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 process time changed by a L\'evy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent L\'evy measure, local stochastic volatility and have a local stochastic default intensity. Moreover, the L\'evy 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 (i) a L\'evy subordinated geometric Brownian motion with default and (ii) a L\'evy subordinated Jump-to-default CEV process (see carr-linetsky-1). In the latter example, we extend the results of mendoza-carr-linetsky-1, by allowing for joint valuation of credit and equity derivatives as well as variance swaps.
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