import numpy as np

Parameters given in the problem

r0 = 0.05 # initial short rate (5%) u = 1.1 # up factor d = 0.9 # down factor q = 0.5 # probability of up move t_expiration = 6 # expiration time for the option strike_price = 80 # strike price for the option

Build the short-rate binomial tree for 10 periods

n = 10 r = np.zeros((n+1, n+1)) r[0, 0] = r0

for i in range(1, n+1): for j in range(i+1): if j == 0: r[j, i] = r[j, i-1] * u else: r[j, i] = r[j-1, i-1] * d

Initialize bond prices at maturity t = 10 (ZCB with face value 100)

bond_prices = np.zeros((n+1, n+1)) bond_prices[:, n] = 100

Backward induction to compute bond prices at each node

for i in range(n-1, -1, -1): for j in range(i+1): bond_prices[j, i] = (q * bond_prices[j, i+1] + (1-q) * bond_prices[j+1, i+1]) / (1 + r[j, i])

Initialize the option values at expiration (t=6) based on the strike price

option_values = np.maximum(bond_prices[:, t_expiration] - strike_price, 0)

Backward induction to compute the option price at t=0, considering early exercise

for i in range(t_expiration-1, -1, -1): for j in range(i+1):

Discounted expected option value if held to the next period

hold_value = (q * option_values[j+1] + (1-q) * option_values[j]) / (1 + r[j, i])

Immediate exercise value

exercise_value = bond_prices[j, i] - strike_price

Option value is the maximum of holding and exercising

option_values[j] = max(hold_value, exercise_value)

The price of the American call option at time 0 is option_values[0]

american_call_price_at_time_0 = round(option_values[0], 2) print("The price of the American call option is:", american_call_price_at_time_0)