Execute the following steps to price American options using the Least Squares Monte Carlo method.

1. Import the libraries:

import numpy as np
from chapter_6_utils import (simulate_gbm, 
                             black_scholes_analytical, 
                             lsmc_american_option)

2. Define the parameters:

S_0 = 36
K = 40
r = 0.06
sigma = 0.2
T = 1 # 1 year
N = 50 
dt = T / N 
N_SIMS = 10 ** 5 
discount_factor = np.exp(-r * dt)
OPTION_TYPE = 'put'
POLY_DEGREE = 5 

3. Simulate the stock prices using GBM:

gbm_sims = simulate_gbm(s_0=S_0, mu=r, sigma=sigma, n_sims=N_SIMS,
                        T=T, N=N)

4. Calculate the payoff matrix:

payoff_matrix = np.maximum(K - gbm_sims, np.zeros_like(gbm_sims))

5. Define the value matrix and fill in the last column (time T):

value_matrix = np.zeros_like(payoff_matrix)
value_matrix[:, -1] = payoff_matrix[:, -1]

6. Iteratively calculate the continuation value and the value vector in the given time:

for t in range(N - 1, 0 , -1):
    regression = np.polyfit(gbm_sims[:, t], 
                            value_matrix[:, t + 1] *
                            discount_factor, 
                            POLY_DEGREE)
    continuation_value = np.polyval(regression, gbm_sims[:, t])
    value_matrix[:, t] = np.where(
        payoff_matrix[:, t] > continuation_value,
        payoff_matrix[:, t],
        value_matrix[:, t + 1] * discount_factor
    )

7. Calculate the option premium:

option_premium = np.mean(value_matrix[:, 1] * discount_factor)

The premium on the specified American put option is 4.465.

8. Calculate the premium of a European put with the same parameters:

black_scholes_analytical(S_0=S_0, K=K, T=T, r=r, sigma=sigma, 
                         type='put')

The price of the European put option with the same parameters is 3.84.

9. As an extra check, calculate the prices of the American and European call options:

european_call_price = black_scholes_analytical(S_0=S_0, K=K, T=T, 
                                               r=r, sigma=sigma)
american_call_price = lsmc_american_option(S_0=S_0, K=K, T=T, r=r, 
                                           sigma=sigma, 
                                           n_sims=N_SIMS, 
                                           option_type='call', 
                                           poly_degree=POLY_DEGREE)

The price of the European call is 2.17, while the American call's price (using 100,000 simulations) is 2.10.