Option valuation: Meaning, Criticisms & Real-World Uses

What Is Option Valuation?

Option valuation refers to the systematic process of determining the theoretical fair value of an option contract, a type of derivatives instrument. This process falls under the broader discipline of quantitative finance, which employs mathematical and statistical methods to analyze financial markets and securities. Accurate option valuation is crucial for traders, investors, and financial institutions to make informed decisions regarding buying, selling, or writing options, as it helps assess whether an option is underpriced or overpriced in the market. The value of an option is influenced by several factors, including the price of the underlying asset, the strike price, the time remaining until expiration date, the volatility of the underlying asset, and prevailing interest rates.

History and Origin

The systematic approach to option valuation gained significant traction with the publication of the Black-Scholes model. Prior to this, option pricing was largely based on intuition, simplified calculations, or crude approximations. In 1973, Fischer Black and Myron Scholes published their groundbreaking paper, "The Pricing of Options and Corporate Liabilities," in the Journal of Political Economy. This seminal work provided a rigorous mathematical framework for valuing European-style options. Their model revolutionized financial markets by offering a consistent and widely applicable method for determining option prices, coinciding with the establishment of the Chicago Board Options Exchange (CBOE) in the same year. Robert C. Merton further developed and generalized the model. Black, Scholes, and Merton are widely credited with laying the foundation for the modern derivatives market, with Merton and Scholes receiving the Nobel Memorial Prize in Economic Sciences in 1997 for their work.⁶,⁵

Key Takeaways

Option valuation is the process of estimating the fair theoretical price of an option contract.
The value of an option is primarily influenced by the underlying asset's price, strike price, time to expiration, volatility, and interest rates.
The Black-Scholes model is a cornerstone of modern option valuation, providing a mathematical formula for European-style options.
Accurate option valuation supports informed trading decisions, risk management, and portfolio optimization.
Option valuation models have limitations, particularly concerning their assumptions about market behavior.

Formula and Calculation

The most widely recognized formula for option valuation is the Black-Scholes model, designed for European-style options. While there are extensions for American options, the original model provides a foundational understanding.

For a non-dividend-paying stock, the price of a European call option (C) and a European put option (P) are given by:

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

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

Where:
(d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}})
(d_2 = d_1 - \sigma \sqrt{T})

And:

(S_0) = Current price of the underlying asset
(K) = Strike price of the option
(T) = Time to expiration date (in years)
(r) = Risk-free rate (annualized)
(\sigma) = Volatility of the underlying asset's returns
(N(x)) = Cumulative standard normal distribution function
(e) = Euler's number (approximately 2.71828)
(\ln) = Natural logarithm

This formula calculates the theoretical value based on the inputs, allowing market participants to compare it against the actual market price. Another important model is the binomial option pricing model, which breaks down the time to expiration into discrete steps.

Interpreting Option Valuation

Interpreting the results of option valuation involves comparing the calculated theoretical value with the actual market price of the option. If the theoretical value of an option valuation model is higher than its market price, the option may be considered undervalued, suggesting a potential buying opportunity. Conversely, if the theoretical value is lower than the market price, the option might be overvalued, potentially indicating a selling opportunity or a reason to avoid purchasing.

Market participants often use option valuation models to identify potential arbitrage opportunities or to gauge the market's expectation of future volatility, known as implied volatility. Differences between theoretical and market prices can also arise from factors not fully captured by models, such as liquidity or market sentiment.

Hypothetical Example

Consider an investor evaluating a call option on XYZ Company stock with the following characteristics:

Current Stock Price ((S_0)): $100
Strike Price ((K)): $105
Time to Expiration ((T)): 0.5 years (6 months)
Risk-Free Rate ((r)): 3% (0.03)
Volatility ((\sigma)): 20% (0.20)

Using the Black-Scholes formula for a call option:

First, calculate (d_1) and (d_2):
(d_1 = \frac{\ln(100/105) + (0.03 + 0.20^2/2) \times 0.5}{0.20 \sqrt{0.5}})
(d_1 = \frac{\ln(0.95238) + (0.03 + 0.02) \times 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})
(d_2 = -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 = 100 \times N(0.1682) - 105 \times e^{(-0.03 \times 0.5)} \times N(0.3096))
(Note: since (d_1) and (d_2) were negative, we need to use (N(d_1)) and (N(d_2)) with the positive values if the formula is for N(x) where x is positive, or use N(-x) if the values are negative. For a call, we need N(d1) and N(d2). Re-evaluating N(x) for positive values.)

