Option pricing

What Is Option Pricing?

Option pricing is the process of determining the theoretical fair value of an option contract. This valuation falls under the broader financial category of derivatives, which are financial instruments whose value is derived from an underlying asset. The price of an option, often referred to as its premium, is influenced by several factors, including the price of the underlying asset, the strike price, the expiration date, the asset's volatility, and prevailing interest rates. Effective option pricing allows market participants to assess whether an option is overvalued or undervalued, facilitating informed trading and hedging strategies.

History and Origin

The mathematical modeling of option prices has roots in early 20th-century work, notably Louis Bachelier's 1900 thesis which applied Brownian motion to derivative pricing. However, modern option pricing theory gained significant traction in the 1970s. The most influential breakthrough came with the publication of "The Pricing of Options and Corporate Liabilities" by Fischer Black and Myron Scholes in the Journal of Political Economy in 1973. This seminal paper introduced what is now widely known as the Black-Scholes model, providing the first widely accepted mathematical method to calculate the theoretical value of an option contract.¹⁹, ²⁰ Robert C. Merton further contributed to the model, leading to it sometimes being referred to as the Black-Scholes-Merton (BSM) model. The model's development coincided with the opening of the Chicago Board Options Exchange (CBOE) in April 1973, revolutionizing the derivatives market and providing mathematical legitimacy to options trading.¹⁸

Key Takeaways

• Option pricing aims to determine the theoretical fair value of an option contract.
• The Black-Scholes model, developed in 1973, is a foundational framework for pricing European options.
• Key inputs for option pricing include the underlying asset's price, strike price, time to expiration, volatility, and the risk-free rate.
• Option prices consist of two main components: intrinsic value and time value.
• While influential, option pricing models like Black-Scholes operate under specific assumptions that may not always hold in real-world market conditions.

Formula and Calculation

The Black-Scholes model is a partial differential equation that can be solved to yield a formula for pricing a non-dividend-paying European call option ($C$) and put option ($P$).

For a call option:
$C = S_0 N(d_1) - K e^{-rT} N(d_2)$

For a 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 (in years)
• $r$ = Risk-free rate (annualized)
• $\sigma$ = Volatility of the underlying asset's returns
• $N(x)$ = Cumulative standard normal distribution function (representing the probability that a standard normal random variable will be less than or equal to $x$)
• $e$ = Euler's number (approximately 2.71828)
• $\ln$ = Natural logarithm

This formula helps determine the theoretical fair value by accounting for the various factors that influence the option's potential profitability and the time value of money.

Interpreting the Option Price

Interpreting the price of an option involves understanding its two primary components: intrinsic value and time value. Intrinsic value is the immediate profit that could be realized if the option were exercised instantly. For a call option, it is the difference between the underlying asset's price and the strike price, if positive. For a put option, it is the difference between the strike price and the underlying asset's price, if positive. Options with no intrinsic value are considered out-of-the-money.

The time value, also known as extrinsic value, is the portion of an option's premium beyond its intrinsic value. This reflects the possibility that the option's intrinsic value could increase before expiration. Factors such as time remaining until the expiration date and the volatility of the underlying asset significantly influence time value. Options with more time until expiration and higher expected volatility generally have greater time value. As an option approaches its expiration, its time value erodes, a phenomenon known as time decay.

Hypothetical Example

Consider a hypothetical call option on Company XYZ stock with a strike price of $100, expiring in six months. The current stock price is $105. Assume the annualized risk-free rate is 2% and the stock's volatility is 30%.

1. Identify Inputs:

   • $S_0 = \$105$
   • $K = \$100$
   • $T = 0.5$ years (six months)
   • $r = 0.02$
   • $\sigma = 0.30$

2. Calculate $d_1$:
   $d_1 = \frac{\ln(105/100) + (0.02 + 0.30^2/2) \times 0.5}{0.30\sqrt{0.5}}$
   $d_1 = \frac{\ln(1.05) + (0.02 + 0.045) \times 0.5}{0.30 \times 0.7071}$
   $d_1 = \frac{0.04879 + 0.0325}{0.21213} \approx \frac{0.08129}{0.21213} \approx 0.3832$

3. Calculate $d_2$:
   $d_2 = 0.3832 - 0.30\sqrt{0.5} = 0.3832 - 0.21213 \approx 0.1711$

4. Find N($d_1$) and N($d_2$):
   Using a standard normal distribution table or calculator:
   $N(0.3832) \approx 0.6492$
   $N(0.1711) \approx 0.5680$

5. Calculate Call Option Price ($C$):
   $C = 105 \times 0.6492 - 100 \times e^{-0.02 \times 0.5} \times 0.5680$
   $C = 68.166 - 100 \times e^{-0.01} \times 0.5680$
   $C = 68.166 - 100 \times 0.99005 \times 0.5680$
   $C = 68.166 - 56.235 \approx \$11.93$

In this example, the theoretical fair value, or premium, for the call option is approximately $11.93.

