Path integrals are a well-known tool in quantum mechanics and statistical & mathematical physics. They could be used to derive the propagator or kernel of stochastic processes, analogous to solve the Fokker-Planck equation. In finance, they become an alternative tool to address the option pricing problem , not only in the vanilla case but also to address path- dependent options  and stochastic volatility models . Moreover, path integrals also are useful in the context of volatility derivatives . In line with the latter, we use path integrals for the pricing of variance swaps under the Constant Elasticity of Variance (CEV) model. We derive the path-dependent propagator in close-form, but also we arrive at an analytical approximation using semiclassical arguments . Alternatively, we follow the hedging formula of the realized variance by means of the log contract, obtaining its expected value in both exact closed-form and by the semiclassical approximation. Our result proves that the semiclassical method provides an alternative computation that shows a high level of accuracy but at the same time lower computational times. References  E. Bennati, M. Rosa-Clot, S.Taddei, International Journal of Theoretical and Applied Finance, 2 (1999) 381-407.  J. Devreese, D. Lemmens, J. Tempere, Physica A, 389 (2010) 780-788.  D. Lemmens, M. Wouters, J. Tempere, S. Foulon, Physical Review E, 78 (2008) 016101.  L. Liang, D. Lemmens, J. Tempere, Physical Review E, 83 (2011) 056112.  A. Araneda, M. Villena. Journal of Computational and. Applied Mathematics, 388 (2021) 113244.