Understanding the Binomial Option Pricing Model

A Guide With Examples for Learning This Key Idea in Options Trading

Reviewed by Samantha SilbersteinReviewed by Samantha Silberstein

Developed in the 1970s, the binomial option pricing model is a deceptively simple approach to a notoriously complex problem. How do you value options, the derivatives that give a buyer the right to buy or sell a stock or other assets at a certain price on or before a given date? The model imagines a world where stock prices can only move up or down, like the branches of a tree. This basic assumption gives rise to a tool of remarkable accuracy and flexibility.

Valuing options is a notoriously knotty problem that makes valuing stocks seem easy by comparison. Given multiple variables, nonlinear relationships, and uncertain future outcomes, you’re not just speculating on a price going up or down. Options values depend on the asset price, the strike price, the time until expiration, interest rates, and the ever-slippery factor of volatility. Add in the potential for early exercise with American-style options, and you’re facing a multidimensional puzzle that would confound even seasoned traders.

Pricing options without a good model is like trying to hit a moving target while riding a roller coaster. You’re not just predicting where the asset price will be but how it will move along the way and how that movement affects the option’s value at each moment.

Key Takeaways

• The binomial model breaks down option pricing into a series of discrete time steps, making it easier to understand and carry out than more complex continuous-time models.
• The model can price both American and European options, handle dividends, and be adapted for various underlying assets, making it a versatile tool for different market scenarios.
• Understanding the binomial model provides crucial insights into option pricing mechanics, helping investors and financial professionals better assess and manage market risks.
• With the model, there are two possible outcomes with each iteration—a move up or a move down that follow a binomial tree.
• The model is intuitive and is used more often than the well-known Black-Scholes model.

The importance of this model extends far beyond the academic circles where it was invented. It’s a crucial tool for risk management, helping banks and corporations hedge against market volatility. Understanding the binomial model can lead to more informed trading decisions and potentially higher returns for investors. Despite its power, the basics of the model are surprisingly accessible. 

In the sections ahead, we’ll walk through its inner workings and some real-world examples of how it’s used. We’ll also see how it compares to another famous options trading model from Black-Scholes. Whether you’re an options trader, risk manager, or just curious about financial modeling, understanding the binomial option pricing model is valuable.

Understanding the Binomial Option Pricing Model

Options are financial contracts that give the buyer the right, but not the obligation, to buy or sell an underlying asset, like a stock, at a preset price on or before a certain date. For example, a call option allows the holder to buy a stock at a specific price, while a put option allows selling at a specific price. Figuring out a fair price to pay for these options is important for anyone trading them.

There are several methods to value options, each with its own strengths and weaknesses. One of the most widely used is the Black-Scholes model. Developed in 1973, Black-Scholes uses complex math to estimate option prices. While powerful, it had some major limitations. It assumes that the underlying asset price follows a smooth path and that volatility remains constant over the option’s life. It also struggles to handle American-style options that can be exercised before expiry.

The binomial options pricing model, developed by John Cox, Stephen Ross, and Mark Rubinstein in 1979, offers a different approach that addresses some of Black-Scholes’ limitations. The binomial model for pricing options relies on a simple yet elegant idea: breaking down a complex problem into many smaller, more manageable pieces, namely smaller periods. At each step, the model assumes the underlying asset (like a stock) can only do one of two things, namely move up or down in price.

By laying out all the possible paths the stock price could take and working backward from the option payoff at expiry to the present, the binomial model determines a fair value for the option. Let’s compare this to a regular security. Suppose you’re trying to value options for the price of a stock over the next year. There are countless factors that could affect the price: economic conditions, company performance, market sentiment, and so on. Trying to account for all these variables at once would be overwhelming.

The binomial model simplifies this problem by making a key assumption: at any given moment, the price can only move in one of two directions—either up or down. This might seem like an oversimplification, but by repeating this simple either/or choice many times over a series of discrete periods, the model can approximate the complex behavior of real stock prices.

How the Binomial Option Pricing Model Works

Here’s how to do it:

1. Divide the time to expiry into a large number of small time intervals or steps.
2. At each step, assume the stock price can only do one of two things:
3. Move up by a certain factor (u)
4. Move down by a certain factor (d)
5. The factors “u” and “d” are chosen based on the stock’s volatility (how much its price tends to move around) and the length of each time step.
6. By following this up/down process over many steps, the model creates a binomial tree, a diagram depicting all the possible paths the stock price could take from now until the option expires. See the example below.

