BS_Vega_Implied

from math import log, sqrt, exp
from scipy import stats
import pandas as pd

S0 = 100.; K = 105.; T = 1.0; r = 0.05; sigma = 0.2

def bsm_call_value(S0, K, T, r, sigma):    
    S0 = float(S0)
    d1 = (log(S0 / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
    d2 = (log(S0 / K) + (r - 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
    value = (S0 * stats.norm.cdf(d1, 0.0, 1.0) - K * exp(-r * T) * stats.norm.cdf(d2, 0.0, 1.0))
    return value
# Vega function
def bsm_vega(S0, K, T, r, sigma):
    S0 = float(S0)
    d1 = (log(S0 / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
    vega = S0 * stats.norm.cdf(d1, 0.0, 1.0) * sqrt(T)
    return vega

# Implied volatility function
def bsm_call_imp_vol(S0, K, T, r, C0, sigma_est, it=100):
    for i in range(it):
        sigma_est -= ((bsm_call_value(S0, K, T, r, sigma_est) - C0) / bsm_vega(S0, K, T, r, sigma_est))
    return sigma_est


##### MONTECARLO ####
bsm_call_value(S0, K, T, r, sigma)