Expected payout of an option
I'd like someone to explain to me why the following argument, which leads me to a slightly different answer than the one above, is wrong. And so to price it today, one would discount this expected payout by the risk-free rate.
The market will make those disappear because they constitute "free money". What is then the right price to charge for the forward contract? For more details on this, you can check my answer to question " Reference for why a derivative is a derivative and not say an insurance contract ".
Still, if someone were to sell enough of these options on identical but independent stocks for the law of averages to win out, then this would be a correct valuation. The fact is that, by valuing the derivative that way, he will be bleeding money away by leaving market participants arbitrage him at each contract.
Intuitively, you can think of it this way: it is not the rate of return of the underlying asset that matters, but rather the rate at which you can borrow the underlying asset to hedge yourself.
Mathematically, the shortest way to understand this is through a PDE approach.