In this paper, we discuss and value variance, volatility, covariance, and correlation swaps in the Vasicek, Schwartz one-factor, and Heston models using a continuous-time regime. The data used is primarily 2019 natural gas and crude oil futures closing prices due to the liquidity and size of the options market in the commodity energy sector. We derive approximations for covariance and correlation swap fair strikes in the Heston model following the continuous time regime, using the discrete regime for reference. We check the accuracy of our approximation using simulated error distributions of the calibrated parameters from the CIR component of the Heston model. We present the effect of varied parameters on the value of the fair strikes for covariance and correlation swaps. Finally, we evaluate the fair strikes of covariance and correlation swaps using three different approximations, yielding values and error bounds of dramatically varying sizes, demonstrating the limitations of the GARCH(1,1) calibration of the Heston model.