We show how to price and replicate a variety of barrier-style claims written on the \log price X and quadratic variation \<X \> \langle \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, a zero-coupon bond , and European calls and puts with the same maturity as the barrier-style claim. We consider both single-barrier and double barrier claims in three varieties: knock-in, knock-out and rebate  .
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 .
D
\<X \>
A
X
D
a
R
bond
bonds
D
both single-barrier and double barrier claims in three varieties:
A
claims in single and double barrier varieties