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Call option
Call option
from Wikipedia
Profits from buying a call.
Profits from writing a call.

In finance, a call option, often simply labeled a "call", is a contract between the buyer and the seller of the call option to exchange a security at a set price.[1] The buyer of the call option has the right, but not the obligation, to buy an agreed quantity of a particular commodity or financial instrument (the underlying) from the seller of the option at or before a certain time (the expiration date) for a certain price (the strike price). This effectively gives the buyer a long position in the given asset.[2] The seller (or "writer") is obliged to sell the commodity or financial instrument to the buyer if the buyer so decides. This effectively gives the seller a short position in the given asset. The buyer pays a fee (called a premium) for this right. The term "call" comes from the fact that the owner has the right to "call the stock away" from the seller.

Price of options

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Option values vary with the value of the underlying instrument over time. The price of the call contract must act as a proxy response for the valuation of:

  • the expected intrinsic value of the option, defined as the expected value of the difference between the strike price and the market value, i.e., max[S−X, 0].[3]
  • the risk premium to compensate for the unpredictability of the value
  • the time value of money reflecting the delay to the payout time

The call contract price generally will be higher when the contract has more time to expire (except in cases when a significant dividend is present) and when the underlying financial instrument shows more volatility or other unpredictability. Determining this value is one of the central functions of financial mathematics. The most common method used is the Black–Scholes model, which provides an estimate of the price of European-style options.[4]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia
from Grokipedia
A call option is a financial derivative contract that gives the buyer the right, but not the obligation, to purchase an underlying asset—typically shares of stock, an index, or another security—at a predetermined strike price on or before a specified expiration date, in exchange for paying a premium to the seller (also known as the writer). The underlying asset's value determines the option's profitability: if the asset's market price rises above the strike price by expiration, the option is "in-the-money" and can be exercised for a gain, or sold for profit; otherwise, it expires worthless, limiting the buyer's loss to the premium paid. This structure provides leverage, allowing investors to control a larger position with less capital compared to buying the asset outright. Key features of call options include their standardization on exchanges, where each contract typically covers 100 units of the underlying asset, and the distinction between American-style options, which can be exercised at any time up to expiration, and European-style options, exercisable only at expiration. The premium reflects factors such as the underlying asset's current price, time to expiration, volatility, interest rates, and dividends, often valued using models like Black-Scholes. Sellers of call options bear the obligation to deliver the asset if exercised, exposing them to potentially unlimited risk if the asset price surges, while buyers face limited downside but forgo the premium if the option expires unexercised. Call options originated in informal markets centuries ago but were standardized with the establishment of the Chicago Board Options Exchange (CBOE) in 1973, which initially traded calls on individual stocks to facilitate organized, transparent trading and . This innovation, supported by the Black-Scholes pricing model introduced that same year, spurred the growth of derivatives markets for hedging against price movements, speculating on upside potential, and generating income through strategies like covered calls. As of 2024, call options are integral to global financial markets, traded on major exchanges like the CBOE, with annual traded notional values in the tens of trillions of dollars, though they carry significant risks including time decay and volatility.

Fundamentals

Definition

A call option is a financial that grants the buyer, known as the holder, the right but not the obligation to purchase an underlying asset—such as a , index, or —at a predetermined on or before a specified expiration date. The seller, or , of the call option is obligated to sell the asset if the holder chooses to exercise the option. This structure allows the holder to potentially benefit from an increase in the asset's price without committing to the full purchase upfront, paying only a premium for the option . This structure provides upside asymmetry for the buyer, as the maximum loss is limited to the premium paid while potential gains are unlimited if the underlying asset price rises significantly. Unlike a , which confers the right but not the obligation to sell the underlying asset at the , a call option specifically enables the purchase of the asset, making it a tool for bullish market positions. The concept of call options originated in the on the , where traders actively dealt in options on shares of the , marking the first known instance of exchange-traded financial . These early contracts, described in Joseph de la Vega's 1688 work Confusion de Confusiones, included calls with features like fixed expiration dates and were used to manage in volatile markets. Modern call options were standardized in 1973 with the founding of the Chicago Board Options Exchange (CBOE), the first organized exchange for trading such contracts, which introduced uniform terms, clearing mechanisms, and centralized trading to enhance and reduce counterparty . At expiration, the payoff of a call option is determined by the difference between the underlying asset's price and the , if positive, otherwise zero. This is expressed as: max(0,STK)\max(0, S_T - K) where STS_T denotes the asset price at expiration and KK is the .

