Variance Swaps on Defaultable Assets and Market Implied Time-Changes

TL;DR

This paper develops a comprehensive model for valuing variance swaps on defaultable assets, incorporating market-implied time-changes, jumps, stochastic volatility, and default risk, with explicit formulas and joint credit-equity derivative valuation.

Contribution

It introduces a flexible framework for variance swap valuation on defaultable assets using Markov processes time-changed by Lévy subordinators, extending previous models to include joint credit and equity derivatives.

Findings

Explicit formulas for variance swap values in Lévy subordinated models

Extension to joint credit and equity derivative valuation

Framework allows market-implied time-change inference from option prices

Abstract

We compute the value of a variance swap when the underlying is modeled as a Markov process time changed by a L\'{e}vy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent L\'{e}vy measure, local stochastic volatility and have a local stochastic default intensity. Moreover, the L\'{e}vy 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 \citet{carr-linetsky-1}). {In the latter example, we extend} the results of \cite{mendoza-carr-linetsky-1}, by allowing for joint valuation of credit and equity derivatives as well as variance swaps.