Option (finance)

Options are derivative contracts that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price within a specified period or on a specified date.

The specified price is called the strike price or exercise price. The buyer of an option generally pays the seller, or writer, a premium for this right.

Options are widely used for hedging, speculation, income strategies, portfolio management, and risk management.

Types of options

The two principal types of options are call options and put options.

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

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

The basic distinction is:

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

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

Option terminology

Important terms associated with options include:

• underlying asset — the asset or financial instrument on which the option is based;
• strike price — the price at which the holder can exercise the option;
• expiration date — the date after which the option can no longer be exercised;
• premium — the price paid by the option buyer to the option seller;
• option holder — the party that owns the option;
• option writer — the party that sells or writes the option;
• exercise — the act of using the rights provided by the option;
• assignment — the process through which an option writer is required to fulfill the contract following exercise.

Call option payoff

The payoff of a long call option at expiration can be represented as:

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

where:

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

The net profit after accounting for the premium paid is:

$\text{Call Profit}=max(S_T-K,0)-C$

where $C$ is the call option premium.

The maximum loss for the buyer of a standard long call is generally limited to the premium paid.

Put option payoff

The payoff of a long put option at expiration can be represented as:

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

The net profit after accounting for the premium paid is:

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

where $P$ is the put option premium.

The maximum loss for the buyer of a standard long put is generally limited to the premium paid.

Intrinsic value

The intrinsic value of a call option is:

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

The intrinsic value of a put option is:

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

An option's intrinsic value represents the value that would result from immediate exercise, subject to the applicable exercise terms.

Time value

An option premium can be represented conceptually as:

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

Time value reflects the additional value associated with the possibility that the option may become more profitable before expiration.

Time value generally declines as an option approaches expiration, all else being equal. This phenomenon is commonly referred to as time decay.

Moneyness

Options are commonly classified according to their relationship with the underlying asset price and strike price.

For a call option:

$S>K\Rightarrow \text{In the Money}$

$S<K\Rightarrow \text{Out of the Money}$

For a put option:

$S<K\Rightarrow \text{In the Money}$

$S>K\Rightarrow \text{Out of the Money}$

An option whose strike price is approximately equal to the underlying asset price is described as at the money.

Exercise styles

Options can differ according to when they may be exercised.

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

An American option can generally be exercised at any time up to and including its expiration date.

Other exercise conventions exist for certain specialized contracts.

Settlement

Options can be settled in different ways.

Physical settlement involves delivery of the underlying asset or instrument.

Cash settlement involves payment of the difference between the relevant settlement value and the applicable strike or contract value.

The settlement method depends on the terms and specifications of the particular option contract.

Option pricing

The value of an option depends on several variables.

A simplified representation is:

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

where:

• $V$ = option value;
• $S$ = underlying asset price;
• $K$ = strike price;
• $T$ = time to expiration;
• $\sigma$ = volatility of the underlying asset;
• $r$ = interest rate;
• $q$ = applicable dividend yield or other carry adjustment.

The relationship between these variables is central to option pricing.

Black–Scholes model

The Black–Scholes model is a mathematical model used to estimate the theoretical value of certain European-style options under specified assumptions.

For a European call option:

$C=S_{0}N(d_1)-K{e}^{-rT}N(d_2)$

where:

$d_1=\frac{\ln (S_0/K)+(r-q+{\sigma }^2/2)T}{\sigma \sqrt{T}}$

and:

$d_2=d_1-\sigma \sqrt{T}$

where:

• $C$ = theoretical call value;
• $S_0$ = current underlying asset price;
• $K$ = strike price;
• $T$ = time to expiration;
• $r$ = continuously compounded risk-free interest rate;
• $q$ = continuous dividend yield;
• $\sigma$ = volatility;
• $N(\cdot )$ = cumulative standard normal distribution function.

Option Greeks

Option Greeks are measures used to describe the sensitivity of an option's value to changes in different variables.

Common Greeks include:

• delta;
• gamma;
• theta;
• vega;
• rho.

For a call option, delta can be represented under the Black–Scholes framework as:

${\Delta }_{\text{call}}=N(d_1)$

For a put option:

${\Delta }_{\text{put}}=N(d_1)-1$

These measures are model-dependent and may differ according to the option-pricing framework used.

Volatility

Volatility is an important determinant of option value.

Higher expected volatility generally increases the value of both call and put options, because greater price uncertainty increases the probability of favorable outcomes for the option holder.

A simplified relationship is:

$\sigma \uparrow \Rightarrow \text{Option Value}\uparrow$

subject to other variables remaining constant.

Hedging

Options can be used to reduce or modify investment risk.

For example, an investor holding an asset can purchase a put option to provide protection against a decline in the asset's price.

The protective position can be represented as:

$\text{Protected Portfolio}=\text{Underlying Asset}+\text{Long Put}$

The put establishes a minimum effective sale value at expiration, subject to the strike price and premium paid.

Covered call

A covered call involves holding an underlying asset while selling a call option against it.

A simplified position is:

$\text{Covered Call}=\text{Long Underlying}+\text{Short Call}$

The strategy can generate premium income but limits some of the potential upside of the underlying asset.

Speculation

Options can be used to obtain leveraged exposure to changes in an underlying asset.

A call buyer may benefit from a substantial increase in the underlying asset price without purchasing the underlying asset directly.

However, the option buyer can lose the entire premium paid if the option expires without sufficient value.

Option writers can face substantially greater losses depending on the position and contract structure.

Options and leverage

Options can provide economic exposure to an underlying asset with a relatively small initial premium.

A simplified leverage relationship is:

$\text{Leverage}\approx \frac{\text{Notional Exposure}}{\text{Capital Invested}}$

The actual economic leverage of an option position depends on factors including delta, volatility, time to expiration, and the option's strike price.

Options and risk management

Options can be used to manage several types of financial risk, including:

• equity-price risk;
• commodity-price risk;
• currency risk;
• interest-rate risk;
• volatility risk;
• portfolio downside risk.

The effectiveness of an option-based hedge depends on the characteristics of the option, the underlying exposure, and the timing and magnitude of market movements.

Risks

Options involve several risks.

For option buyers, the premium paid can be lost entirely if the option expires without sufficient value.

For option writers, potential losses can be substantially larger than the premium received, depending on the position.

Other risks include:

• volatility risk;
• Liquidity risk;
• Counterparty risk;
• Model risk;
• Basis risk;
• Gap risk;
• Assignment risk;
• Time decay.

Options can therefore be used for risk management while simultaneously creating other forms of financial risk.

Exchange-traded and over-the-counter options

Options can be traded on organized exchanges or through over-the-counter (OTC) markets.

Exchange-traded options generally have standardized contract terms and are subject to exchange and clearing arrangements.

OTC options can be customized according to the requirements of the counterparties.

Customization can include:

• strike price;
• expiration date;
• notional amount;
• settlement method;
• underlying asset;
• exercise conditions.

Importance

Options are widely used financial derivatives that allow market participants to obtain conditional exposure to underlying assets. They can be used for hedging, speculation, portfolio management, income generation, and risk transfer. Their value depends on the underlying asset price, strike price, time to expiration, volatility, interest rates, dividends, and other contract-specific characteristics.

See also

• Derivative (finance)
• Call option
• Put option
• Option pricing
• Black–Scholes model
• Option premium
• Strike price
• Volatility (finance)
• Hedging
• Risk management
• Covered call
• Protective put
• Futures contract
• Forward contract
• Swap (finance)
• Delta (finance)
• Gamma (finance)
• Theta (finance)
• Vega (finance)
• Rho (finance)
• Time decay