Let's re-calculate (N(d_1)) and (N(d_2)) assuming a typical table for N(x) for positive x. For negative values, (N(x) = 1 - N(-x)).
(N(d_1) = N(-0.1682) = 1 - N(0.1682) \approx 1 - 0.5668 = 0.4332)
(N(d_2) = N(-0.3096) = 1 - N(0.3096) \approx 1 - 0.6215 = 0.3785)

The call option price is:
(C = 100 \times 0.4332 - 105 \times e^{-0.015} \times 0.3785)
(C = 43.32 - 105 \times 0.9851 \times 0.3785)
(C = 43.32 - 39.19 \approx 4.13)

The theoretical value of the call option is approximately $4.13. If the market price for this call option is $3.50, the model suggests it might be slightly undervalued. If the market price is $4.80, it might be overvalued.

Practical Applications

Option valuation models are fundamental tools with diverse practical applications across financial markets:

Trading and Investment Decisions: Traders use valuation models to determine whether an option is fairly priced, overpriced, or underpriced relative to its theoretical value, guiding their buy or sell decisions. This helps in identifying potential trading opportunities or avoiding unfavorable trades.
Hedging Strategies: Financial institutions and corporations employ option valuation techniques to design and execute hedging strategies that mitigate various forms of financial risk, such as currency risk, commodity price risk, or interest rate risk.
Risk Management: Option valuation contributes to robust risk management by allowing firms to assess their exposure to option positions and stress-test portfolios under different market scenarios. Regulatory bodies, such as FINRA, also implement rules governing options trading, including position limits and reporting requirements, which are often informed by valuation considerations to protect investors and maintain market integrity.⁴
Derivatives Pricing: Beyond simple options, the methodologies developed for option valuation are extended to price more complex derivatives and structured products.
Implied Volatility Calculation: By taking the market price of an option and reversing the valuation formula, practitioners can calculate the market's consensus on future volatility, known as implied volatility. This metric is a key indicator of market sentiment and expectations. The surge in options trading, particularly among retail investors, has become a notable market trend, with call options representing a significant portion of trading volumes.³

Limitations and Criticisms

While revolutionary, option valuation models, particularly the Black-Scholes model, operate under several simplifying assumptions that can limit their accuracy in real-world scenarios. Critics often point to these assumptions as potential drawbacks:

Constant Volatility: The model assumes that the volatility of the underlying asset remains constant until expiration, which is rarely true in dynamic markets. Volatility tends to fluctuate significantly, leading to discrepancies between theoretical and actual prices.²
Constant Risk-Free Rate: Similarly, the assumption of a constant risk-free rate is often violated, as interest rates change over time.
No Dividends (for the basic model): The original Black-Scholes model does not account for dividends paid out by the underlying asset. While extensions exist to incorporate dividends, it adds complexity.
European-Style Options Only: The basic formula is strictly for European options, which can only be exercised at expiration date. Most exchange-traded options are American-style, allowing early exercise, which adds a layer of complexity not captured by the basic formula.
No Arbitrage Opportunity: The model assumes perfectly efficient markets where arbitrage opportunities are immediately exploited. In reality, minor arbitrage opportunities might exist briefly due to market inefficiencies or transaction costs.
Normal Distribution of Returns: The model assumes that the returns of the underlying asset follow a log-normal distribution, implying that asset prices are normally distributed. This assumption often fails to account for "fat tails" (extreme price movements) observed in financial markets, leading to mispricing of out-of-the-money options.¹

Despite these limitations, option valuation models remain essential for providing a theoretical baseline and a common language for market participants. Adjustments and more complex models, like jump-diffusion models or local volatility models, have been developed to address some of these shortcomings.

Option Valuation vs. Option Pricing

While often used interchangeably by beginners, "option valuation" and "option pricing" have distinct nuances in finance. Option valuation refers to the broader analytical process of determining the fair theoretical value of an option contract using mathematical models and various inputs. It is the act of assessing what an option should be worth based on established financial theory and market parameters. The output of option valuation is a theoretical price. Option pricing, on the other hand, typically refers to the actual market price at which an option is currently trading on an exchange or in the over-the-counter market. This price is determined by supply and demand forces among buyers and sellers. While option valuation models provide a benchmark, the actual option pricing in the market can deviate due to factors like market sentiment, liquidity, and imbalances in trading activity, demonstrating that while valuation aims for theoretical correctness, pricing reflects real-time market dynamics.

FAQs

What are the main factors influencing option valuation?

The key factors influencing option valuation include the current price of the underlying asset, the strike price of the option, the time remaining until the expiration date, the expected future volatility of the underlying asset, and the prevailing risk-free rate.

Is the Black-Scholes model the only way to value options?

No, while the Black-Scholes model is the most famous and widely used, especially for European-style options, other models exist. The binomial option pricing model is another common method, particularly useful for American-style options due to its flexibility in handling early exercise. More complex models are also used for exotic options or to address specific market conditions not captured by Black-Scholes.

Why is option volatility so important in option valuation?

Volatility is a critical input in option valuation because it quantifies the expected magnitude of price fluctuations in the underlying asset. Higher volatility increases the probability that an option will finish in-the-money, thus increasing its value. This applies to both call option and put option prices, as greater price swings create more opportunities for profitable exercise.