Payoff of European Call Option with Transactioncosts

Question

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?

Answer 1

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.