So, how does this help value an option? The key is to start at the end of the tree (the expiry date) and work backward.

• At expiration, we will know exactly what the option will be worth at each final possible stock price. That’s because, for a call option, it’s the stock price minus the strike price (or zero if the stock is below the strike). For a put, it’s the strike price minus the stock price (or zero if the stock is above the strike).
• Now, we step back to the second-to-last period. For each pair of up/down nodes, we can calculate the expected value of the option in the next period (taking into account the probabilities of up and down moves).
• Discounting this expected value back one period at the risk-free interest rate gives us the option value at each node in the present period.
• We repeat this process, working backward to the present until we have a single value at the starting node. This is the model’s estimate of the fair value of the option today.

Why the Model Works

By breaking down the time to expiry into many small steps and assuming the stock price can only move up or down by a certain amount at each step, the model creates a binomial tree that can approximate a wide range of possible price paths. As the number of steps increases, the tree starts to resemble the continuous price moves we see in the real world.

But can the model handle the fact that investors are often risk-averse? The key is in the careful choice of the up and down factors (the sizes of the “u” and “d”) and the probabilities assigned to them (p and 1-p). These are calibrated based on the stock’s observed volatility and the risk-free interest rate to build in the risk premium that investors would demand without explicitly modeling their risk preferences. This clever approach is known as risk-neutral valuation.

This allows the model to value the option as a simple discounted expected value for future payoffs. While imperfect, it’s proven robust and intuitive for option pricing, combining mathematical rigor and practical flexibility.

Black-Scholes vs. Binomial Option Pricing Model

The binomial options pricing model has several strengths that have made it a popular choice among traders and analysts. One of its main advantages is its flexibility. Unlike the Black-Scholes model, which assumes constant volatility and a continuous price process, the binomial model can accommodate changing volatility and discrete price changes.

This makes it more adaptable to real-world conditions. The model is also intuitive and relatively easy to understand, as it’s based on the simple idea of the stock price moving up or down by certain factors at each step.

In practice, many traders and analysts use both models. They might use Black-Scholes for a quick estimate, then refine it with the binomial model for more complex situations. Some even use the binomial model to check the results of the Black-Scholes formula, to ensure they’re in the right ballpark.

Another significant advantage of the binomial model is its ability to value American-style options, which allow early exercise before expiry. The Black-Scholes model, in contrast, is designed for European-style options that can only be exercised at expiry. By working backward through the binomial tree, the model can check whether early exercise would be optimal at each step, making it suitable for a broader range of options contracts.

However, the binomial model also has some constraints. One disadvantage is that it requires many computations, especially for options with long expiry times. As the number of steps in the tree increases, the model can become computationally intensive. In addition, while the model is flexible, it still relies on some simplifying assumptions. For example, it typically assumes that the risk-free interest rate and the stock’s volatility are constant over the option’s life, which isn’t always true.

Despite these limitations, the binomial model is a valuable tool for option pricing. It balances the computational complexity of more advanced models and the restrictive assumptions of simpler models like Black-Scholes.

Binomial

• Discrete (up/down moves)

• Can accommodate changing volatility

• Can value American and European options

• Relatively intuitive and easy to understand

• Can be computationally intensive for many steps

• More flexible and adaptable to real-world conditions

Black-Scholes

• Continuous

• Assumes constant volatility

• Designed to value European options

• Based on more complex mathematics

• Less computationally intensive

• More restrictive assumptions

Binomial Options Valuation Example

Suppose a stock is trading at $100. We have an at-the-money call option on this stock with a strike price of $100, expiring in one year. Assume we believe that in one year, the stock will either go up to $110 or down to $90.

Two traders, Alice and Bob, agree on these possible future prices but disagree on the probabilities:

• Alice thinks there’s a 60% chance the stock will go up to $110.
• Bob thinks there’s only a 40% chance the stock will go up to $110.

Who do you think would pay more for the call option? You might think Alice since she’s more optimistic about the stock going up. But here’s where the binomial model reveals something interesting.

