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This is probably a very easy question but I am new to the field and couldn't find an answer.

Assuming that I am building a hedged portfolio with a long option and going short delta on the underlying. Then, if I understood correctly tomorrow's value of the portfolio will be the same as today's. (Please correct if this is a wrong understanding.)

My question is: If tomorrow's value is the same as today's (and by extension the day after tomorrow and so on until the option expires) how does this strategy make profits?

Thank you

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An Investment Bank earns a profit by selling you an option at a slightly higher price than the theoretical price, or buying it back from you at a slightly lower price. They call this "earning a spread". Then they hedge the option, so as not to make any [further] gains or losses on it (other than the risk free rate).

Another way they could earn a profit is if they have a more accurate estimate of volatility than other people have. But that is not easy to do consistently.

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A perfectly hedged portfolio should not make any profits different from the risk free interest rate. However, you won't be able to hedge perfectly in the real world. Delta hedging for example requires continous trading and adjusting (this is one way to derive the black -scholes formula: thex hedge the stock perfectly and therefore obtain a risk -free rate deterministic return) - continous trading can not be realized. Furthermore trading costs will distort your return. Therefore, you will not obtain a perfect hedge and leave room for making profits (and loses )

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