# How the Binomial Option Pricing Model Works

Learn About an Important Method for Valuing Derivatives and Other Assets

![Binomial Option Pricing Model](https://www.investopedia.com/thmb/-ACy5WFxwSWNHilQtzRb5BB1xaE=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/binomialoptionpricing.asp-53b6c346fa1248d5b2985e297af44002.png)

*— Investopedia / Mira Norian*

> **Definition**
>
> The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, between the valuation date and the option's expiration date.

The binomial option pricing model is an options valuation method. Developed in the 1970s by economists John Cox, Stephen Ross, and Mark Rubinstein, the binomial model offers a more intuitive alternative to the famous Black-Scholes formula. It breaks down the life span of an option into discrete periods, assuming that the price of the underlying asset can only move up or down by a certain amount in each step. This simplification allows for a more flexible approach to valuing options, particularly for complex scenarios that its predecessors struggled to handle.

More formally put, the [binomial option pricing model](https://www.investopedia.com/articles/investing/021215/examples-understand-binomial-option-pricing-model.asp) uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option's expiration date.

Below, we explore the mechanics of this tool, probe its applications in real-world trading scenarios, and examine its limitations, as well as ongoing debates about its effectiveness in an era of high-frequency trading (HFT) and machine learning (ML) algorithms.

### Key Takeaways

*   The binomial option pricing model is a technique used to value options by simulating possible paths the underlying asset's price could take over the option's life.
*   It assumes the price of the underlying asset can only move up or down by a certain amount in each time, creating a "binomial tree" of possible price movements.
*   This model is particularly useful for valuing American-style options, which can be exercised before expiration since it evaluates the best exercise times.
*   The model becomes more accurate as the number of time steps increases, eventually converging with the Black-Scholes model in the limit of infinite steps.
*   Key inputs for the model include the stock price, strike price, time to expiration, risk-free interest rate, and the asset's volatility.

The model uses a stepwise binary tree to estimate the changing prices of options, accommodating American-style options that can be exercised at any time before expiration. A simplified example of a [binomial tree](https://www.investopedia.com/terms/b/binomial_tree.asp) might look something like this:

![Image 1](https://www.investopedia.com/thmb/8wDjSmnApyeOoNd5udvpiRL08Rc=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/BinomialOptionPricingModel1_2-d49c047e5bd346d99c54e8bd3378bcdd.png)

**

## Basics of the Binomial Option Pricing Model

With binomial option price models, the assumptions are that there are two possible outcomes—hence, the binomial part of the model. With a pricing model, the two outcomes are a move up, or a move down. The major advantage of a binomial option pricing model is that they’re mathematically simple. Yet these models can become complex in a multi-period model.

In contrast to the [Black-Scholes model](https://www.investopedia.com/terms/b/blackscholes.asp), which provides a numerical result based on inputs, the binomial model allows for the calculation of the asset and the option for multiple periods, along with the range of possible results for each period.

The advantage of this multi-period view is that the user can visualize the change in asset price from period to period and assess the option based on decisions made at different points in time. For American-style [options](https://www.investopedia.com/terms/a/americanoption.asp), which can be exercised at any time before the expiration date, the binomial model can clarify when exercising the option is best and when it should be held for longer periods.

By looking at the binomial tree of values, a trader can determine in advance when a decision on an exercise may occur. If the option has a positive value, it can be exercised. However, if the option has a negative value, it should be held for longer periods.

> **Tip**
>
> The binomial option pricing model is based on the idea that the equilibrium price of an option is equal to the value of a replicating portfolio constructed so it has the same cash ﬂow as the option.

## Calculating Price with the Binomial Model

The basic method of calculating the binomial option model is to use the same probability each period for success and failure [until the option expires](https://www.investopedia.com/articles/optioninvestor/03/090303.asp). However, a trader can incorporate different probabilities for each period based on new information obtained as time passes.

A binomial tree is a useful tool when pricing American-style options and [embedded options](https://www.investopedia.com/terms/e/embeddedoption.asp). Its simplicity is its advantage and disadvantage at the same time. The tree is easy to model out mechanically, but the problem lies in the possible values the underlying asset can take in one period. In a binomial tree model, the underlying asset can only be worth exactly one of two possible values, which is not realistic, as assets can be worth any number of values within any given range.

> **Fast Fact**
>
> Unlike the Black-Scholes Model, the binomial model can handle complex situations such as varying volatility and dividend payments.

For example, there may be a 50/50 chance that the underlying asset price can increase or decrease by 30% in one period. For the second period, however, the probability that the underlying asset price will increase may grow to 70/30.

For example, if an investor is evaluating an [oil well](https://www.investopedia.com/terms/e/exploratory-well.asp), that investor is not sure what the value of that oil well is, but there is a 50/50 chance that the price will go up. If oil prices go up in the first period, making the oil well more valuable, and market [fundamentals](https://www.investopedia.com/terms/f/fundamentals.asp) now point to continued increases in oil prices, the probability of further appreciation in price may now be 70%. The binomial model allows for this flexibility; the Black-Scholes model doesn't.