Using the binomial model (and assuming a risk-free interest rate of 5%), we calculate that the fair price of this call option is about $7.14, regardless of what Alice or Bob think about the probabilities.

Why? Because the binomial model uses something called “risk-neutral pricing.” It doesn’t matter what the actual probabilities are. What matters is the relationship between the possible future prices and the current price.

Binomial Options Calculations

Here’s how to do it:

1. Create a “risk-free” portfolio: Suppose you could create a combination of the stock and the option that would give you the same outcome regardless of the stock price. This is called a “risk-free” portfolio.

2. Build the portfolio: Suppose you buy a certain number of shares (we’ll call this number “d”) and sell one call option.

3. Consider both scenarios:

• If the stock goes up to $110, Your shares are worth 110 × d. You lose $10 on the option: -10 —> Total value: 110d – 10.
• If the stock drops to $90, Your shares are worth 90 × d. The option expires worthless: 0 —> Total value: 90d.

4. Make the portfolio risk-free: For this to be genuinely “risk-free,” both scenarios should give the same result, so you can set them equal: 110d – 10 = 90d.

5. Solve for d: 20d = 10 simplifies to d = 0.5. This means you need to buy half a share of stock for every option you sell to create a risk-free portfolio.

6. Calculate the portfolio value: The value of this risk-free portfolio is 90 × 0.5 = 45

7. Apply the risk-free interest rate: Since this portfolio is risk-free, it should earn the risk-free interest rate (let’s say 5%) over the year. So, its present value is 45 / (1 + 0.05) = 42.86.

8. Calculate the option price: The value of the portfolio (42.86) should equal the value of half a share minus the option price: 42.86 = 0.5 × 100 – option price —> Option price = 7.14

So, the fair price of the call option is $7.14.

This method shows us how to price the option without needing to guess the probability of the stock going up or down. It’s based on the idea that if there were an obvious profit to be made, traders would quickly take advantage of it, making such prospects disappear.

But where is the much-hyped volatility in all these calculations, an essential and sensitive factor that affects options pricing? The volatility is already included in the problem itself. Assuming two price levels ($110 and $90), volatility is implicit in this assumption and included automatically (10% either way in this example).

Are you worried about the math? The Options Clearing Corporation and other platforms offer free calculators to find option values using this and different pricing models.

But how does this approach compare with the commonly used Black-Scholes pricing model? An online options calculator (courtesy of the Options Industry Council) closely matches the computed value:

Binomial Option Pricing Math

Unfortunately, the real world is not as simple as “only two states.” The stock can reach several price levels before the time to expiry.

Is it possible to include all these multiple levels in a binomial pricing model restricted to only two levels? Yes, it is possible, but understanding it takes some simple mathematics.

To generalize this problem and solution:

Say “X” is the market price of a stock, and “X × u” and “X × d” are the future prices for up and down moves “t” years later. Factor “u” will be greater than one, indicating an up move, and “d” will lie between zero and one. For the above example, u = 1.1 and d = 0.9.

The call option payoffs are “P_up” and “P_dn” for up and down moves, respectively, at the time of expiry.

If you build a portfolio of “s” shares bought today and short one call option, then after time “t” you have the following:

$begin{aligned} &text{VUM} = s times X times u – P_text{up} \ &textbf{where:} \ &text{VUM} = text{Value of portfolio in case of an up move} \ end{aligned}$

​VUM=s×X×u−Pup​where:VUM=Value of portfolio in case of an up move​

$begin{aligned} &text{VDM} = s times X times d – P_text{down} \ &textbf{where:} \ &text{VDM} = text{Value of portfolio in case of a down move} \ end{aligned}$

​VDM=s×X×d−Pdown​where:VDM=Value of portfolio in case of a down move​

For a similar valuation in either case of a price move, use the following:

$s times X times u – P_text{up} = s times X times d – P_text{down}$

s×X×u−Pup​=s×X×d−Pdown​

$begin{aligned} s &= frac{ P_text{up} – P_text{down} }{ X times ( u – d) } \ &= text{The number of shares to purchase for} \ &phantom{=} text{a risk-free portfolio} \ end{aligned}$

s​=X×(u−d)Pup​−Pdown​​=The number of shares to purchase for=a risk-free portfolio​

The future value of the portfolio at the end of “t” years will be:

