债券价格在到期时的票面价值等于1。我们证明辅助自由边界问题中存在唯一的行波。采用归纳法来拟合信用评级的多样性。然后我们证明了原始自由边界问题的解收敛到辅助自由边界问题中的行波。回到具有多个信用评级迁移和随机利率的定价模型，我们得出结论，可以通过行权波动模型结合债券的价格曲线来获取债券价格特征，该债券的担保债券价格在到期时的票面价值等于1。