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I am on part c), and I want to know:

Since I am the seller of this call option, I sold it for 0.44, and the option is worth 1 dollar at expiration. Assuming the buyer wants to make profit he will exercise this option to receive the stock worth 19 dollars (he makes a profit of 56 cents.) what does this mean for me?

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That means you have to hedge your option using the underlying stock and cash. If you correctly hedge at the time of the sale, it is irrelevant what the price will be at expiration. But what portfolio do you need to acquire to hedge the option properly?

EDIT Hedging an instrument (e.g. hedging an option) means acquiring a portfolio which exactly reproduces the payoff of that instrument in all market conditions. This is also sometimes called replicating and the resulting portfolio is called the replicating portfolio.

Then, the arbitrage-free price of the instrument becomes the price of the replicating portfolio.

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what does "hedge" mean? –  terrible at math Feb 13 at 5:03
    
@terribleatmath edited –  gt6989b Feb 13 at 5:09
    
so many terms to understand.. this is the difficulty of this area of math. as the writer of the stock, does this mean I also have the ability to distribute the stocks, so i can make money off of them? Im familair with the buyer position but not the seller. I also dont know what you mean by "portfolio do you need to acquire" –  terrible at math Feb 13 at 5:16
    
my book says this: gyazo.com/93d9e32e49e395afb0ceba7934ac0f49.png why the heck would you ever write a call option then, if your max payoff is 0? –  terrible at math Feb 13 at 5:31
    
The most common reason to do something you don't want to do: someone pays you to do it –  Callus Feb 13 at 6:06
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