change increases.

[![Mean exercise time of a perpetual binary call under drift risk. We represent the mean lifetime of the option, in terms of the moneyness (${S_0}/{K_0}$), for different values of $\lambda$. The values for the parameters are: $\tilde{\theta}_a=1.5\%$, $\tilde{\theta}_b=3\%$ and $\sigma_a=10\%$.[]{data-label="figMFPT"}](figure4.eps "fig:"){width="60.00000%"}]{}

From (\[MFPT\]) it becomes evident that the drift of the process is determinant in computing the mean first-passage time, whereas the volatility plays a marginal role: in fact the value of $\sigma_b$ appears only through $\tilde{\theta}_b$. This outcome is no longer true if we focus our interest on the second moment of the first-passage time. For the sake of simplicity we will assume that $\tilde{\theta}_a=\tilde{\theta}_b=\tilde{\theta}$, but with $\sigma_a\neq\sigma_b$. We can observe how, even under our previous assumptions, the second moment depends explicitly on the value of $\sigma_a$ and $\sigma_b$: $$\begin{aligned}
{\mathbb{E}}[(t^{+}_0-t_0)^2|{\mathcal{F}\/}(t_0)] = \\
\frac{1}{\tilde{\theta}^2}\left\{x_0\left[x_0+\frac{\sigma^2_b}{\tilde{\theta}}\right]-\frac{\sigma^2_b-\sigma^2_a}{\lambda}\left[1-e^{-\frac{x_0}{\sigma^2_a}\left( \sqrt{\tilde{\theta}^2+2\lambda \sigma^2_a}- \tilde{\theta}\right)}\right] \right\},\label{VFPT}\end{aligned}$$ an expression that reduces to $$\begin{aligned}
{\mathbb{E}}[(t^{+}_0-t_0)^2|{\mathcal{F}\/}(t_0)]\approx  \frac{x_0}{\tilde{\theta}^2} \left[x_0+\frac{\sigma^2_a}{\tilde{\theta}}\right]\left\{1+ \frac{\lambda}{2}\left(\frac{\sigma^2_b-\sigma^2_a}{\tilde{\theta}^2}\right) \right\}, \label{VFPTb}\end{aligned}$$ if we are concerned about small values of $\lambda$. In fact, if we confront (\[MFPTb\]) and (\[VFPTb\]) we realize that the first correction to the mean first-passage time is governed by the undisturbed second moment of the process.

[![Standard deviation of the exercise time of a perpetual 