Put Option
A contract between a buyer and seller giving a buyer the right, but not the obligation, to sell an underlying asset at a predetermined price
What Is A Put Option?
A put option is a financial settlement between a buyer and a vendor that gives the buyer the right, however no longer the obligation, to sell an underlying asset at a predetermined price (strike rate) within a centered time period. It can be of either the American type or European type.
Put options are typically utilized by buyers as a form of hedging or protection toward a potential decline in the cost of an asset.
An American put option can be defined as the right to sell an underlying asset at the exercise price on or before the exercise date.
A European put option can be defined as the right to sell an underlying asset at the exercise price on the exercise date only.
It can be traded on various financial assets, including stocks, commodities, and currencies. Various factors, including the rate of the underlying asset, the strike charge, the time to expiration, and the extent of volatility inside the market, stimulate the charge of a put option.
American option gives the put buyer more choices as to when the option can be exercised as opposed to the European options.
Key Takeaways
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A put option gives the buyer the right to sell an asset at a fixed price within a set time.
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Terminal value depends on stock price at maturity; exercised if lower than exercise price.
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Put option price before maturity = Intrinsic value + Time value.
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Writing a put option has limited gains/losses based on exercise or expiration.
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Puts are used for hedging against price declines, different from call options (bullish outlook).
Rationale for put options
They are useful when a hedger is concerned with a decrease in prices, considering a hedger who owns a stock and plans to sell it in the future.
Their challenge is that the price of the stock may decrease. They would like to sell at a higher market price if the price increases.
However, if the price decreases, they would like to fix the minimum price they would receive by selling the stock in the future by buying it.
Since it provides the right to sell the stock at the exercise price, the hedger would exercise it only if the exercise price is more than the market price.
Thus, irrespective of the price movement of the stock, the hedger is assured of the minimum price they would receive, which would be the exercise price, through the purchase of it.
Additionally, purchasing this can also be used as a speculative strategy to profit from a market downturn. In this scenario, the investor purchases an option on an asset, which will decrease in cost, and if the asset charge does certainly drop, the investor can sell the put option at a profit.
NOTE
Put options are also available on interest rates, currency exchange rates, commodities, and future contracts.
Terminal value of a put option
Consider a European option on the date of maturity. When an investor buys a put option, they get the right to sell, and there is no obligation on their part to exercise the right. Thus, the buyer will exercise the option and sell the share at a fixed price only if it is profitable for them to do so.
When will it be exercised? It is clear that it will not be exercised as long as the market price of the share is more than the exercise price. This is because the share can be sold for a much higher price in the market than exercising the option.
For example, if the exercise price is $2450 and the market price of a share is $2,550, the option holder will receive $2,450 if they exercise the option and sell the share.
On the other hand, they can sell the share in the market for $2550. Thus, it is not profitable to exercise the option if the share price is more than the exercise price.
If the share price is lower than the exercise price, it is advisable to exercise the option and sell the share at the exercise price rather than sell it at the lower market price. If the option holder exercises this option, they will gain the difference between the exercise and the market prices.
For example, if the market price of a share is $2,300, the holder of the put option can buy this share at $2,300 in the market and then exercise and sell the share at $2,450 to the option writer to make a gain of $150.
This shows that its terminal value depends on the relationship between the share and exercise prices.
- Case 1: If the exercise price (SX) < terminal stock price (ST) ⇒ do not exercise put ⇒ the value of the put option is zero.
- Case 2: If the exercise price (SX) > terminal stock price (ST) ⇒ exercise put ⇒ the value of the put option is (SX - ST).
Therefore, the minimum value of it will be zero, and the maximum value will be the difference between the exercise price and the stock price. Thus, the terminal value can be written as follows:
Terminal value = (SX - ST) if SX > ST
Or,
Terminal value = 0 if SX < ST
Valuing a Put Option Before Maturity
Consider a March put option on a stock ABC with the exercise price of $520. The current stock price is $490. During the three months before the March expiration date, the ABC share price can move in either direction, above $520 or below $490.
Suppose on January 31, the stock ABC price is $470. On this date, the share price is below the exercise price. Whenever the stock price is below the exercise price, the put option is said to be in-the-money.
The in-the-money or intrinsic value of it is calculated as the difference between the exercise and share prices.
In our example, the in-the-money value or the intrinsic value of the ABC put on January 31 is $50 ($520- $470).
What happens if the share price of ABC on January 31 is $530? The share price is above the exercise price, and the put option is considered out-of-money. When the option is out-of-money, its intrinsic value is zero, as the value of an option can never be negative.
If the stock price is close to the exercise price, the option is said to be near-the-money or at-the- money. Thus, the option will be at the money if the ABC stock price is around $520. The intrinsic value of a put option at any time t can be written as
Intrinsic value of a put = Max (SX - ST, 0)
What will be the price of the put option on January 31 when the share price is $470? As we saw earlier, the intrinsic value of the put is $50. If the option is an American put option, an investor can exercise the put and make $50 per share.
