A Structural Credit Risk Model Driven by the Lévy Process under Knightian Uncertainty

Abstract

The classic credit risk structured model assumes that risky asset values obey geometric Brownian motion. In reality, however, risky asset values are often not a continuous and symmetrical process, but rather they appear to jump and have asymmetric characteristics, such as higher peaks and fat tails. On the other hand, there are real Knight uncertainty risks in financial markets that cannot be measured by a single probability measure. This work examined a structural credit risk model in the Lévy market under Knight uncertainty. Using the Lévy–Laplace exponent, we established dynamic pricing models and obtained intervals of prices for default probability, stock values, and bond values of enterprise, respectively. In particular, we also proved the explicit solutions for the three value processes above when the jump process is assumed to follow a log-normal distribution. Finally, the important impacts of Knightian uncertainty on the pricing of default probability and stock values of enterprise were studied through numerical analysis. The results showed that the default probability of enterprise, the stock values, and bond values were no longer a certain value, but an interval under Knightian uncertainty. In addition, the interval changed continuously with the increase in Knightian uncertainty. This result better reflected the impact of different market sentiments on the equilibrium value of assets, and expanded decision-making flexibility for investors.