Key Components

A call option is defined by several essential components that determine its terms and functionality. The underlying asset is the or on which the option is based, granting the buyer the right to purchase it under specified conditions; common examples include individual for equity options, stock indices like the , commodities such as or , and currencies in options. The , often denoted as KK, represents the predetermined fixed price at which the holder of the call option can buy the underlying asset if they choose to exercise the contract. This price serves as the reference point for determining whether the option has value at expiration, as the payoff depends on the difference between the underlying asset's market price and the strike price. The expiration date specifies the final date on which the call option can be exercised; after this date, if the option remains unexercised, it expires worthless, eliminating any further rights or obligations for the parties involved. This temporal boundary is crucial for assessing the option's time-sensitive nature and potential profitability. The premium is the upfront payment made by the buyer to the seller (or writer) of the call option in exchange for acquiring the right to buy the underlying asset; it compensates the seller for the risk undertaken and is typically quoted per unit of the underlying asset. This cost is non-refundable and influences the breakeven point for the buyer. The contract size outlines the quantity of the underlying asset controlled by a single option contract, standardizing trading and settlement; for instance, in U.S. equity options, one contract typically covers 100 shares of the underlying stock, allowing for efficient scaling of positions. Finally, the exercise style defines the timing flexibility for exercising the option, distinguishing between styles that permit action at various points up to expiration or only at the end; this component shapes the strategic use of the contract but varies by market and product.

Types

European Call Options

A European call option is a that grants the holder the right, but not the obligation, to purchase an underlying asset at a predetermined solely on the option's , distinguishing it from other styles by prohibiting exercise at any earlier time. These options are prevalent in index derivatives, such as those on the , and in over-the-counter (OTC) markets, where their fixed exercise timing facilitates standardized settlement and reduces complexity in trading broad market exposure. Key advantages include simpler valuation processes, as models need not account for early exercise decisions, leading to lower premiums without an early exercise premium component; additionally, they incur reduced administrative costs for issuers and exchanges due to the absence of ongoing monitoring for premature assignments. The payoff diagram for a European call option at expiration illustrates a hockey-stick shape: zero value if the underlying asset's price is at or below the strike price, transitioning to a linear increase in profit (asset price minus strike price) as the asset price rises above the strike, reflecting unlimited upside potential offset by the initial premium paid. This aligns with the basic payoff structure of max(S_T - K, 0), where S_T is the asset price at expiration and K is the strike. For instance, consider a European call option on the index with a of 4,500 points, expiring in three months, and a premium of 50 points (equivalent to $5,000 for a standard contract multiplier of 100); if the index closes at 4,700 points on expiration, the holder receives a cash settlement of 200 points ($20,000), netting a profit after the premium.

American Call Options

American call options grant the holder the right to buy the underlying asset at the specified at any time on or before the . This exercisability distinguishes them from European call options, which can only be exercised at expiration. In the U.S. equity options markets, American-style contracts predominate, with most options on individual and exchange-traded funds (ETFs) following this structure. This prevalence reflects the market's emphasis on flexibility for traders responding to events like corporate actions. Early exercise of an American call is typically suboptimal for non-dividend-paying , as the remaining time value in the option exceeds any immediate benefit from conversion to the underlying asset. However, for dividend-paying , it becomes relevant immediately before the , allowing the holder to acquire shares and capture the payout, which option holders otherwise forgo. Due to this added exercise flexibility, an American call option holds a value at least equal to that of an identical European call, with the premium potentially higher in scenarios. For example, consider a deep in-the-money American call on a trading at $105 with a $100 , where the is set to pay a $3 the next day; exercising early enables the holder to buy the shares and receive the , outweighing the minor loss of remaining time value if the option has little extrinsic value left.

