In this lecture we will cover the maximum profit or loss of a put option, for both the buyer and the seller. As with any trade, it is important to be aware of the risk you’re taking before placing the trade.

Firstly, this table shows how to calculate the profit or loss of a put option position for either the buyer or seller. For now, to keep things simple, we’ve left out the position size i.e. the contract multiplier and number of contracts.

What we are doing here to calculate the profit/loss, is calculating the value of the put option at expiry, then adjusting for the premium to give the final profit/loss. The value of a put option that expires in the money is the strike price minus the underlying price at expiry. So we could write this as:
Put Value = Strike – Price

To calculate the buyers profit/loss we then just need to subtract the premium they paid from the value of their put option at expiry. This leads to us calculating:

As we just covered, when the put expires in the money, the put value is ‘Strike – Price’, so we can write as:

So you can see how the formula for the buyer’s PNL is derived.

The seller’s PNL is of course just the negative of this, i.e. multiplied by minus one. So the seller’s PNL can be calculated as:
Put Seller’s PNL = (Put Value – Premium) * -1

= Premium – (Strike – Price)
= Premium + Price – Strike

So you can see how the formula for the seller’s PNL is derived.

When the put option has no value at expiry, because the price expires above the strike, you can substitute in zero for the ‘Put Value’ to give:

And:
Put Seller’s PNL = Premium – Put Value

### PNL example

As a quick example, assume a trader buys a put option with a strike price of \$40, and pays a premium of \$5 per share.

What is the profit and loss for the buyer and seller if the price at expiry is \$25?
We have a:
Price of \$25
Strike of \$40

As the price at expiry of \$25 is below the strike price of \$40, the put has some value at expiry, so we will use the bottom row of formulas.

The buyer of the put option has a profit/loss of:
= Strike – Price – Premium
= 40 – 25 – 5
=\$10

So the put option buyer has a profit of \$10.

The seller of the put option has a profit/loss of:
= Premium + Price – Strike
= 5 + 25 – 40
= -\$10

So the seller of the put option has a loss of \$10.

### Max profit/loss

As well as just being able to calculate the profit or loss for a specific value, it’s also useful to know the maximum profit or loss of any option position you’re thinking of opening.

For example, if we use the previous example with a:
Price of \$25
Strike of \$40

The maximum profit for the buyer is \$35. This is calculated as:
= 40 – 5
= \$35

The maximum loss for the seller is of course also \$35, calculated in the same way.

For the put option buyer, their profit continues to increase for every dollar decrease in the underlying price. Remember the put option buyer’s profit is calculated as: = Strike – Price – Premium
So if we assume a minimum price of \$0, this means their maximum profit is limited to:

When the put option buyer has their maximum profit, the put option seller has their maximum loss of the same amount.

The put option buyer suffers their maximum loss when the underlying price at expiry is above the strike price, rendering the put option worthless. When this is the case they lose the premium they paid for the option, but nothing more.

Similarly for the put option seller, they make their maximum profit when the underlying price at expiry is above the strike price of the put option. The seller gets to keep the premium they collected, and does not have to pay anything out. Their maximum profit is equal to the premium collected.

Note: It is extremely rare, but it is technically possible for an asset price to become negative, which could increase a put option buyer’s profit, and increase a put option seller’s loss past the maximum we’ve listed here.
This can happen for physically settled contracts for assets that cost a lot to take delivery of and store. For example this happened in April 2020 with a WTI Oil contract. With the Covid-19 crisis greatly decreasing demand for oil, oil storage facilities were running very low on storage capacity. Due to this, many people who were left holding the physically settled contract coming into expiry did not want to take physical delivery of the oil, and so were even willing to pay other parties up to \$37.63 a barrel to take the contracts off their hands. A quick google search will give you more details on this event if you want to study it in more detail.

### In summary

The buyer of a put option has a fixed risk, and a profit that is only limited by the underlying price reaching zero.
The seller of a put option has a risk only limited by the underlying price reaching zero, and a fixed maximum profit.

Whether trading call options or put options, or a combination of both, it is always wise to be aware of where your risk lies. It is also important to be aware of the potential magnitude of that risk in a worst case scenario, i.e. your maximum loss.