I understand what an option is and I know that a market maker always publishes a bid and an ask price for which it will buy or sell options on the exchange. Now I heard that market makers always hedge their positions by buying or selling the underlying assets so that whether the market goes up or down, they always make money. And this I don't understand.
So let's go with an example. On the stock exchange
stock X is freely traded. If I am a market maker for options I would publish prices for buying and selling options. Let's say I published an ask price for a call option and somebody buys the call option. If I would not have a call option I would write one. So let's assume the numbers are as follows:
- Stock X costs $100 on the exchange at time of writing the option
- option strike price is $150
- option expiration date is 3 months from now
- I write/sell the option for $5
To hedge my option position I now buy the underlying asset X for $100. That means there are three possible situations:
- At the time of the expiration date, stock X is worth $160 (above the strike price). I sell the stock X (which I bought for $100) for $150 to the holder of the option I wrote. That means I made $5 for the option plus $50 for the price increase of X (minus the transaction costs).
- At the time of the expiration date, stock X is worth $125 (below the strike price but above the price at which I bought stock X). The holder of the option does not execute the option contract. So I sell stock X for $125 on the market. That means I made $5 for the option, plus $25 for the price increase of X (minus the transaction costs).
- At the time of the expiration date, stock X is worth $50 (below the strike price and below the price at which I bought stock X). I sell the stock X (which I bought for $100) for $50 on the market. That means I made $5 for the option minus $50 for the price decrease of X (minus the transaction costs). So in this case I actually lost money.
In case 3 above I could of course also sell the stock X if it drops below $95 and buy again if it increases above $95 again. If stock X is then $50 at the expiration date I would make no profit at all (the $5 I sold the option for is compensated by the $5 loss I made on stock X). If the price of stock X would actually pass the $95 up and down multiple times finally ending at $50, I would actually make a loss because of the transaction costs and the spread I constantly pay for buying and selling stock X at $95.
So what am I missing here? Where do I go wrong in the example I wrote? How do option market makers actually hedge their positions so that they do not have a price risk?