functions

#### print_statistics ####
def print_statistics(a1, a2):
    sta1 = scs.describe(a1)
    sta2 = scs.describe(a2)
    print ('size', sta1[0], sta2[0])
    print ('min', sta1[1][0], sta2[1][0])
    print ('max', sta1[1][1], sta2[1][1])
    print ('mean', sta1[2], sta2[2])
    print ('std', np.sqrt(sta1[3]), np.sqrt(sta2[3]))
    print ('skew', sta1[4], sta2[4])
    print ('kurtosis', sta1[5], sta2[5])


#### BSM functions ####
from math import log, sqrt, exp
from scipy import stats
import pandas as pd
#### Call function ####
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