Exponential Lévy Models Extended by a Jump to Default

This article proposes a new dynamically consistent framework for joint valuation of equity derivatives and credit products, in which uncertainty of the economy is represented by Lévy processes. In the framework, the pre-default stock price of a given firm is presented by an extended exponential Lévy model, while the default arrival rate is presented by the Cox proportional hazard model with stochastic covariates driven by Lévy processes. Under the model, we find the solution of the pricing generator for evaluating equity and credit derivatives, and we derive the pricing formulas of equity call options and credit default swaps by utilizing the pricing generator. In the numerical examples, setting the variance gamma (VG) process and the Brownian motion as driving factors of the model, we compute term structure of credit default swaps and equity implied volatility skews. We also examine the impact of the convexity adjustment on term structure of credit spreads both analytically and numerically.