Robust Replication and Option Pricing with Epsilon Error
Summary
The document asks how to interpret option pricing in an epsilon-arbitrage framework based on robust optimization. The proposed approach searches for a portfolio whose terminal payoff stays within a specified error of the option payoff across realizations in an uncertainty set. The questioner compares this with replication intuition, in which a portfolio's current cost and the option's price must be related through the financing needed to construct the hedge, and wonders whether the cited discussion confuses payoff with price.
No answer or resolution is included, so the document does not establish which interpretation is correct. It provides a small numerical illustration involving a European call, a stock, and a risk-free rate, but this example alone does not settle how the robust model defines its replicating portfolio or its initial cost. The key learning value is the distinction between terminal payoff matching, initial portfolio value, and the option price, alongside the need to check a model's assumptions and financing conventions.
Key ideas
- Robust replication seeks a portfolio whose payoff approximates an option across outcomes in a specified uncertainty set.
- A portfolio's terminal payoff and its initial value are different quantities in option pricing.
- The question raises how the initial cost of a robust replicating portfolio relates to the option price.
- The document contains no answer, so its pricing interpretation remains unresolved.
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Full text
# $\epsilon$-arbitrage model
# $\epsilon$-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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.