Valuation

Intrinsic Value and Time Value

The premium of a call option, which is the price paid by the buyer, comprises two main components: intrinsic value and time value. Intrinsic value represents the immediate profit that could be realized if the option were exercised at the current moment. For a call option, it is calculated as the maximum of zero and the difference between the current price of the underlying asset (S) and the strike price (K): Intrinsic Value=max(0,SK)\text{Intrinsic Value} = \max(0, S - K) This value is zero if the underlying price is at or below the strike price, meaning the option has no immediate exercise value. Time value is the portion of the premium exceeding the intrinsic value, reflecting the market's expectation of potential favorable movements in the underlying asset's price before expiration, influenced by factors such as volatility and remaining time. It is derived as: Time Value=PremiumIntrinsic Value\text{Time Value} = \text{Premium} - \text{Intrinsic Value} Time value is highest for at-the-money options and diminishes as the option moves deeper in or out of . The intrinsic value also determines the option's classification, which indicates its relationship to the . An in-the-money (ITM) call has positive intrinsic value (S > K), providing immediate profitability upon exercise. An at-the-money () call has zero intrinsic value (S ≈ K), with the premium consisting entirely of time value. An out-of-the-money (OTM) call likewise has zero intrinsic value (S < K), relying solely on time value for its premium. Time value erodes over time through a process known as time decay, or , where the option loses value as the approaches, even if the underlying price remains unchanged. This decay is not linear; it typically accelerates in the later stages, with approximately one-third of the time value lost in the first half of the option's life and two-thirds in the second half. At expiration, time value reaches zero, leaving the option's worth equal to its intrinsic value. Furthermore, while a higher underlying stock price generally increases a call option's value due to its positive delta, this increase may not always occur if time decay (theta) offsets the gain, particularly if the price rise is small or the option is near expiration. For the option premium to increase, the stock must rise sufficiently to counteract the daily theta decay. For example, consider a call option with an underlying asset price of $100, a of $95, and a premium of $7. The intrinsic value is max(0,10095)=5\max(0, 100 - 95) = 5, so the time value is $7 - 5 = 2$. If the asset price falls to $94, the intrinsic value becomes zero, and the entire $7 premium would represent time value, assuming no change in other factors. In a bearish trend, where the underlying asset price declines, this reduction in intrinsic value is particularly pronounced for out-of-the-money calls (where S < K), which already have zero intrinsic value and rely entirely on time value. Such price declines can push in-the-money calls out-of-the-money, eliminating their intrinsic value entirely. Moreover, time decay accelerates the overall shrinkage of the premium for these options, as the erosion of time value combines with the lack of intrinsic value recovery in a declining market, leading to faster premium decreases compared to stable or bullish conditions. The breakeven price for a call option is the underlying asset price at which the option buyer achieves zero profit or loss at expiration. It is calculated as the strike price plus the premium paid. For example, with a strike price of $100 and a premium of $0.62, the breakeven price is $100.62. This breakeven point relates directly to the option's intrinsic and time values, as at expiration, the time value is zero, and the intrinsic value must equal the premium paid for the position to break even.

Pricing Models

The Black-Scholes model, introduced in 1973, provides a closed-form solution for pricing European call options under specific assumptions. The model assumes that the underlying asset follows a with constant volatility, no dividends are paid, the risk-free is constant, there are no transaction costs or taxes, and the option cannot be exercised early. It posits that stock prices are lognormally distributed and markets are efficient, allowing for continuous hedging to replicate the option payoff. The core formula for the price CC of a European call option is given by: C=SN(d1)KerTN(d2)C = S N(d_1) - K e^{-rT} N(d_2) where SS is the current stock price, KK is the , rr is the , TT is the time to expiration, σ\sigma is the volatility, N()N(\cdot) is the of the standard normal distribution, and d1=ln(S/K)+(r+σ2/2)TσT,d2=d1σT.d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}.
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