Practical Applications

Option pricing is a fundamental component of modern financial markets, affecting various aspects of investing, risk management, and market analysis. It underpins the valuation of exchange-traded options and over-the-counter (OTC) derivatives.

• Trading and Investment: Investors and traders use option pricing models to identify potentially mispriced options. If an option's market price deviates significantly from its theoretical value, it may present an arbitrage opportunity or a chance to take a directional view. The volume of options trading is substantial, with Cboe Global Markets, for instance, reporting billions of contracts traded annually across its options exchanges.¹⁷
• Risk Management and Hedging: Institutional investors, including pension funds and hedge funds, widely use options for hedging purposes.¹⁴, ¹⁵, ¹⁶ By pricing options accurately, institutions can create positions that offset potential losses in their underlying portfolios, manage exposure to market volatility, or enhance portfolio income.¹², ¹³ For example, a portfolio manager might purchase put options to protect against a decline in the value of their stock holdings.
• Implied Volatility Calculation: Option pricing models can be used in reverse to derive the implied volatility of an underlying asset from the market price of its options. This implied volatility is a forward-looking measure of market expectations for future price swings and is closely watched by market participants, notably through indices like the Cboe Volatility Index (VIX), which is based on S&P 500 options prices.¹¹
• Corporate Finance: Beyond financial market instruments, option pricing methodologies are also applied in corporate finance to value real options embedded in projects, employee stock options, and other corporate liabilities.

Limitations and Criticisms

While revolutionary, option pricing models, particularly the Black-Scholes model, have several notable limitations and have faced criticism, especially during periods of market stress.

One primary criticism is that the Black-Scholes model makes several simplifying assumptions that do not always hold true in real-world markets. These assumptions include:

• Constant Volatility: The model assumes that the volatility of the underlying asset remains constant over the option's life. In reality, volatility fluctuates dynamically.
• Constant Risk-Free Rate: It assumes a constant and known risk-free interest rate, which is not always the case in practice.
• No Dividends (in its original form): The initial model assumed the underlying asset does not pay dividends during the option's life, though extensions have been developed to account for this.
• No Transaction Costs or Taxes: The model presumes no transaction costs or taxes, which are present in actual trading.¹⁰
• Efficient Markets and No Arbitrage: It assumes perfectly efficient markets where continuous trading is possible without friction and no arbitrage opportunities exist.⁹
• Log-Normal Distribution of Returns: The model assumes that the underlying asset's prices follow a log-normal distribution, implying that returns are normally distributed. However, real-world asset returns often exhibit "fat tails" (more extreme events) and skewness (asymmetric distribution) compared to a normal distribution.⁸

These deviations from assumptions can lead to discrepancies between theoretical option prices and actual market prices, particularly for options far out-of-the-money or with long maturities.⁷ During events like the 2008 financial crisis, some argued that the misapplication or blind reliance on such models, without understanding their limitations, contributed to significant financial losses.⁴, ⁵, ⁶ While the Black-Scholes model is a powerful tool, awareness of its inherent simplifications is crucial for its responsible use.³

Option Pricing vs. Implied Volatility

Option pricing and implied volatility are intrinsically linked but represent different aspects of an option's characteristics. Option pricing is the quantitative process of determining the theoretical fair value of an option contract based on a set of known inputs, such as the underlying asset price, strike price, time to expiration, risk-free rate, and estimated volatility. It provides a specific monetary value for the option.

In contrast, implied volatility is not an input to the standard Black-Scholes model but rather an output derived from an option's actual market price. Given the market price of an option and all other input variables (underlying price, strike, time, risk-free rate), the implied volatility is the volatility level that, when plugged into the pricing model, yields the observed market price. It represents the market's collective forecast of the underlying asset's future volatility for the life of that specific option. While option pricing uses volatility to arrive at a price, implied volatility uses the price to infer market-expected volatility. The distinction is crucial: option pricing gives a value, while implied volatility gives a market expectation of future price movement.

FAQs

What factors influence option pricing?

The primary factors influencing option pricing are the current price of the underlying asset, the strike price, the time remaining until the expiration date, the expected volatility of the underlying asset's price, and the prevailing risk-free rate. For dividend-paying assets, expected dividends also play a role.²

Why is option pricing important?

Option pricing is important because it allows market participants to determine a theoretical fair value for options. This helps traders assess whether an option is priced appropriately in the market, aids in designing hedging strategies, and enables the creation of complex derivatives products. It provides a common framework for valuation across the market.

Are all option pricing models the same?

No, not all option pricing models are the same. While the Black-Scholes model is the most well-known, other models exist, such as the Binomial Option Pricing Model, which can be more suitable for pricing American options (options that can be exercised before expiration) or those on dividend-paying stocks. More advanced models also account for phenomena like the "volatility smile" or "skew," which the basic Black-Scholes model does not fully capture.¹