This paper presents closed-form analytical formulas for pricing volatility and variance derivatives with nonlinear payoffs under discrete-time observations. The analysis is based on a probabilistic approach assuming that the underlying asset price follows the Schwartz one-factor model, where the volatility of log-returns is time-varying. A difficult challenge in this pricing problem is to solve an analytical formula under the assumption of time-varying log-return volatility, resulting in the realized variance being distributed according to a linear combination of independent noncentral chi-square random variables with weighted parameters. By utilizing the probability density function, we analytically compute the expectation of the square root of the realized variance and derive pricing formulas for volatility swaps. Additionally, we derive analytical pricing formulas for volatility call options. For the payoff function without the square root, we also derive corresponding formulas for variance swaps and variance call options. Additionally, we study the case of constant log-return volatility; simplified pricing formulas are derived and sensitivity with respect to volatility (vega) is analytically studied. Furthermore,we propose simple closed-form approximations for pricing volatility swaps under the Schwartz one-factor model. The accuracy and efficiency of the proposed methods are demonstrated through Monte Carlo simulations, and the impact of price volatility and the number of trading days on fair strike prices of volatility and variance swaps is investigated across various numerical experiments.