for different stock prices, $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, b:=1.5, c:=0.5$.[]{data-label="fig:comparison"}](comparison.png){width="325px"}

![Option price and intrinsic value for different stock prices, $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, b:=1.5, c:=0.5$.[]{data-label="fig:iv"}](iv.png){width="300px"}

![Performance simulation over $T:=30$ years for the parameters $S_0:=1, b:=2, c:=0.5, r:=0.03, \sigma:=0.15, \nu:=0.1, \mu=\nu+\sigma^2/2.$[]{data-label="fig:sim"}](sim1.png){width="425px"}

![Performance simulation over $T:=30$ years for the parameters $S_0:=1, b:=2, c:=0.5, r:=0.03, \sigma:=0.7, \nu:=0.07, \mu=\nu+\sigma^2/2.$[]{data-label="fig:sim2"}](sim2.png){width="425px"}

The General Discrete Set of Asset Allocations {#several}
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We carry on with the general discrete set $\mathbb{B}:=\{b_1,...,b_n\}\subset\mathbb{R}$ of asset allocations, where the $b_i$ are arranged in increasing order: $b_1<b_2<\cdot\cdot\cdot<b_n$. Thus, we now deal with the payoff $\ V_t^*:=\max\limits_{1\leq i\leq n}V_t(b_i)$. For notational convenience, we will also write $b_0:=-\infty$ and $b_{n+1}:=+\infty$. We let $\Delta b_i:=b_{i+1}-b_{i}$ for $0\leq i\leq n$, and thus $\Delta b_0=\Delta b_{n}=+\infty$. For a given rebalancing rule $b_i\,(1\leq i\leq n)$, the final payoff of the option is equal to $V_T(b_i)$ if and only if $$\frac{b_{i-1}+b_{i}}{2}\sigma\sqrt{T}\leq z_T\leq \frac{b_i+b_{i+1}}{2}\sigma\sqrt{T}.$$ Thus, conditional on the values of time-$t$ variables, $b_i$ will turn out to be the best performer over $[0,T]$ if and only if $$\frac{(b_{i-1}+b_{i})\sigma T/2-\sqrt{t}\cdot z_t}{\sqrt{\tau}}\leq y\leq \frac{(b_i+b_{i+1})\sigma T/2-\sqrt{t}\cdot z_t}{\sqrt{\tau}}.$$ For simplicity, we will write this interval as $y\in[A_{i-1},A_i]$. Thus $A_0=-\infty$ and $A_n=+\infty$. The expected present value of the final payoff with respect to 