Introducing and solving generalized Black-Scholes PDEs through the use of functional calculus

TL;DR

This paper introduces generalized Black-Scholes equations involving fractional derivatives, proves their well-posedness in certain function spaces, and provides explicit solutions using advanced operator theory.

Contribution

It extends classical Black-Scholes models by incorporating fractional derivatives and establishes well-posedness via novel operator-theoretic connections.

Findings

Generalized Black-Scholes equations are well-posed in $(L^1-L^inity)$ spaces.

Explicit integral solutions are derived for these equations.

A new connection between bisectorial and sectorial operators is established.

Abstract

We introduce some families of generalized Black--Scholes equations which involve the Riemann-Liouville and Weyl space-fractional derivatives. We prove that these generalized Black--Scholes equations are well-posed in $(L^1-L^\infty)$-interpolation spaces. More precisely, we show that the elliptic type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial operators and sectorial operators in an abstract setting. Such a connection extends some known results in the topic to a wider family of both operators and the functions involved.