the case of binary options, $H_{{a,b}}={K_0}$. We will assume that the drift terms are both positive, $\theta_{{a,b}}>0$, and that the process starts below the barrier. Therefore, we are computing the mean lifetime of a perpetual binary call: $$\begin{aligned}
{\mathbb{E}}[t^{+}_0-t_0|{\mathcal{F}\/}(t_0)] = \frac{1}{\theta_b} \left\{x_0-\frac{\theta_a-\theta_b}{\lambda}\left[1-e^{-\frac{x_0}{\sigma^2_a}\left( \sqrt{\theta^2_a+2\lambda \sigma^2_a}- \theta_a\right)}\right] \right\}.\label{MFPT}\end{aligned}$$ Note that the result is specially well fitted for the asymptotic analysis of large values of $\lambda$. If we are interested in the case in which the likelihood of a change in the market conditions is very small the following approximate expression becomes more helpful: $$\begin{aligned}
{\mathbb{E}}[t^{+}_0-t_0|{\mathcal{F}\/}(t_0)]\approx  \frac{x_0}{\theta_a} \left\{1+ \frac{\lambda}{2}\left(\frac{1}{\theta_b}-\frac{1}{\theta_a}\right)\left[x_0+\frac{\sigma^2_a}{\theta_a}\right] \right\}. \label{MFPTb}\end{aligned}$$

We must remember that the proper way of performing the computation in (\[moments\]) is by assuming that $r$, $\lambda$, $\theta_a$, $\theta_b$, $\sigma_a$ and $\sigma_b$ are free parameters. Once we have formally set $r=0$ and obtained (\[MFPT\]) we have to recall that this quantity must be evaluated by using the physical not the risk-neutral measure. Therefore we will replace $\theta_{{a,b}}$ with $\tilde{\theta}_{{a,b}}$.

In figure \[figMFPT\] we have depicted the mean lifetime of binary calls under the following market conditions: $\tilde{\theta}_a=1.5\%$, $\tilde{\theta}_b=3\%$ and $\sigma_a=10\%$. Obviously the fact that the drift is bigger after the transition reduces the mean time as the likelihood of the 