A European Put Option gives you the right to sell a stock at a predetermined price at a future date.
You must pay a certain price to have access to this right.
What’s the maximum that price can reach, and still it makes sense to pay?
To answer this question, we look at arbitrage opportunities.
Arbitrage is when you take advantage of mispricing in the market to obtain a risk-free profit with zero investment. It involves buying and selling immediately, and pocketing the difference.
In order to get the upper limit of an option’s price, we figure out the maximum price just before an arbitrage opportunity happens.
What is the Maximum Value of a European Put Option?
A European Put Option cannot be worth more than the maximum profit it can generate.
If you buy a put on a stock, you think it’s likely the stock will go down, right?
A put gives you the right to sell the stock at a certain price fixed beforehand. The lower the price of the stock at expiration, the better for you, as you have the right to sell it for more than what it trades at in the market.
Knowing this, what’s the best-case scenario for you as a put holder at expiration?
That the price of the stock is zero.
Say you buy a put with the strike of 50, and at expiration the company went bankrupt. The stock is now worth 0 dollars.
Well, that sucks for them, but for you? Not really. You have the right to sell something that is worth zero for $50.
This means no one in their right mind would pay $51 for that option. Why? Because even in the best-case scenario (the stock going to zero), you wouldn’t make money.
So, the price of a put can never be worth more than the present value of the strike of that put. This information is fed to the models market makers use to price options in the real world.
But what happens if, for some magical reason, it were possible for a put to have a price higher than its strike? An arbitrage opportunity.
Remember, you can buy and sell options.
In this case, if the put option is pricier than it should be, you should start a short position on it.
Let’s see the steps you should take from now to expiration:
Step 1) Right Now
Sell the put option.
With the money you get from that sale, make a risk-free deposit. How much should you give the bank? The value that corresponds to your strike price. For how long? Until the expiration of the option.
What are you left with as of right now? Well, you received money from the sale of the option, but then you gave away most of it to the bank to make a deposit.
What you have left is a positive value bigger than zero, because remember: the price of the option (that you sold) is incorrectly higher than its strike price (you gave to the bank).
The difference between the two is what you have left as of right now.
Step 2) At Expiration
At expiration, the put can be in-the-money (the stock goes down and the strike of your option is above it) or out-of-the-money (the stock went up and your strike is below it).
No matter what’s the case, you make money. And that’s why this is an arbitrage opportunity. Here’s how:
Put Option is ITM
As the seller of the put, you will need to fulfill the wishes of the person you sold the right to.
The option being ITM is good for them. And (under normal circumstances—which we are not in, as arbitrage is super rare and infinitesimal in the real world) bad for you. But you made a promise, now you have to keep it.
You need to buy the stock from them at the predetermined strike price, which is now above the market price of the stock. Sucks, right?
Well, not really. Remember the deposit you made? You’ll get that cash back now.
You deposited the value of the strike—which is higher than the current stock price, as the option is ITM.
Now, you use that money to buy the stock from your counterparty at the predetermined strike price.
What do you have left? The stocks you just bought!
Even though you bought them for an unfavorable price, they still have value and you can sell them for a profit. (Unless the stock went to zero.)
It’s pretty good considering you made money already in Step 1, and now you either get free stocks or pay zero.
Put Option is OTM
If the option is OTM, your counterparty will not exercise their right. Why would they want the right to sell something for lower than the current market price?
This means you owe them zero.
But there’s someone who owes you something. The bank.
You’ll get your original deposit. How cool is that considering you already made money in Step 1?
Upper Bound on European vs. American Puts
Did you notice how I said the price of a European put cannot surpass the “present value” of its strike?
Why did I say that—present value?
The relationship between European and American puts explains this.
We need to discount the value of the strike to the present because in a European option, you can only exercise your right at expiration.
Discounting something to get its present value makes the price smaller.
This means the maximum price of an American put is higher than that of a European.
This makes sense. The higher flexibility of an American put—you can exercise it anytime before expiration—justifies this relationship.
The upper bound of an American put will at the very least be equal to the upper bound of a European put. The additional exercise opportunities of an American option cannot have a negative value.