$begin{aligned} text{In Case of Up Move} &= s times X times u – P_text{up} \ &=frac { P_text{up} – P_text{down} }{ u – d} times u – P_text{up} \ end{aligned}$

In Case of Up Move​=s×X×u−Pup​=u−dPup​−Pdown​​×u−Pup​​

$begin{aligned} text{In Case of Down Move} &= s times X times d – P_text{down} \ &=frac { P_text{up} – P_text{down} }{ u – d} times d – P_text{down} \ end{aligned}$

In Case of Down Move​=s×X×d−Pdown​=u−dPup​−Pdown​​×d−Pdown​​

The present-day value can be obtained by discounting it with the risk-free rate of return:

$begin{aligned} &text{PV} = e(-rt) times left [ frac { P_text{up} – P_text{down} }{ u – d} times u – P_text{up} right ] \ &textbf{where:} \ &text{PV} = text{Present-Day Value} \ &r = text{Rate of return} \ &t = text{Time, in years} \ end{aligned}$

​PV=e(−rt)×[u−dPup​−Pdown​​×u−Pup​]where:PV=Present-Day Valuer=Rate of returnt=Time, in years​

This should match the portfolio holding of “s” shares at price X and short call value “c” (present-day holding of (s × X – c) should be the answer to this calculation.) Solving for “c” finally gives the following:

Note: If the call premium is shorted, it should be an addition to the portfolio, not a subtraction.

$c = frac { e(-rt) }{ u – d} times [ ( e ( -rt ) – d ) times P_text{up} + ( u – e ( -rt ) ) times P_text{down} ]$

c=u−de(−rt)​×[(e(−rt)−d)×Pup​+(u−e(−rt))×Pdown​]

Another way to write the equation is by rearranging it:

Taking “q” as follows:

$q = frac { e (-rt) – d }{ u – d }$

q=u−de(−rt)−d​

Then the equation becomes:

$c = e ( -rt ) times ( q times P_text{up} + (1 – q) times P_text{down} )$

c=e(−rt)×(q×Pup​+(1−q)×Pdown​)

Rearranging the equation in terms of “q” offers a new perspective.

Now you can interpret “q” as the probability of the up move of the underlying (as “q” is associated with P_up and “1-q” is associated with P_dn). Overall, the equation represents the present-day option price, the discounted value of its payoff at expiry.

Why This ‘Q’ Is Different

How is this probability “q” different from the probability of an up or down move of the underlying asset?

$begin{aligned} &text{VSP} = q times X times u + ( 1 – q ) times X times d \ &textbf{where:} \ &text{VSP} = text{Value of Stock Price at Time } t \ end{aligned}$

​VSP=q×X×u+(1−q)×X×dwhere:VSP=Value of Stock Price at Time t​

Substituting the value of “q” and rearranging, the stock price at time “t” comes to the following:

$begin{aligned} &text{Stock Price} = e ( rt ) times X \ end{aligned}$

​Stock Price=e(rt)×X​

Presuming only two states, the stock price rises by the risk-free rate of return, precisely like a risk-free asset, and hence remains independent of any risk. Investors are indifferent to risk under this model. So, this forms the risk-neutral model.

The example has one essential requirement: The future payoff structure is required with precision (levels $110 and $90). In real life, such clarity about step-based prices isn’t possible. Instead, the price may settle at multiple levels.

To expand the example further, assume that two-step price levels are possible. We know the second-step final payoffs, and we need to value the option today (at the initial step):

Working backward, the intermediate first step valuation (at t = 1) can be made using final payoffs at step two (t = 2), then using these calculated first step valuation (t = 1), the present-day valuation (t = 0) can be reached.

To get option pricing at number two, payoffs at four and five are used. To get pricing for number three, payoffs at five and six are used. Finally, calculated payoffs at two and three are used to get pricing at number one.

Please note that this example assumes the same factor for up (and down) moves at both steps: u and d are applied in a compounded fashion.

Example #1

Assume a put option with a strike price of $110 is trading at $100 and expiring in one year. The annual risk-free rate is 5%. The price is expected to increase by 20% and decrease by 15% every six months.

