Exponential Levy Models Extended by a Jump to Default

Abstract

This paper proposes a new dynamically consistent framework for joint valuation of equity derivatives and credit products, in which uncertainness of the economy is represented by Levy processes. In the framework, the pre-default stock price of a given firm follows an extended exponential Levy model, while the default arrival rate is given by the Cox proportional hazard model with stochastic covariates driven by Levy processes. Under the models, we find the solution of the pricing generator for evaluating equity and credit derivatives, and then 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 process and the Brownian motion as driving factors of the models, 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.

Keywords: Levy processes, jump to default, proportional hazard model, equity option, credit default swap

JEL Classification: G13, C63