In this paper, we introduce novel analytical solutions for valuating volatility derivatives, including volatility options and capped/floored volatility swaps, employing discrete sampling within the framework of the Merton jump-diffusion model, which is driven by a nonhomogeneous Poisson process. The absence of a comprehensive understanding of the probability distribution characterizing the realized variance has historically impeded the development of a robust analytical valuation approach for such instruments. Through the application of the cumulative distribution function of the realized variance conditional on Poisson jumps, we have derived explicit expectations for the derivative payoffs articulated as functions of the extremum values of the square root of the realized variance. We delineate precise pricing structures for an array of instruments, encompassing variance and volatility swaps, variance and volatility options, and their respective capped and floored variations, alongside establishing put-call parity and relationships for capped and floored positions. Complementing the theoretical advancements, we substantiate the practical efficacy and precision of our solutions via Monte Carlo simulations, articulated through multiple numerical examples. Conclusively, our analysis extends to the quantification of jump impacts on the fair strike prices of volatility derivatives with nonlinear payoffs, facilitated by our analytic pricing expressions.

中文翻译：

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在非齐次泊松过程驱动的 Merton 跳跃扩散模型下的离散观察情况下，对波动率期权和具有非线性收益的封顶/下限波动率掉期进行分析定价


在本文中，我们介绍了用于评估波动率衍生品的新型分析解决方案，包括波动率期权和上限/下限波动率掉期，在由非齐次泊松过程驱动的 Merton 跳跃扩散模型的框架内采用离散采样。缺乏对表征已实现方差的概率分布的全面理解，历来阻碍了此类工具稳健分析估值方法的发展。通过应用以泊松跳跃为条件的已实现方差的累积分布函数，我们得出了对导数收益的明确期望，这些导数收益被表述为已实现方差平方根的极值的函数。我们为一系列工具描绘了精确的定价结构，包括方差和波动率掉期、方差和波动率期权，以及它们各自的上限和下限变化，同时为上限和下限头寸建立看跌期权平价和关系。作为理论进步的补充，我们通过蒙特卡洛模拟证实了我们解决方案的实际有效性和精度，并通过多个数值示例进行了阐述。总之，我们的分析扩展到量化具有非线性收益的波动性衍生品对公平执行价格的跳跃影响，这得益于我们的分析定价表达式。