# $\epsilon$-arbitrage model

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!