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European call option

What Is European Call Option?

A European call option is a type of financial derivative contract that grants the holder the right, but not the obligation, to buy an underlying asset at a predetermined price, known as the strike price, on a specific future date, called the expiration date. Unlike its American counterpart, a European call option can only be exercised on its expiration date, making its exercise style more restrictive. This contrasts with other forms of options trading where early exercise might be permitted. Investors typically purchase a European call option when they anticipate that the price of the underlying asset will increase above the strike price before or by the expiration date.

History and Origin

The concept of options dates back to ancient times, with early forms believed to have been used in agricultural markets to manage future harvests. However, modern, standardized options, including the European call option, gained prominence with the establishment of the Chicago Board Options Exchange (CBOE) in 1973. This marked a significant shift from the less regulated over-the-counter (OTC) options that existed previously, bringing standardization and a centralized marketplace to options trading.5,4

A pivotal development for the valuation of European call options was the publication of the Black-Scholes model in 1973 by Fischer Black and Myron Scholes. This groundbreaking mathematical model provided a framework for pricing options, significantly contributing to the growth and legitimacy of the derivatives market.3 The model helped market participants determine a fair value for options, making them more attractive for both hedging and speculative purposes.

Key Takeaways

  • A European call option grants the holder the right, but not the obligation, to buy an underlying asset.
  • Exercise of a European call option is restricted to the specific expiration date, unlike American options which can be exercised anytime before expiration.
  • Investors buy European call options when they expect the underlying asset's price to rise above the strike price.
  • The pricing of European call options is often calculated using models like the Black-Scholes formula, considering factors such as volatility and time to expiration.
  • The maximum loss for the buyer of a European call option is limited to the option premium paid.

Formula and Calculation

The theoretical price of a European call option is commonly determined using the Black-Scholes formula. This formula considers several variables to estimate the fair market value of the option.

The Black-Scholes formula for a European call option (C) is:

C=S0N(d1)−Ke−rTN(d2)C = S_0 N(d_1) - K e^{-rT} N(d_2)

Where:

  • ( S_0 ) = Current price of the underlying asset
  • ( K ) = Strike price of the option
  • ( T ) = Time to expiration date (in years)
  • ( r ) = Risk-free interest rate
  • ( N() ) = Cumulative standard normal distribution function
  • ( e ) = Euler's number (base of natural logarithm)

And ( d_1 ) and ( d_2 ) are calculated as:

d1=ln⁡(S0/K)+(r+σ22)TσTd_1 = \frac{\ln(S_0/K) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}

d2=d1−σTd_2 = d_1 - \sigma \sqrt{T}

Where:

  • ( \ln ) = Natural logarithm
  • ( \sigma ) = Volatility of the underlying asset's returns

Interpreting the European Call Option

Interpreting a European call option involves understanding its potential profitability based on the relationship between the underlying asset's current market price and the option's strike price. When the underlying asset's price is above the strike price, the call option is considered in-the-money, indicating it has intrinsic value and would be profitable if exercised at expiration. Conversely, if the underlying asset's price is below the strike price, the option is out-of-the-money and would expire worthless.

The value of a European call option is also influenced by its remaining time value, which diminishes as the expiration date approaches. Traders analyze these factors to determine whether to hold, sell, or exercise their European call options.

Hypothetical Example

Consider an investor who believes Stock XYZ, currently trading at $50 per share, will increase in value. They decide to purchase a European call option on Stock XYZ with a strike price of $55 and an expiration date three months from now. The option premium is $2.00 per share. Since one option contract typically represents 100 shares, the total cost for this European call option contract is $2.00 * 100 = $200.

  • Scenario 1: Stock XYZ rises to $60 at expiration.

    • The investor's right to buy at $55 is valuable. They can exercise the option, buy 100 shares at $55 each ($5,500 total), and immediately sell them in the market at $60 per share ($6,000 total).
    • The profit from the trade is $6,000 (sale) - $5,500 (purchase) - $200 (premium paid) = $300.

  • Scenario 2: Stock XYZ falls to $48 at expiration.

    • The investor's right to buy at $55 is not valuable, as they could buy shares cheaper in the open market. The option expires worthless.
    • The investor's loss is limited to the initial premium paid, which is $200.

Practical Applications

European call options are utilized in various investment and portfolio management strategies. One common application is for speculation, where investors aim to profit from anticipating price movements in the underlying asset without owning the asset outright. This offers leverage, as a small premium can control a larger value of the underlying asset.

Furthermore, European call options are used in hedging strategies to mitigate potential losses. For example, a company anticipating a future purchase of a foreign currency could buy a European call option on that currency to lock in a maximum exchange rate, protecting against adverse currency fluctuations. Regulatory bodies, such as the U.S. Securities and Exchange Commission (SEC), provide guidance and oversight for options trading to ensure fair and orderly markets and investor protection.2

Limitations and Criticisms

Despite their utility, European call options and the models used to price them, particularly the Black-Scholes model, have certain limitations. A significant criticism of the Black-Scholes model is its underlying assumptions, which do not always reflect real-world market conditions. For instance, the model assumes constant volatility of the underlying asset and that asset prices follow a log-normal distribution, which may not hold true, especially during periods of market turbulence.1 In reality, volatility is dynamic and can fluctuate significantly over time.

Another limitation is that the Black-Scholes model does not account for dividends paid on the underlying asset during the life of the option, nor does it fully incorporate transaction costs. While extensions and modifications to the original model have been developed to address some of these shortcomings, these complexities can impact the accuracy of pricing European call options in practice. The restriction of exercising a European call option only at expiration can also be a limitation for investors who might wish to close their position early to realize profits or cut losses.

European Call Option vs. American Call Option

The primary distinction between a European call option and an American call option lies in their exercise style. A European call option can only be exercised on its specified expiration date. This means the holder must wait until the very end of the option's life to decide whether to buy the underlying asset at the strike price. In contrast, an American call option grants the holder the flexibility to exercise the option at any point up to and including the expiration date. This early exercise feature typically makes American options more valuable than their European counterparts, all other factors being equal, because of the added flexibility they provide to the option holder. Consequently, the option premium for an American call option is usually higher than that of a comparable European call option.

FAQs

What is the maximum loss when buying a European call option?

The maximum loss for the buyer of a European call option is limited to the option premium paid to acquire the contract. If the underlying asset's price does not move favorably, the option will expire worthless, and the investor loses only the initial cost.

Can a European call option be sold before its expiration date?

Yes, a European call option can be sold in the secondary market before its expiration date. While it cannot be exercised early, the holder can close their position by selling the call option to another investor at its current market price.

How does volatility affect the price of a European call option?

Increased volatility in the underlying asset generally increases the theoretical price of a European call option. Higher volatility implies a greater chance of the underlying asset's price moving significantly above the strike price by expiration, thus increasing the potential for profit.

Is a European call option the same as a put option?

No, a European call option and a put option are different types of derivatives. A call option gives the holder the right to buy the underlying asset, while a put option gives the holder the right to sell the underlying asset.