Robust Replication of Volatility and Hybrid Derivatives on Jump Diffusions

TL;DR

This paper develops robust pricing and hedging formulas for derivatives on the log price and quadratic variation of a risky asset, accommodating stochastic volatility and jumps without requiring specific volatility dynamics.

Contribution

It introduces model-independent formulas that work under broad conditions, including non-Markovian volatility and unknown jump distributions.

Findings

Pricing formulas are independent of volatility dynamics.

Hedging uses underlying, bonds, and European options.

Applicable to variance swaps, volatility swaps, and more.

Abstract

We price and replicate a variety of claims written on the log price $X$ and quadratic variation $[X]$ of a risky asset, modeled as a positive semimartingale, subject to stochastic volatility and jumps. The pricing and hedging formulas do not depend on the dynamics of volatility process, aside from integrability and independence assumptions; in particular, the volatility process may be non-Markovian and exhibit jumps of unknown distribution. The jump risk may be driven by any finite activity Poisson random measure with bounded jump sizes. As hedging instruments, we use the underlying risky asset, a zero-coupon bond, and European calls and puts with the same maturity as the claim to be hedged. Examples of contracts that we price include variance swaps, volatility swaps, a claim that pays the realized Sharpe ratio, and a call on a leveraged exchange traded fund.