Here, u = 1.2 and d = 0.85, x = 100, and t = 0.5 using the above derived formula of

$q = frac { e (-rt) – d }{ u – d }$

q=u−de(−rt)−d​

q = 0.35802832

Thus, the value of a put option at point 2 is as follows:

$begin{aligned} &p_2 = e (-rt) times (p times P_text{upup} + ( 1 – q) P_text{updn} ) \ &textbf{where:} \ &p = text{Price of the put option} \ end{aligned}$

​p2​=e(−rt)×(p×Pupup​+(1−q)Pupdn​)where:p=Price of the put option​

In the P_upup condition, the underlying will be = 100 × 1.2 × 1.2 = $144 leading to P_upup = zero.

In the P_updn condition, the underlying will be = 100 × 1.2 × 0.85 = $102 leading to P_updn = $8.

In the P_dndn condition, the underlying will be = 100 × 0.85 × 0.85 = $72.25 leading to P_dndn = $37.75.

p₂ = 0.975 × (0.358 × 0 + (1-0.358) × 8) = 5.008970741.

Similarly, p₃ = 0.975 × (0.358 × 8+(1-0.358) × 37.75) = 26.43.

$p_1 = e ( -rt ) times ( q times p_2 + ( 1 – q ) p_3 )$

p1​=e(−rt)×(q×p2​+(1−q)p3​)

And hence value of put option, p₁ = 0.975 × (0.358 × 5.01+(1-0.358) × 26.43) = $18.29.

Similarly, binomial models allow you to break the entire option duration to even more refined levels. Using computer programs or spreadsheets, you can work backward one step at a time to get the present value of the desired option.

Example #2

Assume a European-type put option with nine months to expiry, a strike price of $12, and an underlying price of $10. Assume a risk-free rate of 5% for all periods. Assume that every three months, the underlying price can move 20% up or down, giving us u = 1.2, d = 0.8, t = 0.25, and a three-step binomial tree.

Red indicates underlying prices, while blue is for the payoff of put options.

The risk-neutral probability “q” computes to 0.531.

Using the above value of “q” and payoff values at t = nine months, the corresponding values at t = six months are computed as follows:

Further, using these computed values at t = 6, values at t = 3, and then at t = 0 are as follows:

That gives the present-day value of a put option as $2.18, pretty close to what you would find doing the computations using the Black-Scholes model, which is $2.30.

When Is it a Good Time to Use the Binomial Options Pricing Model?

If you need to price an American option that can be exercised before expiry, the binomial model is a good choice. It’s also a good model to use to account for changing volatility or dividend payments since it’s flexible enough to allow for these. While more computationally intensive, the binomial model can often provide more accurate prices than simpler models like Black-Scholes.

What Else Can a Binomial Model Be Used for?

In addition to calculating the value of an option, the binomial model can also be used for valuing projects or investments with a high degree of uncertainty, capital budgeting and resource-allocation decisions—as well as projects with multiple periods or an embedded option to either continue or abandon the project at certain points in time. Complex investment decisions, like oil drilling projects, can employ the binomial model by breaking down the process into a series of decision points with different possible outcomes. This approach allows companies to assess risks, quantify potential rewards, and make more informed decisions by considering various scenarios and the flexibility to adapt as new information becomes available.

What Are Embedded Options?

Embedded options are features woven into certain financial securities, giving either the holder or issuer rights to alter the instrument’s cash flows or value under pre-defined conditions. Unlike standalone options that are traded separately, these options are integral components of the security itself. They are more common in fixed-income securities. For instance, a callable bond grants the issuer the right to redeem the bond before its maturity date, typically when interest rates fall. This feature benefits the issuer but can leave investors facing reinvestment risk. Conversely, putable bonds give bondholders the right to sell the bond back to the issuer, providing a form of downside protection.

The Bottom Line

Although computer programs can make these calculations easier, predicting future prices remains a significant limitation of binomial models for option pricing. The finer the time intervals, the more difficult it gets to predict the payoffs at the end of each period with high-level precision.

However, the flexibility to incorporate the changes expected at different periods is a plus, which makes it suitable for pricing American options, including early exercise valuations. The values computed using the binomial model closely match those calculated from other commonly used models like Black-Scholes, suggesting that the binomial models are useful and accurate. Binomial pricing models can be developed according to a trader’s preferences and can work as an alternative to Black-Scholes.

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