binary options since there is no financial reason for holding the option alive once we are in the bonus region. This implies that $H_{{a,b}}={K_0}$ and therefore: $$D^{\pm}(t_0,{S_0};{K_0})={\mathbb{E}}[e^{-r({t^{\pm}}_0-t_0)}|{\mathcal{F}\/}(t_0)]. \label{Dpm}$$ Here it is interesting to note that the magnitude we must compute for obtaining the price of the option is nothing but a typical moment-generating function of the first-passage time interval $({t^{\pm}}_0-t_0)$: $${\mathbb{E}}[({t^{\pm}}_0-t_0)^n|{\mathcal{F}\/}(t_0)] =(-1)^n \left. \frac{\partial^n}{\partial r^n} {\mathbb{E}}[e^{-r({t^{\pm}}_0-t_0)}|{\mathcal{F}\/}(t_0)] \right|_{r=0}, (n>0). \label{moments}$$ In order to conserve this extra functionality in the output of the problem under analysis, we will keep $\theta_a$ and $\theta_b$ unexplicit as much as we can. In this way, we can turn risk-neutral results into physical results merely by replacing $\theta_{{a,b}}$ with $\tilde{\theta}_{{a,b}}$.

The constant case
-----------------

Let us consider in the first place the case in which drift and volatility are constant. For $\bar{\theta}=\theta_{{a,b}}$ and $\bar{\sigma}=\sigma_{{a,b}}$ the probability density function (pdf) of ${t^{\pm}}_0$ can be obtained by invoking the reflection principle of the Brownian motion: $$\begin{aligned}
{\mathbb{P}}\{t<{t^{\pm}}_0\leq t +dt|{\mathcal{F}\/}(t_0)\}=\\\psi^{\pm}_{{a,b}}(t;t_0,{S_0},H_{{a,b}})dt=
\frac{\pm x_{{a,b}}}{\sqrt{2\pi \sigma^2_{{a,b}} (t-t_0)^3}} e^{-\frac{(x_{{a,b}}-\theta_{{a,b}} (t-t_0))^2}{2 \sigma_{{a,b}}^2 (t-t_0)}}dt,\end{aligned}$$ where $x_{{a,b}}=\log(H_{{a,b}}/{S_0})$. Under this assumption, the value of (\[Ppm\]) is well known [@BAW87; @W98]: $$\begin{aligned}
P^{\pm}(t_0,S_0,{\mathbb{K}}) &=&X^{\pm}(H_{{a,b}};{\mathbb{K}}){\mathbb{E}_{{a,b}}}[e^{-r({t^{\pm}}_0-t_0)}|{\mathcal{F}\/}(t_0)]\nonumber \\&=&X^{\pm}(H_{{a,b}};{\mathbb{K}})\left(\frac{{S_0}}{H_{{a,b}}}\right)^{\beta^{\pm}_{{a,b}}}.
\label{Eab}\end{aligned}$$ Here, with ${\mathbb{E}_{{a,b}}}[\cdot|{\mathcal{F}\/}(t_0)]$ we mean that we are using $\psi^{\pm}_{{a,b}}(t;t_0,{S_0},H_{{a,b}})$ in the computation 