Call option

Call option is a type of option that gives the holder the right, but not the obligation, to buy 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 call option generally pays a premium to the option seller for this right.

Concept

A simplified call option can be represented as:

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

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:

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

where:

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

The maximum function ensures that the option holder does not exercise the call when the market price is below the strike price.

Call option buyer

The buyer of a call option benefits when the underlying asset's price increases above the strike price.

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

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

where:

• $C$ = premium paid for the call option.

The break-even price at expiration is:

$S_{\text{Break-even}}=K+C$

ignoring transaction costs and other adjustments.

Call option seller

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

The seller's payoff before considering the premium is:

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

The net profit is:

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

For a conventional uncovered or naked call, potential losses can be theoretically unlimited because the underlying asset's price can rise without a fixed upper limit.

Example

Suppose an investor purchases a call 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 $120 at expiration, the option's intrinsic value per share is:

$max(120-100,0)=\mathrm{\$}20$

The total payoff is:

$100\times 20=\mathrm{\$}2,000$

The net profit is:

$\mathrm{\$}2,000-\mathrm{\$}500=\mathrm{\$}1,500$

If the underlying asset is trading at $90 at expiration:

$max(90-100,0)=0$

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 call 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 call generally has no intrinsic value.

Intrinsic and time value

Before expiration, the market price of a call 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 call option, intrinsic value is:

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

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 call options

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

An American call 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 call option value

The value of a call 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:

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

where:

• $C$ = call 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.

Call option and leverage

A call option can provide leveraged exposure to the underlying asset because the buyer pays a premium that is generally smaller than the total value of the underlying asset represented by the option.

A simplified leverage relationship is:

$\text{Notional Exposure}=S\times Q$

where:

• $S$ = underlying asset price;
• $Q$ = quantity represented by the option contract.

The option buyer's initial cash outlay is generally the premium:

$\text{Premium Paid}=C\times Q$

Leverage can magnify percentage gains and losses relative to the premium invested.

Uses

Call options can be used for:

• speculation on rising asset prices;
• hedging certain exposures;
• obtaining leveraged market exposure;
• constructing option strategies;
• managing portfolio risk.

An investor expecting an asset to appreciate may purchase a call rather than purchasing the underlying asset directly.

Call option versus put 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:

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

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

Covered call

A covered call strategy involves holding the underlying asset while selling a call option against that position.

A simplified structure is:

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

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

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 whether the call is covered or uncovered.

Important risks include:

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

See also

• Option (finance)
• Put option
• Derivative (finance)
• Futures contract
• Forward contract
• Swap (finance)
• Implied volatility
• Volatility (finance)
• Covered call
• Options trading
• Hedging
• Speculation (finance)
• Leverage (finance)