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I was wondering about the following scenario: assume that you have a underlying which trades under a positive bid-ask spread $S^B \leq S^A$ and that there is also a European Call-Option on this underlying available.

Now assume that at maturity we have that $S^B=90\$, S^A=110\$$ and the strike of the option is given by $K=100\$$. Is this option exercised?

On the one hand, the option gives me the right to buy the underlying for $100\$$ (instead of $S^A=110\$$) so the owner of the option would exercise the option and one could argue that the payoff at maturity is given by $(S^A-K)^+$.

On the other hand, if the owner of the option does not want to possess the underlying, then there is no reason to exercise the option: because once he/she pays $100\$$ to buy the underlying, he/she would only get back $S^B=90\$$ after selling the underlying. Thus, one could argue that the payoff at maturity is given by $(S^B-K)^+$.

I realize that the question might not have only true answer. But I would be interested in how this is handled in practice: is there something like a mid-price $S^M \in [S^B,S^A]$ which determines the pay-off, i.e. is the payoff at maturity given by $(S^M-K)^+$? Or is there a completly different approach?

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First let's note that in practice exercise notice (of US equity options) is given after the end of the trading day, when we may have a bid and offer coming in for after-hours trading with very wide spread. That makes your example fairly important.

In the situation you cite, where the bid and ask are $S^B=90\$$ and $S^A=110\$$, the true "fair" mid-market price of the underlying could really be anything in between, so there is no universal prescription for the exercise decision.

It is common to consider the mid-market price $S^M$ to be some combination of the close price (when there was presumably significant volume to help trust it) and the present after-hours mid-price (which is likely to be based on extremely thin markets). Portfolio risk calculations might also alter this subjective mid-price.

As you see, once the option holder has decided on a subjective $S^M$, the subjective exercise value is then $(S^M-K)$ and the option holder will consider themselves to have realized $(S^M-K)^+$ in profit, though only approximately. The true eventual profit can only be determined once the shares have been sold.

Interesting note: This contrasts with options on equity futures contracts, where settlement is into cash and therefore exercise value is far more objective.

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  • $\begingroup$ A small quibble about equity futures contracts: if the futures contract is expiring (i.e. Mar,June,Sep,Dec) then the option settles in cash, but in other months (i.e. non quarterly expiration) it settles into cash plus a futures contract for which there is price uncertainty and you have to wait for trading to resume to get out of it. (So there are a few anxious hours there ;) as you await the reopen). $\endgroup$ – noob2 Jan 30 '17 at 17:15
  • $\begingroup$ Thank you for your answer. I have One more question: in the example I stated above, if the options were cash settled what would the objective payoff be? $\endgroup$ – Cettt Jan 31 '17 at 6:42
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    $\begingroup$ Any cash-settled option has an objective payoff same as the cash-settled futures options (so long as the settlement currency is the final currency you care about). For calls it is exactly equal to $(P-K)^+$ where $P$ is the contractual settlement price. Oh and thanks to @noob2 for the erratum. $\endgroup$ – Brian B Feb 2 '17 at 17:10

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