Binomial Option Pricing Model

The Binomial Option Pricing Model uses a price tree to estimate the theoretical value of an option. At each step, the underlying asset’s price is assumed to move either up or down.

• Time to expiry is divided into fixed intervals.
• Each interval has an upward and downward price movement.
• Option payoffs are calculated at expiry.
• Values are then worked backwards to the present.
• Risk-neutral probabilities are used for valuation.
• The model can account for early exercise in American options.
• More steps generally produce a more detailed estimate.
• The calculated value may differ from the option’s market price.
   

The Binomial Option Pricing Model, or BOPM, is a numerical method used to estimate the theoretical value of an option.

It creates a tree showing different prices that the underlying asset may reach before the option expires. At every stage, the price is assumed to move in one of two directions:

• Up

• Down

  
  

For example, suppose a share is currently priced at ₹100. After one period, the model may assume that its price will either rise to ₹120 or fall to ₹90.

The model calculates the option payoff under both outcomes. It then uses risk-neutral probabilities and the risk-free interest rate to estimate the option’s present value.

The model is especially useful for American options, which may be exercised at any time before expiry.

Additional Read: What is Futures and Options 

The model uses a binomial tree, also called a price tree or lattice, to show possible movements in the underlying asset’s price.

1. Binomial tree construction

The time remaining until expiry is divided into fixed intervals or steps.

At every point in the tree, called a node, the asset price may move upward or downward. The size of these movements is generally based on volatility and the length of each step.

For example, in a two-step model, the price may first rise or fall. From each of those prices, it may rise or fall again during the next step.

2. Risk-neutral valuation

The model uses risk-neutral valuation. Under this approach, the expected return on the underlying asset is linked to the risk-free interest rate for calculation purposes.

Risk-neutral probability is a mathematical input. It does not predict the actual chance of the asset price rising or falling.

3. Option valuation

The option’s payoff is first calculated at the final nodes of the tree.

For a call option:

Call payoff = Higher of ₹0 or underlying price minus strike price

For a put option:

Put payoff = Higher of ₹0 or strike price minus underlying price

Suppose a call option has a strike price of ₹100. If the underlying asset is worth ₹125 at expiry, the payoff is ₹25. If the asset is worth ₹90, the payoff is zero.

The model then works backwards through the tree by calculating and discounting the probability-weighted values at each earlier node.

4. Decision nodes and early exercise

The model can account for the early exercise of American options.

At each node before expiry, it compares:

• The value of exercising the option immediately
• The estimated value of continuing to hold. The higher value is used.

For example, if exercising an American put produce ₹20, while continuing to hold it has an estimated value of ₹17, early exercise may be treated as the better choice at that node.

5. Convergence to the Black–Scholes model

A tree with only one or two steps gives a simplified estimate. As the number of steps increases, the intervals become smaller, and the tree becomes more detailed. Under similar assumptions, the result generally moves closer to the value produced by the Black–Scholes model.

The model is used to estimate option values under different price-movement scenarios.

• Valuing American options

  
  

  It can evaluate whether exercising an American option before expiry may be beneficial.

• Understanding price movements

  
  

  The tree shows how upward and downward movements in the underlying asset may affect the option’s value.

  
  

• Assessing option sensitivity

  
  

  You can change inputs such as price, volatility, interest rate and time to expiry to see how the estimated option value responds. For example, higher assumed volatility creates a wider range of possible future prices, which may affect the theoretical option value.

  
  

• Evaluating complex payoff structures

  
  

  Modified versions of the model may be used for certain barrier options and path-dependent options. These contracts may require additional calculations because their value can depend on the path followed by the underlying asset.

  
  

• Risk management analysis

  
  

  The model helps traders and analysts compare possible risks and outcomes under different assumptions. However, it cannot predict which outcome will occur.

  
  

• Educational and analytical use

  
  

  Its step-by-step structure makes it useful for understanding how option prices are calculated.

  
  

• Comparing with other models

  
  

  The result can be compared with values produced by models such as Black–Scholes. Differences may arise because the models use different assumptions.

  
  
  

The BOPM uses several simplifying assumptions.

1. Discrete time

The model divides time into separate intervals. At the end of each interval, the underlying asset’s price may move up or down.

2. No arbitrage

The model assumes that there are no opportunities to earn a risk-free profit from price differences between related securities. This assumption helps maintain consistent option pricing.

3. Two possible outcomes

At each step, the underlying asset’s price can either rise by an up factor or fall by a down factor. Actual market prices can move by many different amounts, but the model simplifies them into two possible outcomes.

