Summary: The Black-Scholes semigroup is studied on spaces of continuous functions on which may grow at both 0 and at which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces
with norm , the Black-Scholes semigroup is strongly continuous and chaotic for , with , where is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion [G. Godefroy and J. H. Shapiro, J. Funct. Anal. 98, No. 2, 229–269 (1991; Zbl 0732.47016)].