## What is the Put Option?

Put Option is a contract that gives the buyer the right to sell the option at any point in time on or before the date of contract expiration. This is essential to protect the underlying asset from any downfall of the underlying asset anticipated for a certain period of time or horizon.

### Types of Put Options

#### #1 – Long Put Option (Buy)

Long put is termed when the investor buys a put. It is usually bought if the investor anticipates that the underlying asset will fall during a certain time horizon, the option thereby protects the investor from any downfall or running in loss.

#### #2 – Short Put Option (Sell)

A short put is termed when the investor sells such an option or is also called writing a put. It can be sold if an investor anticipates that the underlying asset will not fall over a certain period of time, and this would also ensure that the seller of the option will generate income if the option buyer doesn’t exercise the option by expiration date.

### Put Option Payoffs

If the current stock price is “S”, the strike price is “X” and the stock price at expiration is “S_{T}”. The premium paid is “p_{0}”. Then the profit for put option buyer and seller can be calculated as below:

**#1 – Put Option Payoff for Buyer:** The put buyer will earn a profit when exercise price exceeds an underlying asset and put premium.

**P**Max (0, X – S

_{T}=_{T})

**Net Profit =**P

_{T}– p

_{0}

**#2 – Put Option Payoff Seller:** The put seller will earn a profit if exercise price moves below the underlying asset or does not have a large movement below the strike price, thereby the seller can earn the premium that he receives from the buyer.

**P**– Max (0, X – S

_{T}=_{T})

**Net Profit =**p

_{0}– P

_{T}

### Examples of Put Option

Below are examples for your better understanding.

#### Example #1

Calculate the profit or payoff for put buyer if the investor owns one put option, the put premium is $0.95, the exercise price is $50, the stock is currently trading at $100 and the stock trading at expiration is $40. Assume one option equals 100 shares.

**Solution:**

As given in the above example, we can interpret that an investor has the right to sell the shares at a strike price of $50 up to the expiration date. As shown in the example, if the stock falls to $40 then the investor can exercise the option by buying the stock at $40 and further selling the shares to the option writer at the agreed strike price of $50 each.

Thus if the investor would like to make a profit then the payoff can be calculated as shown below.

PO = P_{T }= 100 * Max (0, 50 – 40)** = $1000**

Moreover, the gross payoff needs to be subtracted by the premium paid on the put option or less any more commissions paid.

Net Profit = 1000 – (100 * 0.95)

**Net Profit = $905**

Thus the maximum gain is earned until the stocks fall to $0 and the maximum loss is incurred up to the amount paid on put option premium that is $95.

#### Example #2

Calculate the profit or payoff for put writer if the investor owns one put option, the put premium is $0.95, the exercise price is $50, the stock is currently trading at $100 and the stock trading at expiration is $40. Assume one option equals 100 shares.

In this case, the put writer is obligated to purchase the underlying asset or take the delivery. In the given example, if the stock price at expiration had ended up above strike price then

- The put buyer will let the option expire and thereby will not exercise the option as the stock price at expiration is greater than the exercise price.
- The investor who had expected the stock price to rise will now collect the put premium price $95 (Payoff = 100 * 0.95) by selling an option at an exercise price of $50. This is the maximum profit that can be earned by the put writer.

**Solution:**

As shown in the above example, the underlying stock price is less than the strike price. Here the option writer is obligated to buy the shares at $50 even if the underlying stock price falls at $40 or below. Thus the loss calculated for the example is as follows:

PO, P_{T }= – 100* Max (0, 50 – 40)**= -$1000**

Here the net loss is calculated by subtracting premium from put option gross loss.

Net Profit = -$905

Theoretically, the maximum loss can be as high as strike price into the number of shares if the underlying asset that is stock price falls to zero, thus the calculation is shown below:

PO, P_{T =} – 100* Max (0, 50 – 0)**= -$5000**

Here the net loss is calculated by subtracting premium from put option gross loss.

**Net Loss = -$4905**

#### Example #3

Calculate the profit or payoff for put buyer and put writer, the put premium is $5, the exercise price is $80, the stock is currently trading at $110 and the stock trading at expiration is $110.

**Solution:**

Put Buyer:

PO P_{T}= Max (0, 80 – 110) **= $0**

Net Profit will be –

Net Profit = 0 – 5

**Net Profit = -$5**

Put Writer:

PO P_{T}= – Max (0, 80 – 110) **= $0**

Net Profit will be –

Net Profit = 5 – 0

**Net Profit = $5.**

**Conclusion**

- It becomes valuable as the exercise price exceeds the underlying asset. Conversely, the option loses its value as the underlying asset exceeds exercise price.
- While the investor is a buyer or seller in the option, the option can be in the money, at the money or out of the money at expiration.
**In the Money –**This occurs when the strike price is greater than the underlying asset, X > S_{T}.**At the Money –**This occurs when the strike price is equal to the underlying asset, X = S_{T}.**Out of the Money –**This occurs when the underlying asset is greater than the strike price, X < S_{T}.

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