# Payoff of European Call Option with Transactioncosts

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?

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. • 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). – noob2 Jan 30 '17 at 17:15 • 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? – Cettt Jan 31 '17 at 6:42 • 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. – Brian B Feb 2 '17 at 17:10