# Option Pricing Theory Explained: Key Models and Their Goals

## What Is Option Pricing Theory?

Option pricing theory estimates the value of an options contract by assessing the likelihood that the contract will be "in the money" at expiration. Market participants, including traders and market makers, employ various models—such as Black-Scholes, binomial option pricing, and Monte-Carlo simulation—to quantify an option's theoretical fair value. Understanding these models helps traders strategize effectively to maximize their financial outcomes.

Models used to price options account for variables such as current market price, [strike price](https://www.investopedia.com/terms/s/strikeprice.asp), volatility, interest rate, and time to expiration to theoretically value an option. Some commonly used models to value options are [Black-Scholes](https://www.investopedia.com/terms/b/blackscholes.asp), binomial option pricing, and Monte-Carlo simulation.

### Key Takeaways

*   Option pricing theory estimates the value of options contracts by calculating the probability of them being in-the-money at expiration.  
    
*   The Black-Scholes model, binomial option pricing, and Monte-Carlo simulation are widely used methods in option pricing theory.  
    
*   Variables such as underlying asset price, strike price, volatility, interest rate, and time to expiration are crucial in determining an option's theoretical fair value.  
    
*   The "Greeks" are derived from option pricing models and measure an option's sensitivity to various risk factors.  
    
*   Real market option prices may differ from theoretical values, but having these values helps traders predict profitability.

## How Option Pricing Theory Works

The primary goal of option pricing theory is to calculate the probability that an option will be [exercised](https://www.investopedia.com/terms/e/exercise.asp), or be ITM, at expiration and assign a dollar value to it. The [underlying asset](https://www.investopedia.com/terms/u/underlying-asset.asp) price (e.g., a stock price), exercise price, volatility, interest rate, and time to expiration, which is the number of days between the calculation date and the option's exercise date, are [commonly-employed variables](https://www.investopedia.com/terms/b/boolean-algebra.asp) that are input 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 are known as an option's "[Greeks](https://www.investopedia.com/terms/g/greeks.asp)". Since market conditions are constantly changing, the Greeks provide traders with a means of determining how sensitive a specific trade is to price fluctuations, [volatility](https://www.investopedia.com/terms/v/volatility.asp) fluctuations, and the passage of time.

> **Important**
>
> 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 in-the-money (ITM) and profitable as it nears expiration if the investor has 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](https://www.investopedia.com/terms/i/interestrate.asp), too, should translate into higher option prices.

## Key Factors in Option Pricing

Marketable options need different valuation methods compared to non-marketable options. Real traded options prices are determined in the [open market](https://www.investopedia.com/terms/o/open-market.asp) and, as with all assets, the value can differ from a theoretical value. However, having the theoretical value allows traders to 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 is used to derive a theoretical price for financial instruments with a known expiration date. However, this is not the only model. The Cox, Ross, and Rubinstein [binomial option pricing model](https://www.investopedia.com/articles/investing/021215/examples-understand-binomial-option-pricing-model.asp) and [Monte-Carlo simulation](https://www.investopedia.com/terms/m/montecarlosimulation.asp) are also widely used.

## Applying the Black-Scholes Model: Real-World Insights

The original Black-Scholes model required five input variables—the strike price of an option, the current price of the stock, time to expiration, the [risk-free rate](https://www.investopedia.com/terms/r/risk-freerate.asp) of return, and volatility. Direct observation of future volatility is impossible, so it must be estimated or implied. Thus, [implied volatility](https://www.investopedia.com/terms/i/iv.asp) is not the same as historical or realized volatility.

> **Fast Fact**
>
> For many options on stocks, dividends are often used as a sixth input.

The Black-Scholes model, one of the most highly regarded pricing models, assumes stock prices follow a [log-normal distribution](https://www.investopedia.com/terms/l/log-normal-distribution.asp) because asset prices cannot be negative. Other assumptions made by the model are that there are no transaction costs or taxes, that the risk-free interest rate is constant for all [maturities](https://www.investopedia.com/terms/m/maturity.asp), that short selling of securities with the use of proceeds is permitted, and that there are no arbitrage opportunities without risk.

Many of these assumptions are not always true in the real world. For example, the model also assumes volatility remains constant over the option's lifespan. This is unrealistic, and normally not the case, because volatility fluctuates with the level of [supply and demand](https://www.investopedia.com/terms/l/law-of-supply-demand.asp).

Options pricing models often adjust for [volatility skew](https://www.investopedia.com/terms/v/volatility-skew.asp), 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 the implied volatility values for options further [out of the money](https://www.investopedia.com/terms/o/outofthemoney.asp) (OTM) are higher than for those at the strike price closer to the price of the underlying instrument.

Also, the Black-Scholes model assumes the options are [European style](https://www.investopedia.com/terms/e/europeanoption.asp), meaning they can only be exercised at maturity. The model doesn't consider [American style](https://www.investopedia.com/terms/a/americanoption.asp) options, which can be exercised anytime before or on the expiration day. On the other hand, the binomial or [trinomial models](https://www.investopedia.com/terms/t/trinomialoptionpricingmodel.asp) can handle both styles of options because they can check for 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, traders can better anticipate an option's potential profitability. Although theoretical values may differ from market prices, having these insights allows traders to make more informed decisions and manage risk effectively. It is crucial to consider factors such as market conditions and the unique characteristics of different option styles to apply these models successfully.