We analyze the effect of discrete sampling on the valuation of options on the realized variance in the Heston stochastic volatility model. It has been known for some time that, although quadratic variance can serve as an approximation to discrete variance for valuing longer-term options on the realized variance, this approximation underestimates option values for short-term maturities (with maturities up to three months). We propose a method that involves mixing the discrete variance in a lognormal model and the quadratic variance in a stochastic volatility model. This allows us to accurately approximate the distribution of the discrete variance in the Heston model. As a result, we can apply semianalytical Fourier transform methods developed by Sepp for pricing shorter-term options on the realized variance.