Robust replication of barrier-style claims on price and volatility

TL;DR

This paper develops a model-free method to price and hedge various barrier-style options on an asset's price and volatility, accommodating jumps and non-Markovian volatility.

Contribution

It introduces a framework that uses only basic market instruments to replicate complex barrier claims without relying on specific stochastic models.

Findings

Effective replication of barrier claims using minimal instruments

Framework accommodates jumps and non-Markovian volatility

Applicable to knock-in, knock-out, and rebate options

Abstract

We show how to price and replicate a variety of barrier-style claims written on the $\log$ price $X$ and quadratic variation $\langle X \rangle$ of a risky asset. Our framework assumes no arbitrage, frictionless markets and zero interest rates. We model the risky asset as a strictly positive continuous semimartingale with an independent volatility process. The volatility process may exhibit jumps and may be non-Markovian. As hedging instruments, we use only the underlying risky asset, zero-coupon bonds, and European calls and puts with the same maturity as the barrier-style claim. We consider knock-in, knock-out and rebate claims in single and double barrier varieties.