of expected values. We have also introduced constants $\beta^{\pm}_{{a,b}}$, $$\beta^{\pm}_{{a,b}} = \frac{1}{\sigma^2_{{a,b}}}\left(-\theta_{{a,b}} \pm \sqrt{\theta^2_{{a,b}}+2 r \sigma^2_{{a,b}}}\right),
\label{betapm}$$ which differentiate the overall properties of calls and puts since $\beta^{\pm}_{{a,b}}\gtrless 0$ [@BAW87]. Note that we must eventually recover (\[Eab\]) with $\beta^{\pm}_{a}$ ($\beta^{\pm}_b$) for the limiting case $\lambda \rightarrow 0$ ($\lambda \rightarrow \infty$). When the pay-off is (\[payv\]) we will have a perpetual American vanilla option: $$V^{\pm}_b(t_0,{S_0};K)=\pm(H_b - K) \left(\frac{{S_0}}{H_b}\right)^{\beta^{\pm}_b}.$$ The value of $H_b$ shall be obtained by demanding smoothness of the solution, recall (\[smooth\]), what implies here that $${\left.\frac{\partial V^{\pm}_b(t,S;K)}{\partial S} \right|}_{S=H_b}=\pm 1,$$ and therefore [@BAW87]: $$H_b = \frac{\beta^{\pm}_b}{\beta^{\pm}_b - 1} K. \label{Hb}$$ For put options we have guaranteed $0<H_b<K$ because $\beta^{-}_b<0$. For call options it can be proved that $\beta^{+}_b \geq 1$, an therefore we will have $H_b>K$. In fact, for $\beta^{+}_b = 1$, or what is the same, for $\theta_b=r-\sigma^2_b/2$, the value of $H_b$ diverges: there is no optimal boundary, the option is never exercised and, as a consequence, it must quote as its underlying stock, $V^{+}_b(t,S(t))=S(t)$. Thus, if the volatility is constant and the stock pays no dividend, only American puts are meaningful. For them we will have $\beta^{-}_b=-2r/\sigma^2_b$, and therefore $$V^{-}_b(t_0,{S_0};K)=\frac{\sigma^2_b{S_0}}{2 r} \left(\frac{H_b}{{S_0}}\right)^{1+\frac{2r}{\sigma^2_b}},$$ with $$H_b = \frac{K}{1+\frac{\sigma^2_b}{2 r}}.$$

