Many hedge funds and retail investors demand volatility and variance derivatives in order to manage their exposure to volatility and volatility-of-volatility risk associated with their trading positions. The Heston model is a standard popular stochastic volatility model for pricing volatility and variance derivatives. However, it may fail to capture some important empirical features of the relevant market data due to the fact that the elasticity of volatility of volatility of the underlying price takes a special value, i.e., 1/2, whereas it has a merit of analytical tractability. We exploit a multiscale stochastic extension of volatility of volatility to obtain a better agreement with the empirical data while taking analytical advantage of the original Heston dynamics as much as possible in the context of pricing discrete variance swaps. By using an asymptotic technique with two small parameters, we derive a quasi-closed form formula for the fair strike price of variance swap and find useful pricing properties with respect to the stochastic extension parameters.

许多对冲基金和散户投资者需要波动率和方差衍生工具，以管理其承受的波动性以及与交易头寸相关的波动性风险。Heston模型是用于定价波动率和方差导数的标准流行随机波动率模型。但是，由于标的价格的波动率的弹性具有特殊值（即1/2），因此它可能无法捕获相关市场数据的一些重要的经验特征，而它具有分析可控性的优点。 。我们利用波动率的多尺度随机扩展来获得与经验数据更好的一致性，同时在定价离散方差掉期的背景下尽可能多地利用原始Heston动力学的分析优势。