Variance Swaps on Defaultable Assets and Market Implied Time-Changes

Abstract

We compute the value of a variance swap when the underlying is modeled as a Markov diffusion process time changed by a Lévy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent Lévy measure and local stochastic volatility and have a local stochastic default intensity. Moreover, the Lévy subordinator that drives the underlying can be obtained directly by observing European call/put prices. To illustrate our general framework, we provide an explicit formula for the value of a variance swap when the underlying is modeled as a Lévy subordinated jump-to-default constant elasticity of variance process (see [Carr and V. Linetsky, Finance Stoch., 10, pp. 303--330, 2005]). In this example, we extend the results of [Mendoza-Arriaga, Carr, and Linetsky, Math. Finance, 20, pp. 527--569, 2010], by allowing for joint valuation of credit and equity derivatives as well as variance swaps.