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.

Bio: Matt Lorig earned a PhD in physics from UC - Santa Barbara in 2011 under the supervision of Jean-Pierre Fouque. He then moved to Princeton where he spent three years as a postdoc in the Department of Operations Research and Financial Engineering. Presently, he is an Associate Professor in the Department of Applied Mathematics at the University of Washington. Professor Lorig's research interests include both traditional topics within financial mathematics, such as robust option pricing and implied volatility modeling, as well as nontraditional topics such as sports betting markets, technical analysis and decentralized finance.