Generalized measure Black–Scholes equation: towards option self-similar 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.