Risk free profit with a put option

Question

From Terence Tao Blog the price of a put option at time $t_0$ cannot exceed the strike price P at time $t_1$. The reason is that otherwise there would be an arbitrage opportunity. Everyone in the market could build a put option Simply putting aside a certain amount of cash for paying the strike price P. In this way could one make risk-free profit and how?

Answer 1

You are the seller, I am the buyer.

At $t_0$, I pay you $x$ to purchase a put option for $S$ at strike price $P$ at time $t_1$, meaning that at $t_1$ I can choose to force you to buy $S$ at price $P$.

Generally I will exercise this option if the market price of $S$ is far less than $P$ at $t_1$, I can buy $S$ off the market and force you to pay extra for it. (as the buyer of a put option, I am basically betting that the price of $S$ will decrease in the future)

If $x > P$, then you simply set aside $P$ and pocket $x-P$. Then at time $t_1$ even if $S$ is worthless and I force you to buy it you still come out ahead.

Of course, as a buyer I would never accept such a deal because I am guaranteed to lose money, so it's not really an arbitrage opportunity because no one would buy this option. Similarly "You give me \$2 and I will give you between \$0 and \$1 back" is also "risk-free profit" if you can find anyone gullible enough to take your offer.