The Black-Scholes semigroup is studied on spaces of continuous functions on $(0,\infty )$ 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 \[ $Y^{s,\tau }:=\{u\in C((0,\infty )):\;\lim _{x\rightarrow \infty } \frac {u(x)}{1+x^{s}} =0, \; \lim _{x\rightarrow 0}\frac {u(x)}{1+x^{-\tau }} =0\}$ \] with norm $\left \Vert u\right \Vert _{Y^{s,\tau }}=\underset {x>0}{\sup }\left \vert \frac {u(x)}{(1+x^{s})(1+x^{-\tau })}\right \vert >\infty ,$ the Black-Scholes semigroup is strongly continuous and chaotic for $s>1$ , $\tau \geq 0$ with $s\nu >1$ , where $\sqrt 2\nu$ is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.

Let $\Omega$ be a smooth bounded domain in $R^N$ and let \begin{eqnarray} Lu=\sum_{j,k=1}^N \partial_{x_j}\left(a_{jk}(x)\partial_{x_k} u\right), \end{eqnarray} in $\Omega$ and \begin{eqnarray} Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\partial_{\tau_k}\left(b_{jk}(x)\partial_{\tau_j}u\right)=0, \end{eqnarray} on $\partial\Omega$ define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary. Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space. If we have a sequence of such operators $S_0,S_1,S_2,...$ with corresponding coefficients \begin{eqnarray} \Phi_n=(a_{jk}^{(n)},b_{jk}^{(n)}, \beta_n,\gamma_n,q_n) \end{eqnarray} satisfying $\Phi_n\to\Phi_0$ uniformly as $n\to\infty$, then $u_n(t)\to u_0(t)$ where $u_n$ satisfies \begin{eqnarray} i\frac{du_n}{dt}=S_n^m u_n, \end{eqnarray} or \begin{eqnarray} \frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0, \end{eqnarray} or \begin{eqnarray} \frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0, \end{eqnarray} for $m=1,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$. This includes Schrödinger equations, damped and undamped wave equations, and telegraph equations.