exchange $U_1$ for $U_2$, plus an option to exchange $U_2$ for $U_3$, $\cdot\cdot\cdot$ , plus an option to exchange $U_{n-1}$ for $U_n$. At expiration, the trader keeps exchanging $U_{i}$ for $U_{i+1}$ until the maximum $U_i^*$ is reached. Applying the Margrabe Formula (cf. Zhang 1998) in conjunction with linear pricing, we find that the no-arbitrage price of this portfolio (consisting of a unit of $U_1$ plus $n-1$ exchange options) is $$\label{easy}
\boxed{C(U_1,...,U_n,t)=U_1+\sum\limits_{i=1}^{n-1}\{N(d_{1i})U_{i+1}-N(d_{2i})U_i\}},$$where $$\boxed{d_{1i}:=\frac{\log(U_{i+1}/U_i)}{\sigma_{ai}\sqrt{\tau}}+\frac{\sigma_{ai}\sqrt{\tau}}{2}},$$$d_{2i}:=d_{1i}-\sigma_{ai}\sqrt{\tau}$, and $\sigma_{ai}:=\Delta b_i\sigma$ is the aggregate volatility in a two-asset market consisting of $U_i$ and $U_{i+1}$. Collecting terms, we get the linear combination $$\label{combo}
\boxed{C(U_1,...,U_n,t)=N(-d_{21})U_1+\sum_{i=2}^{n-1}[N(d_{1,i-1})-N(d_{2i})]U_i+N(d_{1,n-1})U_n},$$which agrees with equation (\[eq:genprice\]) above. Figure \[payoff2\] plots the option price and intrinsic value for different stock prices under the parameters $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7,$ and $\mathbb{B}:=\{0,0.5,1,1.5,2\}.$ In general there will be several implied volatilities $\sigma$ that could rationalize an observed price of the rebalancing option. Figure \[vol\] plots the option price for different volatilities under the parameters $r:=0.03, T:=10, S_0:=100, t:=5, S_t:=200,$ and $\mathbb{B}:=\{0,0.5,1.5\}$.

![Option price and intrinsic value for different stock prices, $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, \mathbb{B}:=\{0,0.5,1,1.5,2\} $.[]{data-label="payoff2"}](generaliv.png){width="300px"}

![Option prices for different volatilities, $r:=0.03, T:=10, S_0:=100, t:=5, S_t:=200, \mathbb{B}:=\{0,0.5,1.5\} $.[]{data-label="vol"}](vol.png){width="300px"}

In specializing the pricing formula for $t:=0$ and simplifying 