Binomial Tree for Option Pricing¶
For the following data:
We have:
graph LR
S0["S₀ = 100"]
S1H["S₁(H) = 120"]
S1T["S₁(T) = 80"]
S2HH["S₂(HH) = 144"]
S2HT["S₂(HT) = 96"]
S2TH["S₂(TH) = 96"]
S2TT["S₂(TT) = 64"]
S0 --> S1H
S0 --> S1T
S1H --> S2HH
S1H --> S2HT
S1T --> S2TH
S1T --> S2TTWhen using paths, the number of paths in the binomial tree is equal to the \(2^N\)
Note that in the Cox-Ross-Rubinstein (CRR) model (i.e., NumPy style matrix), we only track distinct price levels, not every path- so it’s much more memory efficient in calculation but loses the intuitive path flavor.
Which can be implemented as this :
Python
import numpy as np
def binomial_call(S0, K, T, r, u, d, N):
"""
Computes the price of a European call option using the binomial tree model.
Args:
S0 (float): Initial stock price.
K (float): Strike price of the option.
T (float): Time to maturity (in years).
r (float): Risk-free interest rate (annualized).
u (float): Upward movement factor for the stock price.
d (float): Downward movement factor for the stock price.
N (int): Number of time steps in the binomial tree.
Returns:
float: The price of the European call option.
np.ndarray: The option value tree.
np.ndarray: The stock price tree.
"""
dt = T / N # Time step size
disc = np.exp(-r * dt) # Discount factor for one time step
q = (np.exp(r * dt) - d) / (u - d) # Risk-neutral probability
# Initialize trees for stock prices and option values
S = np.zeros((N + 1, N + 1)) # Stock price tree
C = np.zeros((N + 1, N + 1)) # Option value tree
# Compute terminal stock prices and option payoffs
for i in range(N + 1):
S[N, i] = S0 * (u ** i) * (d ** (N - i)) # Stock price at maturity
C[N, i] = max(S[N, i] - K, 0) # Payoff for a call option at maturity
# Perform backward induction to calculate option values at earlier nodes
for j in range(N - 1, -1, -1): # Iterate over time steps in reverse
for i in range(j + 1): # Iterate over nodes at each time step
S[j, i] = S0 * (u ** i) * (d ** (j - i)) # Stock price at node
# Option value is the discounted expected value of future payoffs
C[j, i] = disc * (q * C[j + 1, i + 1] + (1 - q) * C[j + 1, i])
# Return the option price, option value tree, and stock price tree
return C[0, 0], C, S
And:
\[
S[j, i] = S₀ × u^i × d^{(j−i)}
\]
where:
- j = time step (row)
- i = number of up moves
To find the europian call option price, underlying stock tree and payoffs using binomial model, we can do: