Option's value

A call option gives the holder the right, but not the obligation, to buy an underlying asset at a specified price (the strike price) on or before a certain date (the expiration date). Conversely, a put option grants the holder the right, but not the obligation, to sell an underlying asset at a specified strike price on or before the expiration date. Options derive their value from the price movements of an underlying asset, such as stocks, bonds, commodities, or currencies. This concept is central to derivatives within the field of financial economics. The option's value is influenced by several factors, including the current price of the underlying asset, the strike price, the time remaining until expiration, the volatility of the underlying asset, and prevailing interest rates.

History and Origin

The formal mathematical modeling of option values has roots dating back to the work of Louis Bachelier in 1900, who applied Brownian motion to model stock option prices. However, it was the publication of the Black-Scholes model in 1973 by Fischer Black and Myron Scholes that revolutionized the pricing of options. The publication of their paper, "The Pricing of Options and Corporate Liabilities," in the Journal of Political Economy coincided with the opening of the Chicago Board Options Exchange (CBOE) in April 1973, which further propelled the development and growth of the listed options market.¹¹, ¹² Robert C. Merton also made significant contributions, expanding on the mathematical understanding of the model and coining the term "Black-Scholes options pricing model." The introduction of this model transformed options trading from a more intuitive, supply-and-demand-driven activity to one based on quantitative analysis and established a framework for risk management in financial markets.¹⁰

Key Takeaways

• An option's value is the theoretical price an option contract should trade for, based on various market and intrinsic factors.
• Key determinants include the underlying asset's price, strike price, time to expiration, volatility, and interest rates.
• The intrinsic value of an option is the immediate profit that could be realized if the option were exercised.
• Time value, or extrinsic value, accounts for the possibility that an option's intrinsic value will increase before expiration.
• Option pricing models, such as Black-Scholes, provide a theoretical framework for calculating an option's value.

Formula and Calculation

The Black-Scholes formula is widely used to calculate the theoretical price of European-style call and put options. The formula for a European call option (C) is:

$C = S_0 N(d_1) - K e^{-rT} N(d_2)$

And for a European put option (P):

$P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$

Where:

• (S_0) = Current price of the underlying asset
• (K) = Strike price of the option
• (T) = Time to expiration (in years)
• (r) = Risk-free interest rate (annualized)
• (\sigma) = Volatility of the underlying asset's returns
• (N(x)) = Cumulative standard normal distribution function
• (d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}})
• (d_2 = d_1 - \sigma \sqrt{T})

The variable (N(d_1)) represents the delta of the call option, indicating how much the option's value is expected to change for a one-unit change in the underlying asset's price. The term (K e^{-rT}) discounts the strike price back to the present value.

Interpreting the Option's Value

Interpreting an option's value involves understanding its components: intrinsic value and extrinsic (time) value. The intrinsic value is the immediate profit if the option were exercised. For a call option, this is (\max(S_0 - K, 0)); for a put, it's (\max(K - S_0, 0)). Options that have intrinsic value are considered in-the-money.

The extrinsic value is the difference between the option's market price and its intrinsic value. This portion reflects the potential for the option to gain intrinsic value before expiration. Factors like time to expiration, implied volatility, and interest rates significantly influence extrinsic value. A higher implied volatility generally leads to a higher extrinsic value, as it indicates a greater expected price movement in the underlying asset. Traders assess an option's value by comparing its theoretical price from a model to its current market price to identify potential mispricings.

Hypothetical Example

Consider a hypothetical example for a call option on Stock XYZ.

• Current stock price ((S_0)): $100
• Strike price ((K)): $105
• Time to expiration ((T)): 0.5 years (6 months)
• Risk-free interest rate ((r)): 3% (0.03)
• Volatility ((\sigma)): 20% (0.20)

Using the Black-Scholes formula:

First, calculate (d_1) and (d_2):
[
d_1 = \frac{\ln(100/105) + (0.03 + 0.20^2/2)0.5}{0.20 \sqrt{0.5}}
]
[
d_1 = \frac{\ln(0.95238) + (0.03 + 0.02)0.5}{0.20 \times 0.7071}
]
[
d_1 = \frac{-0.04879 + 0.025}{0.14142}
]
[
d_1 = \frac{-0.02379}{0.14142} \approx -0.1682
]

[
d_2 = d_1 - \sigma \sqrt{T} = -0.1682 - 0.20 \times 0.7071
]
[
d_2 = -0.1682 - 0.14142 \approx -0.3096
]

Next, find (N(d_1)) and (N(d_2)) from a standard normal distribution table:
(N(-0.1682) \approx 0.4332)
(N(-0.3096) \approx 0.3785)

Now, calculate the call option price (C):
[
C = 100 \times 0.4332 - 105 e^{-0.03 \times 0.5} \times 0.3785
]
[
C = 43.32 - 105 e^{-0.015} \times 0.3785
]
[
C = 43.32 - 105 \times 0.9851 \times 0.3785
]
[
C = 43.32 - 39.11 \approx 4.21
]

The theoretical value of this call option is approximately $4.21. This example demonstrates the Black-Scholes model's application in pricing a specific derivative contract.

