$\mathbb{Q}$ and the information available at $t$ is equal a sum of integrals $I_1+\cdot\cdot\cdot+I_n$, where $$I_i:=\frac{\exp(rt-\sigma^2b_i^2T/2+b_i\sigma\sqrt{t}\cdot z_t)}{\sqrt{2\pi}}\int_{A_{i-1}}^{A_i}\exp(-y^2/2+b_i\sigma\sqrt{\tau}\cdot y)dy.$$ Evaluating these integrals, we obtain the general pricing formula $$\boxed{C(S,t)=\sum\limits_{i=1}^n\{N(A_i-\beta_i)-N(A_{i-1}-\beta_i)\}V_t(b_i)},$$where $$\boxed{A_i:=\frac{(b_i+b_{i+1})\sigma T/2-\sqrt{t}\cdot z_t}{\sqrt{\tau}}}.$$and $$\boxed{\beta_i:=b_i\sigma\sqrt{\tau}}.$$ Bearing in mind that $A_0=-\infty$ and $A_n=+\infty$, we can also write $$\begin{gathered}
\label{eq:genprice}
C(S,t)=N(A_1-\beta_1)V_t(b_1)+\sum\limits_{i=2}^{n-1}\{N(A_i-\beta_i)-N(A_{i-1}-\beta_i)\}V_t(b_i)\\
+N(\beta_n-A_{n-1})V_t(b_n).\\\end{gathered}$$The general option price could again have been obtained differently, by an interesting application of Margrabe’s theory of exchange options. Indeed, we could consider the wealth processes $(V_t(b_i))_{i=1}^n$ as separate underlyings $U_i:=V_t(b_i)$ of a multi-asset option whose final payoff is equal to $\max\{U_1,U_2,...,U_n\}$. First of all, we remark that at any given time the ordered sequence of numbers $U_1(t),...,U_n(t)$ is unimodal, or single-peaked. This happens because the $(\log U_i)_{i=1}^n$ trace out a sequence of heights on the parabola $b\mapsto\log V_t(b)$ as we move from left to right over the abscissae $b_1<b_2<\cdot\cdot\cdot<b_n$. The peak occurs for the index $$i^*:=\arg\min\limits_{1\leq i\leq n}|b_i-b(S_t,t)|=\arg\max\limits_{1\leq i\leq n}V_t(b_i).$$Thus, $U_i$ is increasing in $i$ for $i<i^*$ and decreasing in $i$ for $i\geq i^*$. This unimodality in hand, we now have the identity $$\max\{U_1,...,U_n\}=U_1+(U_2-U_1)^++(U_3-U_2)^++\cdot\cdot\cdot+(U_n-U_{n-1})^+,$$where $x^+:=\max\{x,0\}$ denotes the positive part of $x$. Hence, the payoff $\max\limits_{1\leq i\leq n}U_i$ is equivalent to a portfolio consisting of one unit of $U_1$, plus an option to 