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

ABSTRACT

We show how to price and replicate a variety of barrier-style claims written on the log price X and quadratic variation $〈X〉$ 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.

KEYWORDS:

• Robust pricing
• robust hedging
• knock-in
• knock-out
• barrier
• quadratic variation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In a non-zero interest rate environment, our results are valid for derivatives written on the the path of the T-forward price of a risky asset.