Practical Applications

The option's value and the models used to determine it are foundational in various financial applications. In portfolio management, investors use option pricing models to make informed decisions about buying or selling options, incorporating them into strategies for hedging risk or speculating on price movements. Options are key instruments in risk hedging strategies, allowing investors to protect existing positions against adverse price changes. For example, a portfolio manager holding a large stock position might buy put options to hedge against a potential downturn.

Options also play a significant role in arbitrage strategies, where discrepancies between an option's theoretical value and its market price can present opportunities for profit without significant risk. Furthermore, regulators, such as the U.S. Securities and Exchange Commission (SEC), oversee the derivatives market to ensure transparency and mitigate systemic risk, particularly after events like the 2008 financial crisis.⁸, ⁹ This regulatory oversight impacts how option values are assessed and managed within financial institutions. The Chicago Board Options Exchange (CBOE) provides current market statistics and historical data for options, which is crucial for market participants analyzing option values and trends.⁶, ⁷

Limitations and Criticisms

Despite its widespread use, the Black-Scholes model has several notable limitations. One primary criticism is its assumption of constant volatility over the option's life, which is often not the case in real markets.⁵ In reality, volatility tends to fluctuate, leading to phenomena like the "volatility smile" or "volatility skew," where implied volatility varies across different strike prices and maturities.⁴

Another significant limitation is the model's assumption that the underlying asset pays no dividends during the option's life. While modifications exist to account for dividends, the original model's simplicity in this regard can lead to inaccuracies, especially for stocks with regular dividend payouts.³ Furthermore, the Black-Scholes model is designed for European-style options, which can only be exercised at expiration, making it less suitable for American-style options that can be exercised anytime before maturity.² The model also assumes frictionless markets with no transaction costs, taxes, or restrictions on borrowing and lending, which are unrealistic in actual trading environments. These assumptions mean that while the option's value calculated by the Black-Scholes model provides a useful theoretical benchmark, real-world market prices can deviate due to these factors.

Option's Value vs. Premium

While often used interchangeably in casual conversation, an option's "value" and "premium" have distinct meanings in finance, primarily due to how they are derived and used.

An option's value, as calculated by models like Black-Scholes, represents a theoretical fair price based on specific inputs and assumptions. This is often what analysts refer to when discussing the inherent worth of an option. In contrast, the option premium is the actual price that buyers pay and sellers receive in the market. This premium includes both the intrinsic and extrinsic value, but it is also directly influenced by the forces of supply and demand and overall market sentiment. While theoretical value guides traders, the premium is the tangible price at which transactions occur.

FAQs

What is the intrinsic value of an option?

The intrinsic value is the portion of an option's value that is immediately realizable if the option were exercised. For a call option, it's the amount by which the underlying asset's price exceeds the strike price. For a put option, it's the amount by which the strike price exceeds the underlying asset's price. If an option has no immediate profit upon exercise, its intrinsic value is zero. Options with intrinsic value are referred to as being in-the-money.¹

How does time affect an option's value?

Time significantly affects an option's extrinsic value. Generally, as an option approaches its expiration date, its extrinsic value (or time value) decreases. This phenomenon is known as time decay or theta decay. Options with more time until expiration have a greater chance for the underlying asset's price to move favorably, thus commanding a higher time value.

What is the role of volatility in option pricing?

Volatility is a critical factor in determining an option's value. Higher expected volatility in the underlying asset generally leads to a higher option value, as it increases the probability that the option will become in-the-money or further in-the-money. The expected volatility is a key input in option pricing models, representing the market's anticipation of future price fluctuations.

Are all options priced using the same model?

While the Black-Scholes model is a cornerstone of option pricing, it is primarily used for European-style options on non-dividend-paying stocks. Other models and variations exist to account for different option types and market conditions, such as American options, dividend-paying stocks, or options on commodities and currencies. These models often build upon the core principles established by Black-Scholes but incorporate additional complexities.

How does the risk-free rate influence an option's value?

The risk-free interest rate also plays a role in determining an option's value. For call options, an increase in the risk-free rate generally leads to a higher option value because the present value of the strike price (which is paid at expiration) decreases. Conversely, for put options, an increase in the risk-free rate generally decreases the option's value. This is due to the impact of present value calculations in the pricing models.