Price a European call under GBM and return (price, standard_error) from the discounted payoffs:

• draw n_paths // 2 standard normals from np.random.default_rng(seed)
• stack them with their negatives
• compute terminal prices S0 * exp((r - sigma**2/2) * T + sigma * sqrt(T) * z)

Input

S0 = 100.0
K = 100.0
r = 0.05
sigma = 0.2
T = 1.0
n_paths = 20000
seed = 42

Output

(10.480590085144362, 0.10514306245072577)