In the model here described, Bertsimas says that we can use the Robust Optimization to find the replicating portfolio the value of which is such that minimize the difference $|P(\widetilde{S},K)-W_T|=\epsilon$ in the face of all possible realizations of returns of the underlying included in the uncertainty set $U\in \mathbb{R}^L$ (with $P(\widetilde{S},K)$ and $W_T$ the payoff of the option and the payoff of the portfolio, respectively). Thus the problem (8) at page 845.
Now my doubt. He says that the present value of $W_T$ is the value of option. I quote: "The price of the option would thus be the initial value of this replicating portfolio. [...] After finding the portfolio, the price of the option would then be given by $x_0^S + x_0^B$, which is the value of the portfolio at time $t=0$." (page 845, paragraph 1-2). But from replication theory I know that the value in $T$ of a replicating portfolio coincides with its current value without the fair price of the option. For example:
Given $r=0.12$ and $T=3$ months, and knowing that $S_T^+=21$ for probability $p$ and that $S_T^-=18$ and probability $1-p$, a European Call with strike $K=20$ and $S_0=21$ has a replicating portfolio the value of which of $e^{0.12\cdot \frac{3}{12}}4.5=4.367$ (for $\Delta=0.25$ stocks) coincides with its current value of $20\cdot 0.25=5$ without the fair price of the option, i.e. $4.367=5-c \Rightarrow c=0.633$.
Instead, as I understand, for Bertsimas the price of option should be directly $4.367$ and not $0.633$. Is it possible that the author might be confusing payoff and price of an option? What am I missing?
Thanks in advance for any help!
