\frac{{S_0}}{1+ \lambda/\delta_a} \left[\frac{1}{\gamma^{+}_a}\left(\frac{{S_0}}{H_a}\right)^{\gamma^{+}_{a}-1} +\frac{\lambda}{\delta_a}\right]<{S_0},$$ for ${S_0}\leq H_a$. If we consider now the limit $\delta_a\rightarrow 0$ we can see that we recover $H_a\rightarrow \infty$ and the option is never exercised also in this case. The opposite situation, in which $\delta_a=0$ but $\delta_b\neq0$, is much more complex. We have evidences pointing to the conclusion that $H_a\rightarrow \infty$ for any combination of the remaining parameters —see \[SB\]. This implies that the stock must pay dividends after the change or in both periods in order to differ call options from their underlying with certainty.

In the case of American puts we can still consider volatility and dividend risks separately: $H_b\neq 0$ and it is bounded since $\beta^{+}_b<0$, and $H_a=0$ does not solve equation (\[HH\]) for $\lambda \neq 0$. Moreover, when $\lambda =0$ the expression for $H_a$ trivially reduces to: $$H_a = \frac{\beta^{\pm}_a}{\beta^{\pm}_a - 1} K.
\label{Ha}$$ For vanilla put options, as well as for call options with $\beta^{+}_b\neq 1$, the most relevant feature of transcendental equation (\[HH\]) is the evident fact that it formally admits multiple solutions. A closer analysis shows that (\[HH\]) may have two real and positive solutions [*at the most*]{}. In \[SB\] we explain how in such a case we can 