Convergence of the discrete variance swap in time-homogeneous diffusion models

Abstract

In stochastic volatility models based on time-homogeneous diffusions, we provide a simple necessary and sufficient condition for the discretely sampled fair strike of a variance swap to converge to the continuously sampled fair strike. It extends Theorem 3.8 of Jarrow et al. [Discretely sampled variance and volatility swaps versus their continuous approximations. Financ. Stoch., 2013, 17(2), 305–324]. In particular, it gives an affirmative answer to a problem posed in that paper in the case of the stochastic volatility model. We also give precise conditions (not based on asymptotics) when the discrete fair strike of the variance swap is higher than the continuous one and discuss the convex order conjecture proposed by Griessler and Keller-Ressel [Convex order of discrete, continuous and predictable quadratic variation and applications to options on variance. SIAM J. Financ. Math., 2014, 5(1), 1–19] in this context.