A Step-by-Step Guide to Understanding BOPM for Option Valuation
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.
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.
The binomial model supports American-style options, which can be exercised early. Its step-by-step, tree-based structure also offers greater flexibility for modeling real-world conditions.
More time steps increase accuracy, especially for long-dated options. Typically, 50 to 100 steps provide reliable results without excessive computation.
Yes. For European options, binomial and Black-Scholes models produce nearly identical outcomes when a sufficient number of steps is used.
Yes. Many trading platforms, spreadsheets, and institutional tools use the binomial model for its adaptability and ability to handle complex option features.
Yes. The model adjusts for dividends by factoring them into the up/down movement probabilities or reducing the stock price to reflect expected payouts.