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The writer is selling a european call option with $K=S_{0}$, $S=S_{0}$ ($payoff_{T} = (S_{T} -k)_{+}$), time to maturity $T$, with a twist:

With some probability, $Pr(l) \geq 0,$ $\forall t,$ $0 < t < T$ the option holder may 'exercise' some portion of the option $n$, $ 0 < n <= 1$ and receive $n \cdot BS(S,T-t, \sigma , k, r)$, where $BS(S,T-t, \sigma , k, r)$ is the B-S market value of a euro call option at the time of exercise.

The question I'm wrangling with is this, I believe (and please correct me if I am wrong) is that the Risk-Neutral price of the above option is $BS(S_{0},T, \sigma , k, r)$ regardless of $l$ because a replicating portfolio would be to purchase an option and liquidate $n$ of that option every time the holder exercised.

However, does this remain true when $\sigma_{T-t,Strike-Ratio}$ has a term structure and volatility smile?

My intuition says yes, because the replicating call portfolio still holds but I wanted to confirm.

Thanks!

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    $\begingroup$ The taxology of a derivative is separate from the model you use to analyze the derivative. Term structure or not, skewness or not, vol of vol or not - the classification of the derivative rests on its contractual provisions. An example of something which is not a European vanilla option is a barrier option, like knockout or one touch. $\endgroup$
    – stans
    Aug 8, 2023 at 11:04

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If the 'exercise' means receiving the option and not its extrinsic value (such as in an American option) then yes I agree with you the replicating portfolio is just the call option. You have to be careful though that when exercised the option is not being exercised at the initial implied vol. Because if so, then in that case the replicating portfolio is not so easy as there is forward vol risk. In other words, the term sheet should be read very carefully.

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A martingale's immediate exercise value always equals the continuation value (=average value in the future) and therefore one is always indifferent to when it is stopped.

Since the BS price is a martingale this early exercise feature in not consequential.

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