Generalized measure Black-Scholes equation: Towards option self-similar pricing

TL;DR

This paper extends the Black-Scholes model by incorporating investor uncertainty through a measure, using advanced mathematical tools to establish well-posedness and analyzing self-similar cases numerically.

Contribution

It introduces a generalized Black-Scholes framework that accounts for uncertainty via measures, employing Dirichlet forms and PDE theory for rigorous analysis.

Findings

Well-posedness of the generalized model established

Numerical analysis conducted for self-similar measures

Model captures investor uncertainty effects in option pricing

Abstract

In this work, we give a generalized formulation of the Black-Scholes model. The novelty resides in considering the Black-Scholes model to be valid on 'average', but such that the pointwise option price dynamics depends on a measure representing the investors' 'uncertainty'. We make use of the theory of non-symmetric Dirichlet forms and the abstract theory of partial differential equations to establish well posedness of the problem. A detailed numerical analysis is given in the case of self-similar measures.