I learned how to price a European call option using this video lecture. The considered case is very simple. The call option gives the right to buy 100 Euros for 100 Dollars in one month from now. The 100 Euros cost 100 Dollars now 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, then we borrow \$47.03 (the interest rate is 1% for 1 month). We use these \$50 (47.03 + 2.97) to buy 50 Euros. This procedure guarantees that in one month we will not lose/earn anything independently of the price of the Euros.
Why do we need to borrow? To me it seems to not be beneficial. We take \$47.03 and we need to return \$47.50. So, in total we lose \$0.47 independently of the price of the Euro.
I would propose another strategy. We use our own money (\$47.03) to buy the 50 Euros (we also use \$2.97 to pay for the option). In this case, in one month, we do not need to repay any loan. As a result, we can earn \$0.47 independently of the price of the Euros.
You could say that the second strategy is worse because, if we pay our own \$47.03, we cannot use this money for a whole month. Hence, the money will not earn interest. But we can consider this an investment. We use \$47.03 to earn \$47 in one month. So, the interest rate is 1%. You could say, that lending provides the same interest rate, so we could use the first strategy, and then use the saved \$47.03 to lend them out and earn the same \$0.47.
But what if we use the second strategy (we take no loan to buy the 50 Euros)? Since we do not need to repay more money, we can make our option cheaper. It could cost \$2.50 instead of \$2.97. If we set the price equal to \$2.60, we will get \$0.10 from every option (independently of the price of the Euro). Since our option is cheaper than the options provided by other companies, we can sell a lot of options. We may not earn \$0.47 cents from the option, we will earn only \$0.10 but may be we will sell 10 times more options. Is this possible?