Put option

Put option is a type of option that gives the holder the right, but not the obligation, to sell an underlying asset at a predetermined price on or before a specified date, depending on the option's exercise terms.

The predetermined price is known as the strike price or exercise price. The buyer of a put option generally pays a premium to the option seller for this right.

Concept

A simplified put option can be represented as:

$\text{Put Option}\to \text{Right to Sell Underlying Asset}$

The holder can exercise the option when doing so is economically beneficial, subject to the terms of the contract.

A simplified payoff at expiration is:

$\text{Put Payoff}=max(K-S_T,0)$

where:

• $K$ = strike price;
• $S_T$ = price of the underlying asset at expiration.

Put option buyer

The buyer of a put option benefits when the underlying asset's price declines below the strike price.

The buyer's net profit at expiration can be represented as:

$\text{Net Profit}=max(K-S_T,0)-P$

where:

• $P$ = premium paid for the put option.

The break-even price at expiration is:

$S_{\text{Break-even}}=K-P$

ignoring transaction costs and other adjustments.

Put option seller

The seller, or writer, of a put option receives the option premium and assumes the obligation to buy the underlying asset at the strike price if the holder exercises the option.

The seller's payoff before considering the premium is:

$\text{Put Writer Payoff}=-max(K-S_T,0)$

The seller's net profit is:

$\text{Net Profit}=P-max(K-S_T,0)$

The potential loss for the put writer can be substantial if the underlying asset's price falls significantly, although the loss is limited because the underlying asset's price cannot normally fall below zero.

Example

Suppose an investor purchases a put option with:

• strike price = $100;
• premium = $5 per share;
• contract size = 100 shares.

The total premium paid is:

$100\times 5=\mathrm{\$}500$

If the underlying asset is trading at $70 at expiration, the option's intrinsic value per share is:

$max(100-70,0)=\mathrm{\$}30$

The total payoff is:

$100\times 30=\mathrm{\$}3,000$

The net profit is:

$\mathrm{\$}3,000-\mathrm{\$}500=\mathrm{\$}2,500$

If the underlying asset is trading at $110 at expiration:

$max(100-110,0)=0$

The option expires worthless and the buyer's maximum loss is the $500 premium, ignoring transaction costs.

In-the-money, at-the-money and out-of-the-money

A put option is generally described as in the money (ITM) when:

$S_T<K$

It is at the money (ATM) when the underlying price is approximately equal to the strike price:

$S_T\approx K$

It is out of the money (OTM) when:

$S_T>K$

At expiration, an out-of-the-money put generally has no intrinsic value.

Intrinsic and time value

Before expiration, the market price of a put option can be viewed conceptually as consisting of intrinsic value and time value.

A simplified relationship is:

$\text{Option Premium}=\text{Intrinsic Value}+\text{Time Value}$

For a put option, intrinsic value is:

$\text{Intrinsic Value}=max(K-S,0)$

where $S$ is the current underlying asset price.

Time value reflects the possibility that the option may become more valuable before expiration.

European and American put options

A European put option can generally be exercised only on its expiration date.

An American put option can generally be exercised at any time before or on the expiration date, subject to the contract's terms.

The distinction concerns the exercise opportunity rather than the underlying asset.

Factors affecting put option value

The value of a put option can be affected by several variables, including:

• underlying asset price;
• strike price;
• time to expiration;
• volatility;
• risk-free interest rate;
• dividends or other distributions;
• market conditions.

A simplified relationship is:

$P=f(S,K,T,\sigma ,r,q)$

where:

• $P$ = put option value;
• $S$ = underlying asset price;
• $K$ = strike price;
• $T$ = time to expiration;
• $\sigma$ = volatility;
• $r$ = risk-free interest rate;
• $q$ = dividend or income yield, where applicable.

Uses

Put options can be used for:

• speculation on declining asset prices;
• hedging existing positions;
• limiting downside risk;
• portfolio protection;
• constructing option strategies.

An investor holding an asset may purchase a put option to establish a minimum effective selling price, subject to the option's strike price and premium.

Protective put

A protective put strategy involves holding the underlying asset while purchasing a put option.

A simplified structure is:

$\text{Protective Put}=\text{Long Underlying}+\text{Long Put}$

The put can provide protection against a decline in the underlying asset below the strike price, subject to the premium paid and the option's terms.

Put option versus call option

A call option gives the holder the right to buy the underlying asset.

A put option gives the holder the right to sell the underlying asset.

A simplified distinction is:

$\text{Call}\to \text{Right to Buy}$

$\text{Put}\to \text{Right to Sell}$

Risks

For the buyer, the primary risk is that the option expires without sufficient value to recover the premium paid.

For the seller, the risk depends on the underlying asset and the position structure.

Important risks include:

• market risk;
• volatility risk;
• time-decay risk;
• liquidity risk;
• assignment risk;
• leverage risk;
• gap risk.

See also

• Option (finance)
• Call option
• Derivative (finance)
• Futures contract
• Forward contract
• Swap (finance)
• Implied volatility
• Volatility (finance)
• Protective put
• Options trading
• Hedging
• Speculation (finance)
• Leverage (finance)