| Description |
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 [1], not only in the vanilla case but also to address path- dependent options [2] and stochastic volatility models [3]. Moreover, path integrals also are useful in the context of volatility derivatives [4]. 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 [5]. 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 [1] E. Bennati, M. Rosa-Clot, S.Taddei, International Journal of Theoretical and Applied Finance, 2 (1999) 381-407. [2] J. Devreese, D. Lemmens, J. Tempere, Physica A, 389 (2010) 780-788. [3] D. Lemmens, M. Wouters, J. Tempere, S. Foulon, Physical Review E, 78 (2008) 016101. [4] L. Liang, D. Lemmens, J. Tempere, Physical Review E, 83 (2011) 056112. [5] A. Araneda, M. Villena. Journal of Computational and. Applied Mathematics, 388 (2021) 113244.
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