Closed-Form Variance Swap Prices under General Affine GARCH Models and Their Continuous-Time Limits

In this paper, we derive fully explicit closed-form expressions for the fair strike prices of discrete-time variance swaps under general affine GARCH type models that have been risk-neutralized with a family of variance dependent pricing kernels. The methodology relies on solving differential recursions for the coefficients of the joint cumulant generating function of the log price and the conditional variance processes. An alternative derivation is provided in the case of Gaussian innovations. Using standard assumptions on the asymptotic behavior of the GARCH parameters as the sampling frequency increases, we derive the diffusion limit of a Gaussian GARCH model and we further investigate the convergence of the variance swap prices to its continuous-time limit. Numerical examples on the term structure of the variance swap rates and on the convergence results are also presented