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

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