What Is Option Pricing Theory?

What Is Option Pricing Theory?

Let me explain option pricing theory to you directly: it estimates the value of an options contract by assessing the likelihood that the contract will be 'in the money' at expiration. As a trader or market maker, you would use various models—such as Black-Scholes, binomial option pricing, and Monte-Carlo simulation—to quantify an option's theoretical fair value. Understanding these models allows you to strategize effectively and maximize your financial outcomes.

These models account for variables like the current market price, strike price, volatility, interest rate, and time to expiration to theoretically value an option. Some commonly used ones are Black-Scholes, binomial option pricing, and Monte-Carlo simulation.

How Option Pricing Theory Works

The primary goal here is to calculate the probability that an option will be exercised, or be in-the-money (ITM), at expiration and assign a dollar value to it. You input variables like the underlying asset price (such as a stock price), exercise price, volatility, interest rate, and time to expiration—the number of days between the calculation date and the option's exercise date—into mathematical models to derive an option's theoretical fair value.

Options pricing theory also derives various risk factors or sensitivities based on those inputs, which we know as an option's 'Greeks'. Since market conditions change constantly, the Greeks give you a way to determine how sensitive a specific trade is to price fluctuations, volatility fluctuations, and the passage of time.

Remember, the more likely an option will finish in-the-money (ITM) and be profitable, the more valuable it is, and vice versa. An option becomes more likely to be ITM and profitable as it nears expiration if you have more time to exercise it—this means, all else equal, longer-dated options are more valuable. Similarly, the more volatile the underlying asset, the greater the odds that it will expire ITM. Higher interest rates should translate into higher option prices too.

Key Factors in Option Pricing

Marketable options require different valuation methods compared to non-marketable ones. Real traded options prices are determined in the open market, and as with all assets, the value can differ from a theoretical value. However, having the theoretical value lets you assess the likelihood of profiting from trading those options.

The current options market evolved significantly due to the 1973 pricing model by Fischer Black and Myron Scholes. The Black-Scholes formula derives a theoretical price for financial instruments with a known expiration date. But this isn't the only model—the Cox, Ross, and Rubinstein binomial option pricing model and Monte-Carlo simulation are also widely used.

Applying the Black-Scholes Model: Real-World Insights

The original Black-Scholes model requires five input variables: the strike price of an option, the current price of the stock, time to expiration, the risk-free rate of return, and volatility. You can't directly observe future volatility, so it must be estimated or implied—thus, implied volatility isn't the same as historical or realized volatility. For many options on stocks, dividends are often used as a sixth input.

The Black-Scholes model assumes stock prices follow a log-normal distribution because asset prices cannot be negative. Other assumptions include no transaction costs or taxes, a constant risk-free interest rate for all maturities, permission for short selling of securities with use of proceeds, and no arbitrage opportunities without risk.

Many of these assumptions don't always hold in the real world. For example, the model assumes volatility remains constant over the option's lifespan, which is unrealistic because volatility fluctuates with supply and demand levels. Options pricing models often adjust for volatility skew, which shows how implied volatilities vary across different strike prices for options with the same expiration date. The resulting shape often shows a skew or 'smile' where implied volatility values for options further out of the money (OTM) are higher than for those at the strike price closer to the underlying instrument's price.

Also, the Black-Scholes model assumes the options are European style, meaning they can only be exercised at maturity. It doesn't consider American style options, which can be exercised anytime before or on the expiration day. On the other hand, binomial or trinomial models can handle both styles because they check the option's value at every point in time during its life.

The Bottom Line

Option pricing theory provides a framework for calculating the fair value of an options contract using variables such as stock price, strike price, volatility, interest rate, and time to expiration. By understanding models like Black-Scholes, binomial pricing, and Monte-Carlo simulations, you can better anticipate an option's potential profitability. Although theoretical values may differ from market prices, these insights allow you to make more informed decisions and manage risk effectively. It's crucial to consider factors such as market conditions and the unique characteristics of different option styles to apply these models successfully.