In American style exotic options, the holder is often faced with choices at certain times during the life span of the option. Following the/an optimal choice allows the user to maximize the value of the option. The price of the option, in my understanding, comes from assuming the decision that maximizes the conditional expectation of the payoff. How is the delta of such options usually computed? What is the relationship between the delta and the optimal choice at a given time, if there is any?

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    $\begingroup$ Usually the delta is computed by a discrete approximation. You compute (perhaps by a complicated numerical procedure) the price of the option $V_{S+h}$ assuming the stock price is $S+h$, then you separately compute the price $V_{S-h}$ assuming stock price is $S-h$ (with optimal decision made in both cases) and finally you compute $\Delta\approx \frac{V_{S+h}-V_{S-h}}{2h}$. There is no simple relation between the optimal decision and $\Delta$ AFAIK. $\endgroup$ – noob2 Jul 8 at 17:12
  • $\begingroup$ You can probably look up "free boundary problem". I forgot what the results are but it will give you the point of optimal exercise. There should be some relation between optimal decision and delta because the option at the very least should be ITM, meaning it should have delta > ~50. $\endgroup$ – confused Jul 29 at 8:23

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