parameters of the problem: $$\begin{aligned}
\ell^{\pm} &=& \left(\beta^{\pm}_a-\beta^{\pm}_b\right)\left(\frac{r}{\beta^{\pm}_a}+\frac{1}{2}\sigma^2_a\beta^{\pm}_b\right), 
\label{ell}\end{aligned}$$ which cancels if no change in the market is induced by the point process $\tau$. When $H_{{a,b}}={K_0}$, as in the case of binary options, the sum of the two expectations lead to $${\mathbb{E}}[e^{-r({t^{\pm}}_0-t_0)}|{\mathcal{F}\/}(t_0)]= \frac{1}{\lambda+\ell^{\pm}} \left[\lambda\left(\frac{{S_0}}{{K_0}}\right)^{\beta^{\pm}_b}  
+\ell^{\pm}\left(\frac{{S_0}}{{K_0}}\right)^{\gamma^{\pm}_{a}} \right].\label{moment-generating}$$ It is easy to check that in the two limiting cases, $\lambda \rightarrow 0$ and $\lambda \rightarrow \infty$, we recover the right expressions. In the case in which the pay-off function is (\[payv\]) we will have $$\begin{aligned}
V^{\pm}(t_0,{S_0};K) =
\pm(H_a - K)\left(\frac{{S_0}}{H_a}\right)^{\gamma^{\pm}_{a}} \nonumber \\  \pm(H_b - K)\frac{\lambda}{\lambda+\ell^{\pm}} \left[\left(\frac{{S_0}}{H_b}\right)^{\beta^{\pm}_b} -\left(\frac{H_a}{H_b}\right)^{\beta^{\pm}_b} \left(\frac{{S_0}}{H_a}\right)^{\gamma^{\pm}_{a}} \right].
\label{Vpm}\end{aligned}$$ The value of $H_a$ must fulfil the transcendental equation that follows from the smoothness condition (\[smooth\]), $$H_a (\gamma^{\pm}_a-1)-(H_b-K)  \frac{\lambda}{\theta_a+\sigma^2_a(\gamma^{\pm}_a+\beta^{\pm}_b)/2} \left(\frac{H_a}{H_b}\right)^{\beta^{\pm}_b}-\gamma^{\pm}_a K=0, \label{HH}$$ where $H_b$ was introduced in (\[Hb\]). We have seen above how an American call option quotes as its underlying if the volatility remains constant and the share pays no dividend. This conclusion is still valid in our framework, even though $\sigma_a\neq \sigma_b$. Let us consider in the first place $\delta_a\neq0$ and $\delta_b=0$. Here $\beta^{+}_b=1$, $H_b\rightarrow \infty$, and $\ell^{+}=\delta_a$. This reduces (\[HH\]) to an algebraic equation in $H_a$, which takes the simple form: $$H_a = \frac{\gamma^{+}_a(1+\lambda/\delta_a)}{\gamma^{+}_a-1} K > K.$$ Then $$V^{+}(t_0,{S_0};H_a)= 