I learned how to price an European call option using this [video lecture][1]. The considered case is very simple. The call option gives the right to buy 100 Euros for 100 Dollars in one month from now. Now the 100 Euros cost 100 Dollars but in 1 month they will cost either 105 Euros or 95 Euros (the probability of both prices is the same). The proposed strategy is the following: We, as the option seller, sell the option for 2.97 Dollars, then we take a 47.03 Dollars loan from the bank (the interest rate of the return is 1% for 1 month). We use these 50 Dollars (47.03 + 2.97) to buy 50 Euros. This procedure guarantees that in one month we will not lose/earn anything independently on the price of the Euros.

Why do we need to take a loan? To me it seems as non beneficial operation. We take 47.03 dollars and we need to return 47.5 Dollars. So, in total we loose 47 cents independently on the price of the Euro.

I would propose another strategy. We use our own money (47.03 Dollars) to buy the 50 Euros (we also use 2.97 Dollars paid for the option). In this case, in one month, we should not return anything to the bank. As a result, we can earn 47 cent independently on the price of the Euros.

You could say that the second strategy is worse because, if we pay our own 47.03 Dollars, we cannot use this money for a whole month. So, the money will not earn an interest. But I do not agree since we can consider this as an investment. We use 47.03 Dollars, to earn 47 in one month. So, the interest rate is 1%. You could say, that the bank provide the same interest rate, so we could use the first strategy, and then use the saved 47.03 Dollars to put them into the bank and earn the same 47 cents. OK. Here I would agree with you.

But what if we use the second strategy (we take no loan to buy the 50 Euros). Since we do not need to return a larger amount of moneys to the bank, we can make our option cheaper. It could cost 2.50 instead of 2.97. So, if we set the price equal to 2.60, we will get 10 cent from every option (independently on the price of the Euro). And since our option is cheaper than the options provided by other companies, we can sell a lot of options. Yes, we will not earn 47 cents from the option, we will earn only 10 cents (about 5 times less) but may be we will sell 10 times more options. Is this possible?

  [1]: https://umconnect.umn.edu/fmpresentation1/