Pricing European Put Option
# Pricing European Put Option
import numpy as np
import matplotlib.pyplot as plt
def binomial_option_pricing(S, K, T, r, sigma, n, option_type):
delta_t = T / n
u = np.exp(sigma * np.sqrt(delta_t))
d = 1 / u
p = (np.exp(r * delta_t) - d) / (u - d)
stock_tree = np.zeros((n + 1, n + 1))
option_tree = np.zeros((n + 1, n + 1))
stock_tree[0, 0] = S
for i in range(1, n + 1):
stock_tree[i, 0] = stock_tree[i - 1, 0] * u
for j in range(1, i + 1):
stock_tree[i, j] = stock_tree[i - 1, j - 1] * d
if option_type == "call":
option_tree[-1, :] = np.maximum(0, stock_tree[-1, :] - K)
elif option_type == "put":
option_tree[-1, :] = np.maximum(0, K - stock_tree[-1, :])
else:
raise ValueError("Invalid option type. Please choose 'call' or 'put'.")
for i in range(n - 1, -1, -1):
for j in range(i + 1):
option_tree[i, j] = np.exp(-r * delta_t) * ((1-p) * option_tree[i + 1, j + 1] + p * option_tree[i + 1, j])
return option_tree[0, 0], stock_tree, option_tree
def plot_binomial_tree(stock_tree, option_tree):
n = len(stock_tree) - 1
plt.figure(figsize=(10, 6))
for i in range(n):
for j in range(i + 1):
plt.plot([i, i + 1], [stock_tree[i, j], stock_tree[i + 1, j]], color='b')
plt.plot([i, i + 1], [stock_tree[i, j], stock_tree[i + 1, j + 1]], color='b')
plt.text(i, stock_tree[i, j], f'{stock_tree[i, j]:.2f}', ha='right', color='black', fontsize=8)
plt.text(i+0.5, stock_tree[i, j]+2, f'{option_tree[i, j]:.2f}', ha='center', color='red', fontsize=8)
plt.xlabel('Time Steps')
plt.ylabel('Stock Price')
plt.title('Binomial Tree for Option Pricing')
plt.grid(True)
plt.show()
# Example usage
initial_stock_price = 100
strike_price = 105
time_to_expiration = 1
risk_free_rate = 0.05
volatility = 0.2
time_steps = 5
option_type = "put"
option_price, stock_tree, option_tree = binomial_option_pricing(initial_stock_price, strike_price, time_to_expiration,
risk_free_rate, volatility, time_steps, option_type)
print(f"The price of the European {option_type} option is: {option_price:.4f}")
plot_binomial_tree(stock_tree, option_tree)Output
The price of the European put option is: 8.0068
Pricing American Put Option
# Pricing American Put Option
import numpy as np
import matplotlib.pyplot as plt
def binomial_american_option_pricing(S, K, T, r, sigma, n, option_type="call"):
delta_t = T / n
u = np.exp(sigma * np.sqrt(delta_t))
d = 1 / u
p = (np.exp(r * delta_t) - d) / (u - d)
stock_tree = np.zeros((n + 1, n + 1))
option_tree = np.zeros((n + 1, n + 1))
stock_tree[0, 0] = S
for i in range(1, n + 1):
stock_tree[i, 0] = stock_tree[i - 1, 0] * u
for j in range(1, i + 1):
stock_tree[i, j] = stock_tree[i - 1, j - 1] * d
if option_type == "call":
option_tree[-1, :] = np.maximum(0, stock_tree[-1, :] - K)
elif option_type == "put":
option_tree[-1, :] = np.maximum(0, K - stock_tree[-1, :])
else:
raise ValueError("Invalid option type. Please choose 'call' or 'put'.")
for i in range(n - 1, -1, -1):
for j in range(i + 1):
if option_type == "call":
option_tree[i, j] = max(stock_tree[i, j] - K, np.exp(-r * delta_t) * ((1-p) * option_tree[i + 1, j + 1] + p * option_tree[i + 1, j]))
elif option_type == "put":
option_tree[i, j] = max(K - stock_tree[i, j], np.exp(-r * delta_t) * ((1-p) * option_tree[i + 1, j + 1] + p * option_tree[i + 1, j]))
return option_tree[0, 0], stock_tree, option_tree
def plot_binomial_tree(stock_tree, option_tree):
n = len(stock_tree) - 1
plt.figure(figsize=(10, 6))
for i in range(n):
for j in range(i + 1):
plt.plot([i, i + 1], [stock_tree[i, j], stock_tree[i + 1, j]], color='b')
plt.plot([i, i + 1], [stock_tree[i, j], stock_tree[i + 1, j + 1]], color='b')
plt.text(i, stock_tree[i, j], f'{stock_tree[i, j]:.2f}', ha='right', color='black', fontsize=8)
plt.text(i+0.5, stock_tree[i, j]+2, f'{option_tree[i, j]:.2f}', ha='center', color='red', fontsize=8)
plt.xlabel('Time Steps')
plt.ylabel('Stock Price')
plt.title('Binomial Tree for American Option Pricing')
plt.grid(True)
plt.show()
# Example usage
initial_stock_price = 100
strike_price = 105
time_to_expiration = 1
risk_free_rate = 0.05
volatility = 0.2
time_steps = 5
option_type = "put"
option_price, stock_tree, option_tree = binomial_american_option_pricing(initial_stock_price, strike_price, time_to_expiration,
risk_free_rate, volatility, time_steps, option_type)
print(f"The price of the American {option_type} option is: {option_price:.4f}")
plot_binomial_tree(stock_tree, option_tree)Output
The price of the American put option is: 8.7253
