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Scenario: stock trading at 100 today, 80% chance it will trade at 110 tomorrow, 20% chance it will trade at 90 tomorrow

A new 100 strike call option on this stock is worth 8 today (assuming no discounting).

Assume you buy this call option. Now, you can hedge this option by selling 50 shares today, giving a payoff of -3 tomorrow. This is clearly a poor trade.

Logically, shouldn’t the payoff of the hedged position be zero in this scenario? What am I missing here?

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You are missing the fact that the option is worth 5 not 8. You cannot use the 80/20 probabilities to value the option, you have to use the risk/neutral probabilities , which are 50/50.

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    $\begingroup$ To add, the risk neutral probabilities are calculated as $q$ and $1-q$ where $q = (S_\mathrm{now} - S_\mathrm{down})/(S_\mathrm{up} - S_\mathrm{down})$. $\endgroup$ – Bob Jansen Sep 22 '17 at 6:13

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