portfolio [*secondary derivatives*]{}, derivatives with the same underlying stock but different contract specifications. The immediate consequence of such constraint is that the risk premium coming from $\tau$ is arbitrary to a certain extent, because investors can evaluate it on the basis of their own perceptions. We will avoid entering into the discussion of the financial consequences of this arbitrariness now: we leave it to \[SA\], where the hedging-portfolio approach is taken. The most relevant point to be noted here is that we can use (\[PP\]) in order to define a risk-neutral measure for $S$, $$\begin{aligned}
{\mathbb{P}}\{s<S(t)\leq s +ds|{\mathcal{F}\/}(t_0)\}=\left\{
\frac{e^{-\lambda (t-t_0)}}{s \sqrt{2\pi \sigma^2_a (t-t_0)}} e^{-\frac{[\log(s/S(t_0))-\theta_a(t-t_0)]^2}{2 \sigma_a^2 (t-t_0)}} \right.\\
+\left. \int_0^{t-t_0} du  \frac{\lambda e^{-\lambda u}}{s \sqrt{2\pi \bar{\sigma}^2(u,t-t_0) (t-t_0)}} e^{-\frac{[\log(s/S(t_0))-\bar{\theta}(u,t-t_0)(t-t_0)]^2}{2  \bar{\sigma}^2(u,t-t_0)(t-t_0)}}\right\}ds,\end{aligned}$$ where we have introduced some quantities depending on the risk-free interest rate $r$, the volatilities $\sigma_{{a,b}}$, and the dividend pay-offs $\delta_{{a,b}}$: $$\begin{aligned}
\theta_{{a,b}}=r-\delta_{{a,b}}-\sigma_{{a,b}}^2/2, \\
\bar{\theta}(u,v)=\frac{\theta_{a} u+\theta_{b}(v-u)}{v} \mbox{ and} \\
\bar{\sigma^2}(u,v)=\frac{\sigma^2_{a} u+\sigma^2_{b}(v-u)}{v}.\end{aligned}$$ The use of this measure guarantees that $$F(t)=e^{-\int_{t_0}^{t}(r-\delta)dt'} S(t)
\label{defF}$$ fulfils $${\mathbb{E}}[F(t)|{\mathcal{F}\/}(t_0)]=F(t_0)=S(t_0). 
\label{martF}$$ Note that physical and risk-free measures coincide when $\tilde{\theta}_{{a,b}}=\theta_{{a,b}}$.

American option as a first passage problem {#S3}
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Options are contracts between two parties, sold by one party to another, that give the buyer the right, but not 