Semi-Robust Replication of Barrier-Style Claims on Price and Volatility

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 $ 〈X〉 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.