American option

What Is American Option?

An American option is a type of option contract that grants the holder the right, but not the obligation, to exercise the option at any time between the purchase date and the expiration date. This flexibility distinguishes it within the broader category of derivatives and makes it particularly valuable for certain trading strategies. Unlike its European counterpart, which can only be exercised at expiration, the American option offers the potential for early exercise, allowing holders to lock in profits or mitigate losses based on market movements. This feature is especially relevant for options on assets that pay dividends, as early exercise might be advantageous just before a dividend payment.

History and Origin

Before the advent of standardized exchanges, options were primarily traded over-the-counter (OTC), with terms often negotiated individually between buyers and sellers, leading to a lack of transparency and liquidity. A pivotal moment in the history of options trading occurred with the establishment of the Chicago Board Options Exchange (CBOE) in 1973. The CBOE revolutionized the market by introducing standardized option contracts and creating a centralized marketplace, making options accessible to a wider range of investors. Joe Sullivan, the founding president of Cboe, played a crucial role in bringing this vision to fruition, transforming what was once a complex, manual process into a structured, exchange-traded system.⁵ This innovation laid the groundwork for the modern options market, including the widespread adoption and trading of American options.

Key Takeaways

• An American option allows for exercise at any time up to and including the expiration date.
• This flexibility for early exercise is a key differentiator from other option types.
• The decision to exercise an American option early often depends on factors like dividend payments or significant price movements of the underlying asset.
• American options can offer greater strategic flexibility for speculation and hedging strategies.
• Valuing American options is typically more complex than valuing European options due to the early exercise feature.

Formula and Calculation

The valuation of American options, particularly those on dividend-paying stocks, is often performed using numerical methods, as there is no simple closed-form solution like the Black-Scholes model for European options. The binomial option pricing model is a widely used discrete-time model for valuing American options. It constructs a "tree" of possible future stock prices and works backward from expiration to determine the option's value at each node, considering the possibility of early exercise.

The general approach involves:

1. Constructing the Binomial Tree: Projecting the possible upward (u) and downward (d) movements of the underlying asset price over discrete time steps.

   • (S_t) = Stock price at time t
   • (u = e^{\sigma \sqrt{\Delta t}}) (upward movement factor)
   • (d = e^{-\sigma \sqrt{\Delta t}}) (downward movement factor)
   • (\sigma) = Volatility of the underlying asset
   • (\Delta t) = Length of a single time step

2. Calculating Option Payoffs at Expiration: At the final nodes of the tree, the intrinsic value of the option is calculated.

   • For a call option: (Max(0, S_T - K))
   • For a put option: (Max(0, K - S_T))
   • K = Strike price

3. Working Backward through the Tree: At each earlier node, the option's value is determined by comparing two values:

   • The expected discounted value of continuing to hold the option (derived from the values of the subsequent nodes).
   • The intrinsic value if the option were exercised at that node.

   For an American option, the value at each node (V_i) is:
   $V_i = \max(\text{Intrinsic Value at } i, e^{-r \Delta t} [pV_{up} + (1-p)V_{down}])$
   Where:

   • (p) = Risk-neutral probability of an upward movement
   • (V_{up}) = Option value at the upward node in the next step
   • (V_{down}) = Option value at the downward node in the next step
   • (r) = Risk-free interest rate

While powerful, the binomial model can sometimes undervalue American options due to how dividend yields are incorporated or suboptimal early exercise decisions within the model's structure.⁴

Interpreting the American Option

The primary interpretation of an American option centers on its flexibility. The ability to exercise an American option at any point before the expiration date means its value will always be at least its intrinsic value. In contrast, a European option's value can fall below its intrinsic value before expiration if there is no advantage to exercising it at that moment.

For a call option, early exercise is generally considered only when the underlying asset pays a dividend, and exercising before the ex-dividend date allows the holder to capture that dividend. For a put option, early exercise might be considered if the underlying asset's price has fallen significantly, allowing the holder to sell the asset at the higher strike price and reinvest the proceeds. The decision to exercise early is a nuanced one, balancing the immediate payoff against the loss of the option's remaining time value and the potential for further favorable price movements.

Hypothetical Example

Consider an investor, Sarah, who buys an American call option on Company XYZ stock.

• Strike Price (K): $50
• Expiration Date: 3 months from now
• Current Stock Price (S): $52
• Option Premium: $3.00 per share

A month after purchasing the option, Company XYZ announces an unexpectedly large dividend payment, with the ex-dividend date next week. The stock price, anticipating the dividend, remains around $53.

