  We price and replicate a variety of claims written on the log price $X$ and
quadratic variation $[X]$ of a risky asset, modeled as a positive
semimartingale, subject to stochastic volatility and jumps. The pricing and
hedging formulas do not depend on the dynamics of volatility process, aside
from integrability and independence assumptions; in particular, the volatility
process may be non-Markovian and exhibit jumps of unknown distribution. The
jump risk may be driven by any finite activity Poisson random measure with
bounded jump sizes. As hedging instruments, we use the underlying risky asset,
a zero-coupon bond, and European calls and puts with the same maturity as the
claim to be hedged. Examples of contracts that we price include variance swaps,
volatility swaps, a claim that pays the realized Sharpe ratio, and a call on a
leveraged exchange traded fund.
