Analytical Path-Integral Pricing of Moving-Barrier Options under non-Gaussian Distributions

TL;DR

This paper introduces an analytical path-integral approach for pricing barrier options under non-Gaussian distributions, extending previous models to include higher-order cumulants and drift, and calibrates it with market data.

Contribution

It generalizes existing path-integral models to account for non-Gaussian features and moving barriers, providing a new analytical pricing framework for vanilla and barrier options.

Findings

The model accurately prices vanilla options without volatility smile assumptions.

Barrier option prices from the model align well with those from the relative entropy model.

Calibration with market data demonstrates practical applicability.

Abstract

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 possible non-gaussian distributions of the underlying object. We adapt to our problem a model originally proposed to describe the formation of galaxies in the universe of De Simone et al (2011), which uses cumulant expansions in terms of the Gaussian distribution, and we generalize it to take into acount 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 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.