We price and replicate a variety of claims written on the log price and quadratic variation 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.

中文翻译：

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波动性和混合衍生品在跳跃扩散上的稳健复制

我们定价和复制写在对数价格和二次变化上的各种声明风险资产，建模为正半鞅，受随机波动和跳跃的影响。除了可积分性和独立性假设外，定价和对冲公式不依赖于波动过程的动态；特别是，波动率过程可能是非马尔可夫的，并表现出未知分布的跳跃。跳跃风险可能由具有有界跳跃大小的任何有限活动泊松随机测度驱动。作为对冲工具，我们使用标的风险资产、零息债券以及与要对冲的债权具有相同到期日的欧洲看涨期权和看跌期权。我们定价的合约示例包括方差掉期、波动性掉期、支付已实现的夏普比率的索赔以及对杠杆式交易所交易基金的要求。