Analytical Path-Integral Pricing of Deterministic Moving-Barrier Options Under Non-Gaussian Distributions
Electronic version of an article published as [International Journal of Theoretical and Applied Finance, Volume 23, Number 1, 2020, 52 pages] [DOI/10.1142/S0219024920500053] © [copyright World Scientific Publishing Company]
Posted: 1 Feb 2021
Date Written: February 14, 2020
In this work, we present an analytical model, based on the path-integral formalism of statistical mechanics, for pricing options using first-passage time problems involving both fixed and deterministically moving absorbing barriers under possibly non-Gaussian distributions of the underlying object. We adapt to our problem a model originally pro- posed by De Simone et al. (2011) to describe the formation of galaxies in the universe, which uses cumulant expansions in terms of the Gaussian distribution, and we generalize it to take into account drift and cumulants of orders higher than three. From the probability density function, we obtain an analytical pricing model, not only for vanilla options (thus removing the need of volatility smile inherent to the Black & Scholes (1973) model), but also for fixed or deterministically moving barrier options. Market prices of vanilla options are used to calibrate the model, and barrier option pricing arising from the model is compared to the price resulted from the relative entropy model.
Keywords: Non-Gaussian distribution, stochastic processes, first-passage time, moving barrier, Black & Scholes model, cumulant expansion, path integral, Breeden–Litzenberger theorem, relative entropy, Gram–Charlier expansion, Edgeworth expansion
JEL Classification: C73, G12
Suggested Citation: Suggested Citation