In this paper we present an analytical formula for pricing discretely-sampled variance swaps with the realized variance being defined in terms of squared log return of the underlying asset. The dynamics of the underlying asset price follows the Schwartz’s two-factor model which can be used to describe commodity prices and allows the convenience yields to be stochastic. A partial differential equation formulated for the pricing $n$^th moment swaps is analytically solved in real space to obtain our solution as a special case for variance swaps when $n=2$. Interestingly, we successfully manage to establish an interrelationship equation for variance swap prices and futures prices that would be beneficial for market practitioners who prefer to hedge price volatility risk using futures contracts. We further discuss the validity of our solution as well as propose a methodology to characterize a feasible parameter subspace, ensuring the model parameters estimated from market data produce finiteness and strict positiveness of variance swap prices when they are applied to our solution. We also demonstrate that our solution approach is quite versatile and can be adopted for pricing new generation of variance and volatility derivatives as well. Numerical tests are provided to confirm the correctness and efficiency of our solution as well as investigate how sensitive our solution is to the change of the model parameters. Finally, we apply our solution to quantify variance risk premia in gold using historical price data of gold futures obtained from the Thailand Futures Exchange.

中文翻译：

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随机便利收益率下大宗商品衍生产品市场的价格变动掉期分析

在本文中，我们提供了一种用于对离散采样方差掉期进行定价的解析公式，其中，已实现方差是根据基础资产的对数收益平方确定的。基础资产价格的动态遵循Schwartz的两因素模型，该模型可用于描述商品价格并允许便利收益率是随机的。为定价$ n $ ^th制定的偏微分方程当n = 2 $时，矩交换在现实空间中被解析求解，以获得我们的解，作为方差交换的特例。有趣的是，我们成功地为差异掉期价格和期货价格建立了一个相互关系方程，这对于希望使用期货合约对冲价格波动风险的市场从业者来说将是有益的。我们进一步讨论了我们的解决方案的有效性，并提出了一种表征可行参数子空间的方法，以确保根据市场数据估算的模型参数在应用于我们的解决方案时能够产生方差掉价的有限性和严格的正性。我们还证明，我们的解决方案方法用途广泛，也可用于为新一代方差和波动率衍生产品定价。提供数值测试来确认我们的解决方案的正确性和效率，并研究我们的解决方案对模型参数变化的敏感程度。最后，我们使用从泰国期货交易所获得的黄金期货的历史价格数据，将我们的解决方案应用于量化黄金的方差风险溢价。