4. Constant volatility

A basic version of the model assumes that volatility remains constant during the option’s life. Actual volatility may change. More advanced versions may use different volatility assumptions at different stages.

5. Interest-rate assumptions

The model generally uses a known risk-free interest rate to discount future option values. Basic models may assume that this rate remains constant throughout the valuation period.

6. Dividend treatment

A basic example may assume that the underlying asset does not pay dividends. However, adjusted versions of the model can include expected dividends or dividend yields because dividend payments may affect the asset price and early-exercise decisions.

The main inputs used in the model include:

• Current underlying asset price
• Strike price
• Time to expiry
• Volatility
• Risk-free interest rate
• Expected dividends, where applicable
• Up factor
• Down factor
• Risk-neutral probability

1. Calculation of up and down factors

The up factor, represented by u, shows how much the asset price may rise during one step.

The down factor, represented by d, shows how much it may fall.

If the current asset price is S:

• Upward price = S × u
• Downward price = S × d

2. Risk-neutral probability

The risk-neutral probability, represented by p, is used to weight the upward outcome. The downward outcome is weighted by 1 − p.

In a basic model without dividends:

p = (risk-free growth factor − d) ÷ (u − d)

The result normally needs to fall between zero and one for the inputs to support a valid no-arbitrage calculation.

3. Option valuation at final nodes

At expiry, the payoff is calculated at every final node.

For a call option:

Maximum of ₹0 or asset price minus strike price

For a put option:

Maximum of ₹0 or strike price minus asset price

4. Discounting future values

The final option values are weighted using risk-neutral probabilities and discounted at the risk-free rate.

For a one-period model:

Option value today = Discounted value of [p × upward payoff + (1 − p) × downward payoff]

For a multi-step model, the process is repeated backwards through the tree.

Suppose shares of XYZ Ltd. are trading at ₹150. A one-year call option has a strike price of ₹160.
The assumptions are:

• Current share price: ₹150
• Strike price: ₹160
• Time to expiry: 1 year
• Risk-free rate: 6%
• Up factor: 1.30
• Down factor: 0.80

1. Constructing the binomial price tree

Price movement    Calculation    Possible price
Upward movement    ₹150 × 1.30    ₹195
Downward movement    ₹150 × 0.80    ₹120

2. Calculating option prices at final nodes

Share price    Calculation    Call payoff
₹195    ₹195 − ₹160    ₹35
₹120    Maximum of ₹0 or ₹120 − ₹160    ₹0

3. Calculating today’s option price

The risk-neutral probability is:
p = (1.06 − 0.80) ÷ (1.30 − 0.80)
p = 0.52
The upward outcome is weighted by 0.52, while the downward outcome is weighted by 0.48.
The expected future payoff is:
(₹35 × 0.52) + (₹0 × 0.48) = ₹18.20
The value is then discounted:
₹18.20 ÷ 1.06 = approximately ₹17.17
Therefore, the estimated present value of the call option is approximately ₹17.17.
This is a theoretical estimate. The actual market price may differ because of changing volatility, liquidity, transaction costs, demand and supply.
 

• Transparent calculations
  The tree clearly shows how the underlying asset and option values change under different scenarios.
• Multi-period view
  The time to expiry can be divided into several steps, providing a detailed view of possible price movements.
• Flexibility for American options
  The model can compare immediate exercise with continued holding at each node.
• Adjustable assumptions
  Modified versions can include dividends, different volatility levels and other contract features.
• Scenario analysis
  The inputs can be changed to study how different assumptions affect the theoretical option price.
   

• Computational complexity

  
  

  The number of calculations increases as more steps are added. Large trees are generally calculated using software.

  
  

• Dependence on inputs

  
  

  The result depends on assumptions such as volatility, interest rates, dividends and price-movement factors. Incorrect assumptions may produce an inaccurate estimate.

  
  

• Simplified movements

  
  

  The basic model allows only two price movements at each step. Actual market prices can move by different amounts or experience sudden changes.

  
  

• Different model variations

  
  

  Different methods for selecting up factors, down factors and probabilities may produce different results.

  
  

• Difference from market prices

  
  

  The model provides a theoretical value. The actual option price is determined by buyers and sellers and may be affected by liquidity, bid–ask spreads and market sentiment.

  
  
  

The Binomial Option Pricing Model estimates an option’s theoretical value by mapping possible upward and downward movements in the underlying asset’s price.
It calculates the option payoff at expiry and then works backwards through the tree using risk-neutral valuation. The model is particularly useful for American options because it can assess early exercise before expiry.
However, the result depends on assumptions such as volatility, interest rates and expected price movements. It should therefore be understood as a valuation estimate rather than an exact or guaranteed market price.