Sarah faces a choice:

1. Hold the American option: She would receive no dividend, and the stock price might drop by the dividend amount after the ex-dividend date, potentially reducing the option's value.
2. Exercise the American option early: Sarah could exercise her option, buying XYZ stock at $50 per share (her strike price). She would then own the stock and be eligible to receive the dividend. After receiving the dividend, she could sell the shares in the open market.

If the dividend is substantial, exercising early and receiving the dividend might yield a higher net profit than holding the option until expiration, even considering the loss of any remaining time value. This scenario highlights the specific advantage of the American option's early exercise feature, particularly in relation to dividend capture.

Practical Applications

American options are widely used in financial markets for various purposes, including hedging, speculation, and income generation. Their flexibility makes them suitable for dynamic trading strategies.

• Hedging: Portfolio managers might use American put options to protect against a decline in the value of their stock holdings. The ability to exercise the put at any time provides immediate protection if the underlying stock drops sharply, allowing them to sell shares at the strike price before further losses occur.
• Speculation: Traders speculate on price movements of an underlying asset. For instance, an investor bullish on a stock might buy an American call option, hoping to profit from a rise in its price. The early exercise feature means they don't have to wait until expiration to capitalize on significant gains.
• Income Generation: Selling (writing) American options, particularly covered calls, is a common strategy to generate premium income. However, writers of American options face the risk of early assignment, meaning they could be forced to deliver or take delivery of the underlying asset at any time.

The trading of options, including American options, is subject to extensive regulation to ensure fair and orderly markets and protect investors. In the U.S., regulatory bodies like the Securities and Exchange Commission (SEC) and the Financial Industry Regulatory Authority (FINRA) establish rules governing options trading practices, position limits, and disclosure requirements.³,² This regulatory framework is crucial for maintaining market integrity and facilitating effective risk management for market participants.

Limitations and Criticisms

While American options offer flexibility, they also come with certain limitations and complexities:

• Valuation Complexity: Due to the early exercise feature, valuing American options is computationally more intensive than valuing European options. This complexity can lead to minor discrepancies or challenges in applying theoretical models, particularly for options on dividend-paying assets or those with long expirations.¹
• Loss of Time Value: Exercising an American option early means forfeiting any remaining time value, which is the portion of the option's premium attributable to the time left until expiration and the possibility of favorable price movements. This is often the main reason why early exercise is not optimal unless specific conditions (like dividend capture for calls or significant downside for puts) make it financially advantageous.
• Difficulty for Novice Traders: The nuanced considerations for early exercise and the more intricate valuation models make American options somewhat more challenging for inexperienced traders to fully understand and utilize effectively compared to simpler financial instruments.
• Early Assignment Risk for Writers: For those who sell or "write" American options, there is a constant risk of early assignment. This means the option holder can choose to exercise at any time, forcing the writer to fulfill their obligation (e.g., sell shares in the case of a written call, or buy shares in the case of a written put option), which can sometimes be disruptive to a trading strategy.

American Option vs. European Option

The fundamental difference between an American option and a European Option lies in when they can be exercised. An American option grants the holder the right to exercise at any time up to and including the expiration date. Conversely, a European option can only be exercised on its expiration date.

This distinction has significant implications for their valuation and practical application. Because the American option offers more flexibility (the option to exercise early), it will theoretically always be worth at least as much as, and usually more than, an otherwise identical European option. This is especially true for call options on dividend-paying stocks or put options that are deep in-the-money, where the value of early exercise can be realized. The additional flexibility of American options means their pricing models must account for the optimal decision to exercise at any point in time, leading to more complex valuation techniques compared to European options.

FAQs

Q1: Can I always exercise an American option early?

Yes, an American option can be exercised at any time between the purchase date and the expiration date. However, exercising it early is not always the most financially advantageous decision, as it means forfeiting any remaining time value of the option.

Q2: Why would someone choose an American option over a European option?

Investors choose American options primarily for the flexibility of early exercise. This feature can be particularly useful for strategies involving dividend capture for call options or for quickly realizing profits/limiting losses on a deeply in-the-money put option.

Q3: Are American options more expensive than European options?

Generally, yes. All else being equal (same underlying asset, strike price, and expiration date), an American option will typically be priced higher than a European option. This is because the added flexibility of early exercise provides a greater potential benefit to the holder, making the American option more valuable.