However, the price of the put option will be higher than $50 in the market. Why?
The option matures in March, and it is only January 31. Since the option has a life of around two months on January 31, there is a good chance that the share price can go below the current price of $470.
The purchasers of this will consider the possibility of lower share prices by March 31 and, therefore, would be willing to pay some amount for this possibility. The amount the option buyers are willing to pay for the possible decrease in the stock price over time is called the time value of the put option.
Thus, the value of the put option before maturity is made of two components:
- The intrinsic value of the put option
- The time value of the put option
Price of put=Intrinsic value + Time value
This relationship shows that a put option will always have a positive value, as the time value is always positive as long as maturity is not on the same date.
NOTE
That the time value depends on the probability that the share price will decrease, and it depends on the variability of the stock price and the time remaining until maturity.
Terminal value of put from the writer’s point of view
If the buyer does not exercise the put, the writer has no obligation to buy the share and hence is not affected. Thus, the value of a written put is zero as long as it is not exercised.
If the put is exercised, it is clear that it is in-the-money. Thus, the share price in the market is lower than the exercise price. The put writer would be forced to buy the stock from the option buyer at a price higher than what it would have cost if the stock were bought in the market.
Thus, the value of the written put will be the difference between SX and ST.
Hence, the value of the written put can be written as
Pw = Min (ST - SX, 0)
This shows that the terminal value of a written put will be negative if the put is exercised, and the value will be the difference between the stock price and the exercise price.
In case the put is not exercised, the value of the written put will be zero, as the writer has no obligation. The value of the written put will be the minimum of zero and the difference between the stock price and exercise price.
And, the gain and loss from the put where P0 is the premium received would be as follows:
Gw = Min [P0, P0 - (ST - SX)]
When to buy and when to write a put option
When a person buys it, they make gains only if the share price is expected to decrease. However, the put buyer would make a positive gain only when the share price is less than the sum of the exercise price and the price paid for the option.
It would be bought only when the share price ST, is expected to be less than (SX -P0).
- If the exercise price is $520 and the put price is $78:40, a person will buy the put when they expect the stock price to be less than $441.60 ($520 - $78.40).
- If the exercise price is $520 and the put price is $78.40, a person will write a put when they expect the stock price to not go below $441.60 ($520 - $78.40).
Hence a person will write a put only if they believe the terminal stock price ST will not go below (SX -P0).
NOTE
The belief about the movement of stock prices could differ for the put option writer and the put option buyer.
What is the difference between a Put and a Call?
A person would buy a call option when the stock price is expected to increase; their gain would be the difference between the stock price at maturity and the exercise price minus the premium paid for buying the call option.
A person buying a put option when the stock price is expected to decrease their gain would be the difference between the exercise price and the stock price at maturity less the premium paid for buying the put option.
Some key differences between calls and puts are:
1. Earnings capacity: A call option has unlimited earnings ability as the rate of the underlying asset can upward push infinitely, even as a put option has limited earnings ability as the rate of the underlying asset can only fall to zero.
2. Risk: The risk for both call and put options is limited to the premium paid for the contract.
3. Market outlook: A call option is used when the buyer expects the price of the underlying asset to upward push, while a put choice is used when the buyer expects the price of the underlying asset to fall.
4. Payoff profile: The payoff profile of a call option is a positive slope, meaning the profit increases as the price of the underlying asset rises. The payoff profile of a put option is a negative slope, meaning the profit increases as the price of the underlying asset falls.
5. Break Even point: The breakeven point for a call option is the strike price plus the premium paid, while the breakeven point for a put option is the strike price minus the premium paid.
NOTE
It is a common misconception that writing a call would be acceptable when the stock price is anticipated to decline while buying a call is good when the stock price is anticipated to climb.
Conclusion
An option is a zero-sum game in the sense that the gains made by the buyer would equal the loss incurred by the option writes and vice versa.
The value of an option before maturity is the sum of its intrinsic value and time value. The intrinsic value of a put at a particular time is the difference between the exercise price and the stock price at that time or zero, whichever is greater.
The intrinsic value is either zero or positive. If the intrinsic value is positive, the option is said to be in-the-money. If it is zero, the option is said to be out-of-money. The time value is the option value that arises because the option can be in-the-money by the time the option maturity is reached.
It may be optimal to exercise a put option on a stock that pays no dividends if the option is deep-in-the-money and the option premium is less than its intrinsic value.
It is bought when the investor believes the stock price would go below (Exercise price-Call option premium-Put option premium).
It will be written when the stock price is expected to be in the range of (Exercise price Call option premium- Put option premium) to (Exercise price + Call option premium +Put option premium).
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