![Image 2](https://www.investopedia.com/thmb/CagdWYvpJOJei8QHaxl9myf2M3E=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/BinomialOptionPricingModel2_3-1e1c7de15cba4ecda2bbd840ea9406de.png)

**

## How To Use the Binomial Options Pricing Model

The binomial option price mode accommodates a range of financial instruments from standard American and European-style options to more complex derivatives and real options used in corporate finance. Employed not only for pricing and risk management, this model also assists in strategic decision-making and hedging and serves to help understand how options are valued. Its stepwise, tree-based approach provides clear insights into how option values are influenced by market conditions, making it an essential model for analysts, traders, and other professionals in corporate finance.

Financial institutions can use the model to assess the risk associated with holding options. Through multiple market simulations and observing how changes in market variables like interest rates and stock prices affect the option's value, risk managers can better understand potential losses and prepare mitigating strategies.

In addition, traders use the model to devise hedging strategies by understanding how options might perform under various scenarios. The model helps in determining the number of shares needed to hedge against a position in options. This is known as [delta hedging](https://www.investopedia.com/terms/d/deltahedging.asp). Thus, traders are allowed to minimize risk based on predicted market movements. Another use is for valuing exotic options. While primarily designed for standard options, the binomial options pricing model can be adapted to price more complex products, such as exotic options, which have features that standard European or American options do not. The model can be modified to handle path dependent options like Asian and barrier options, although more sophisticated techniques would be needed.

> **Tip**
>
> American-style options can be exercised at any time before their expiration date, providing greater flexibility for the holder. In contrast, European-style options can only be exercised at expiration, making them less flexible but simpler to manage.

Beyond financial markets, the binomial option price mode is applied in real options analysis, which evaluates investment opportunities akin to options in capital budgeting. This approach is used to assess the value of making business decisions, such as expanding, contracting, or deferring investment projects, under uncertainty.

Moreover, the model is a very good educational tool because of its simplicity and step-by-step valuation approach. It helps students and new finance professionals grasp the fundamentals of option pricing before moving on to more complex models like Black Scholes or [Monte Carlo simulations](https://www.investopedia.com/terms/m/montecarlosimulation.asp).

Finally, companies may use the binomial option price mode to price convertible bonds, warrants, and employee stock options. Understanding these instruments' values helps with financing and compensation decisions.

Indeed, the binomial option price model's adaptability to incorporate different types of options and market conditions, coupled with its clear, visual representation of decisions at each step, makes it invaluable in both theoretical finance and practical financial operations.

### Advantages and Disadvantages of Binomial Options Pricing Model

Advantages

*   Flexible
    
*   Easily adjustable
    
*   Intuitive visualization
    

Disadvantages

*   Computationally intensive
    
*   Volatility estimation sensitivity
    
*   Simplistic assumptions
    

The rise of HFT and ML algorithms has helped inform debates about the model's continued relevance. HFT operates on timescales much shorter than traditional option pricing models typically consider, potentially exploiting inefficiencies that the model doesn't capture.

Meanwhile, ML algorithms can process vast amounts of data to identify pricing patterns that may deviate from the binomial model's assumptions. These technologies often require more dynamic, data-driven approaches to option pricing. Nevertheless, the binomial model's fundamental insights remain valuable, even if they need to be supplemented with more sophisticated techniques in certain high-speed or complex market scenarios.

## Other Options Pricing Models

Other options pricing models include the Black-Scholes model, Monte Carlo simulations, and the finite difference method.

*   **Black-Scholes Model**: This model is arguably the most popular options pricing model. The [Black-Scholes Model](https://www.investopedia.com/terms/b/blackscholes.asp) provides a theoretical estimate of the price of European-style options and introduces a continuous time framework. Unlike the binomial option price model, the Black-Scholes model assumes a geometric Brownian motion and is based on the key assumptions that the volatility of the underlying asset is constant and markets are frictionless. It's most suitable for pricing European options where early exercise isn't possible.
*   **Monte Carlo simulations**: This technique uses random sampling and statistical modeling to estimate mathematical functions and simulate the behavior of various assets over time. Monte Carlo simulations are particularly useful for pricing options where the payoff is path dependent, such as Asian options or American options.
*   **Finite difference method**: This model is used to solve differential equations by approximating them with difference equations that finite differences approximate. For options prices, the finite difference method is used to solve partial differential equations typically derived from options pricing formulas, such as those in the Black-Scholes framework. This technique is particularly effective for American options and other derivatives where risk assessment requires an analysis of the effects over time if the boundary conditions can be efficiently modeled.

