import numpy as np
from scipy.stats import norm

def d1(F, K, sigma, T):
    return (np.log(F / K) + sigma**2 * T / 2) / (sigma * np.sqrt(T))

def d2(F, K, sigma, T):
    return d1(F, K, sigma, T) - sigma * np.sqrt(T)

def black(F, K, T, sigma, option_type = "payer"):
    chi = 1 if option_type == "payer" else -1
    if option_type == "payer":
        return F * norm.cdf(d1(F, K, sigma, T)) - K * norm.cdf(d2(F, K, sigma, T))
    else:
       return K * norm.cdf(- d2(F, K, sigma, T)) - F * norm.cdf(- d1(F, K, sigma, T))