Consider a defaultable bond traded in a financial market that is subject to shocks and regime shifts. Its recovery payment has a hybrid structure, comprising two components: one contingent on historical information up to the time of default, and the other an independent variable indexed by the regime at the time of default. The default intensity, interest rate, and reference rate are assumed to be general deterministic functions of certain state variables, while these state variables jointly follow a jump-diffusion process, with drift and volatility coefficients governed by the regime and with jumps induced by shocks. We construct a risk-neutral pricing measure that prices all risk sources in an integrated manner. A rigorous verification of this pricing measure reveals the corresponding time-dependent market prices of these risk sources. The resulting pricing framework is applicable to most defaultable bonds and credit derivatives.

According to the gradient allocation principle based on a positively homogeneous and subadditive risk measure, the capital allocated to a sub-portfolio is the Gâteaux derivative, assuming it exists, of the underlying risk measure at the overall portfolio in the direction of the sub-portfolio. We consider the capital allocation problem based on the higher moment risk measure, which, as a generalization of expected shortfall, involves a risk aversion parameter and a confidence level and is consistent with the stochastic dominance of corresponding orders. As the main contribution, we prove that the higher moment risk measure is Gâteaux differentiable and derive an explicit expression for the Gâteaux derivative, which is then interpreted as the capital allocated to a corresponding sub-portfolio. We further establish the almost sure convergence and a central limit theorem for the empirical estimate of the capital allocation, and address the robustness issue of this empirical estimate by computing the influence function of the capital allocation. We also explore the interplay of the risk aversion and the confidence level in the context of capital allocation. In addition, we conduct intensive numerical studies to examine the obtained results and apply this research to a hypothetical portfolio of four stocks based on real data.