What Is the Binomial Option Pricing Model

The Binomial Option Pricing Model (BOPM) estimates the value of options by modeling price changes over discrete time intervals. It builds a tree of possible asset price paths and computes option values at each node—working backward from expiration to the present. This flexible model is especially useful for valuing American-style options, which allow early exercise.

The model divides the time to expiry into several steps (n). At each step, the asset price can either move up or down by predetermined factors. Using probabilities and discounting future payoffs, the model calculates the option’s theoretical value today.

Core Formulas:

• Up factor (u) = exp(σ × √Δt)
  

• Down factor (d) = 1 / u
  

• Risk-neutral probability (p) = [exp(r × Δt) – d] / (u – d)
  

Where:
σ = volatility,
Δt = time per step (T/n),
r = risk-free interest rate

Step 1: Define Inputs:

• Current stock price (S),
  

• Strike price (K),
  

• Time to expiry (T),
  

• Volatility (σ),
  

• Risk-free rate (r),
  

• Number of time steps (n)
  

Step 2: Calculate u, d, and p:
Determine movement factors and probabilities based on the inputs.

Step 3: Build Price Tree:
Use u and d to chart asset prices at each node over n steps.

Step 4: Calculate Payoffs at Expiry:

• For calls: max(0, S – K)
  

• For puts: max(0, K – S)
  

Step 5: Back-Solve the Tree:
Starting from terminal nodes, recursively compute value at each prior node:
Value = [p × value_up + (1 – p) × value_down] / exp(r × Δt)

Step 6: Final Output:
The value at the root node is the theoretical option price today.

Let’s value a European call option:

• Stock price = ₹100
  

• Strike = ₹100
  

• Volatility (σ) = 30%
  

• Risk-free rate (r) = 5%
  

• Time = 1 year
  

• Steps (n) = 2
  

• Δt = 0.5
  

Calculations:

• u = exp(0.30 × √0.5) ≈ 1.23
  

• d = 1 / 1.23 ≈ 0.813
  

• p ≈ 0.56
  

Use this to build the price tree, find option payoffs at expiry, and work backward to get the present value.

• Flexibility: Ideal for American options with early exercise features
  

• Clarity: Easy to visualize outcomes through its node-based tree
  

• Educational: Breaks down complex pricing into digestible steps

• Computational Load: Accuracy improves with more steps, but requires greater processing
  

• Simplified Assumptions: Assumes constant volatility and interest rates
  

• Slower for Real-Time Trading: Not as efficient as closed-form models like Black-Scholes

The Binomial Option Pricing Model provides a powerful framework for valuing both European and American options. While it may demand more computation, its flexibility and intuitive logic make it a favorite among traders, analysts, and educators alike—especially when modeling early exercise features or pricing in dividend-paying stocks.

This content is for informational purposes only and the same should not be construed as investment advice. Bajaj Finserv Direct Limited shall not be liable or responsible for any investment decision that you may take based on this content.