Variance Swaps under Multiscale Stochastic Volatility of Volatility

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.