always discard one them eventually. We also find the right expression in a practical situation involving put options. The most surprising fact is that there are market conditions for which equation (\[HH\]) has no real solution. Unfortunately we have found no financial reason that motivates this behaviour. In the next section we will consider a sample case that exhibits this handicap but shows no other evident singularity.

Finally note that the above pricing expressions assume that $\gamma^{\pm}_a\neq \beta^{\pm}_b$. In order to obtain the right limiting values when a numerical concordance in the parameters lead to $\gamma^{\pm}_a = \beta^{\pm}_b$, we have to take into account that $\gamma^{\pm}_a \rightarrow \beta^{\pm}_b$ implies that $\ell^{\pm}\rightarrow -\lambda$ too. In such a case one can easily check that, for instance, the solution in (\[Vpm\]) must be replaced by: $$\begin{aligned}
V^{\pm}(t_0,{S_0};H_a,K)=\\ \pm\left[(H_a - K) \left(\frac{{S_0}}{H_a}\right)^{\beta^{\pm}_b} -(H_b - K)\frac{\lambda}{\theta_a+\sigma^2_a \beta^{\pm}_b}\left(\frac{{S_0}}{H_b}\right)^{\beta^{\pm}_b} \log\left(\frac{{S_0}}{H_a}\right)\right].\end{aligned}$$

Some examples and applications {#S5}
==============================

In this section we present some practical examples that may illustrate the consequences of volatility and dividend risk in the properties of perpetual American options. We will consider, in the first place, the pay-off (\[payv\]) and analyse the outcomes related to the price.

[![Option prices for a perpetual vanilla call under 