The Black-Scholes semigroup is studied on spaces of continuous functions on
(
0
,
∞
)
(0,\infty )
which may grow at both 0 and at
∞
,
\infty ,
which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces
\[
Y
s
,
τ
:=
{
u
∈
C
(
(
0
,
∞
)
)
:
lim
x
→
∞
u
(
x
)
1
+
x
s
=
0
,
lim
x
→
0
u
(
x
)
1
+
x
−
τ
=
0
}
Y^{s,\tau }:=\{u\in C((0,\infty )):\;\lim _{x\rightarrow \infty } \frac {u(x)}{1+x^{s}} =0, \; \lim _{x\rightarrow 0}\frac {u(x)}{1+x^{-\tau }} =0\}
\]
with norm
‖
u
‖
Y
s
,
τ
=
sup
x
>
0
|
u
(
x
)
(
1
+
x
s
)
(
1
+
x
−
τ
)
|
>
∞
,
\left \Vert u\right \Vert _{Y^{s,\tau }}=\underset {x>0}{\sup }\left \vert \frac {u(x)}{(1+x^{s})(1+x^{-\tau })}\right \vert >\infty ,
the Black-Scholes semigroup is strongly continuous and chaotic for
s
>
1
s>1
,
τ
≥
0
\tau \geq 0
with
s
ν
>
1
s\nu >1
, where
2
ν
\sqrt 2\nu
is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.