Pricing forward-start variance swaps with stochastic volatility

In this paper, a general approach is presented to price forward-start variance swaps with discrete sampling times, based on the Heston (1993)’s two-factor stochastic volatility model. Using this approach we work out two analytical closed-form formulae for the price of forward-start variance swap with the realized variance being defined by the actual-return realized variance and the log-return realized variance, respectively. The main features of this new approach and the developed formulae as a result include: (1) treating the pricing problem of variance swaps with different definitions of discretely-sampled realized variance in a highly unified way; (2) easily obtaining analytical closed-form solutions for forward-start variance swaps with two popularly-used definitions of discretely-sampled realized variance; (3) enabling the investigation of some important properties of the forward-start variance swaps, utilizing the elegant and simple form of the obtained solutions. With these advantages, we believe that the approach can be applied to price variance swaps based on other stochastic volatility models as well.