We propose robust numerical algorithms for pricing discrete variance options and volatility swaps under general time-changed Lèvy processes. Since analytic pricing formulas of these derivatives are not available, some of the earlier pricing methods use the quadratic variation approximation for the discrete realized variance. While this approximation works quite well for long-maturity options on discrete realized variance, numerical accuracy deteriorates for options with low frequency of monitoring and/or short maturity. To circumvent these shortcomings, we construct numerical algorithms that rely on the computation of the moment generating function of the discrete realized variance under the time-changed Lèvy models. We adopt the randomization of the Laplace transform of the discrete log return with a standard normal random variable and develop a recursive quadrature algorithm to compute the moment generating function of the discrete realized variance. Our pricing approach is rather computationally efficient when compared with the Monte Carlo simulation and works particularly well for discrete realized variance and volatility derivatives with low frequency of monitoring and/or short maturity. The pricing properties of various variance and volatility derivatives under various time-changed Lèvy processes and the Heston model are also investigated.