Each model is chosen based on the specific characteristics of the option being valued and the assumptions that can be reasonably justified for the underlying asset and market dynamics. Here's a summary of these and other options pricing models traders should know:

## Real-World Example of Binomial Option Pricing Model

A simplified example of a [binomial tree](https://www.investopedia.com/terms/b/binomial_tree.asp) has only one step. Assume there is a stock that is priced at $100 per share. In one month, the price of this stock will go up by $10 or go down by $10, creating this situation:

*   **Stock price** = $100
*   **Stock price in one month (up state)** = $110
*   **Stock price in one month (down state)** = $90

Next, assume there is a call option available on this stock that expires in one month and has a strike price of $100. In the up state, this call option is worth $10, and in the down state, it is worth $0. The binomial model can calculate what the price of the call option should be today.

> **Tip**
>
> The binomial model serves as the foundation for more advanced lattice models, which are essential tools in modern financial engineering.

To simplify, assume that an investor purchases a one-half share of stock and writes or sells one call option. The total investment today is the price of half a share less the price of the option, and the possible payoffs at the end of the month are as follows:

*   **Cost today** = $50 - option price
*   **Portfolio value** (up state) = $55 - max ($110 - $100, 0) = $45
*   **Portfolio value** (down state) = $45 - max ($90 - $100, 0) = $45

The portfolio payoff is equal no matter how the stock price moves. Given this outcome, assuming no arbitrage opportunities, an investor should earn the risk-free rate over the course of the month. The cost today must equal the payoff discounted at the risk-free rate for one month. The equation to solve is thus:

*   **Option price** = $50 - $45 × e^(-risk-free rate x T), where e is the mathematical constant 2.7183.

Assuming the risk-free rate is 3% per year, and T equals 0.0833 (one divided by 12), then the price of the call option today is $5.11.

The binomial option pricing model presents two advantages for option sellers over the Black-Scholes model. The first is its simplicity, which allows for fewer errors in the commercial application. The second is its iterative operation, which adjusts prices in a timely manner to reduce the opportunity for buyers to execute arbitrage strategies.

## What are Some of the Limits of the Binomial Options Pricing Model?

One is that the model assumes that volatility is constant over the life of the option. In the real world, markets are dynamic and have spikes during periods of market stress. Another issue is that it's reliant on the simulation of the asset's movements being discrete and not continuous. Thus, the model may not capture rapid price changes effectively, especially if the number of steps is too few.

Lastly, the model overlooks transaction costs, taxes, and spreads. These factors can affect the real cost of executing trades and the timing of such activities, impacting the practical use of the model in real-world trading scenarios.

## How Does the Binomial Options Pricing Model Handle Nonstandard Options?

The binomial trees become more complex as it attempts to handle nonstandard options. Additional parameters, variables, or constraints must be incorporated at each node, and this can make it harder to compute.

## How Transparent and Understandable is the Binomial Options Pricing Model?

The model is arguably one of the more transparent and understandable models, primarily because of its logical and intuitive structure. However, effective communication about its assumptions and limitations is needed to ensure that all stakeholders understand its capabilities and boundaries in practical applications.

## The Bottom Line

The binomial options pricing model stands out for its flexibility and adaptability in handling a variety of options, particularly American-style options. Its stepwise approach in building price paths makes it highly intuitive and capable of accommodating complex scenarios, including variable [dividends](https://www.investopedia.com/terms/d/dividend.asp) and fluctuating interest rates.

However, careful consideration of input parameters and computational resources is required, especially for accurate volatility estimates and maneuvering through many steps. Despite these challenges, the model's ability to visually depict possible price evolutions and adjust for different market conditions makes it a valuable tool for traders' and analysts' arsenal.

> **Article Sources**
>
> 1.  Cox, John C., Ross, Stephen, and Rubenstein, Mark. “[Option Pricing: A Simplified Approach.](https://onlinelibrary.wiley.com/action/getFTRLinkout?url=http%3A%2F%2Fscholar.google.com%2Fscholar_lookup%3Fhl%3Den%26publication_year%3D1978%26author%3DJohn%2BC.%2BCox%26author%3DStephen%2BRoss%26author%3DMark%2BRubinstein%26title%3D%25E2%2580%259COption%2BPricing%253A%2BA%2BSimplified%2BApproach.%25E2%2580%259D&doi=10.1111%2Fj.1540-6261.1979.tb00058.x&linkType=gs&linkLocation=Reference&linkSource=FULL_TEXT)” University of California at Berkeley, September, 1978, Working paper No. 79.
>     
> 2.  Rendleman, Jr., R. J., and Bartter, B. J. "[Two-State Option Pricing](https://onlinelibrary.wiley.com/doi/10.1111/j.1540-6261.1979.tb00058.x)." _Journal of Finance_, vol. 34, no. 5 (1979), pp. 1093-1110.
>     
> 3.  Corporate Finance Institute. "[Option Pricing Models](https://corporatefinanceinstitute.com/resources/knowledge/valuation/option-pricing-models/)."
>     
> 4.  R. S. Johnson. "[Derivatives Markets and Analysis](https://www.wiley.com/en-us/Derivatives+Markets+and+Analysis-p-9781118202692)," Pages 439-504. John Wiley & Sons, 2017.