BS Function

##
# Valuation of European call options in Black-Scholes-Merton model
# incl. Vega function and implied volatility estimation
# bsm_functions.py
#
# Analytical Black-Scholes-Merton (BSM) Formula

#Parameters
#==========
S0 = 100     #initial stock/index level
K = 105      #strike price
T = 1        #maturity date (in year fractions)
r = 0.05     #constant risk-free short rate
sigma = 0.2  #volatility factor in diffusion term
Otype='C'


import numpy as np
import scipy.stats as ss
import time 

#Black and Scholes
def d1(S0, K, r, sigma, T):
    return (np.log(S0/K) + (r + sigma**2 / 2) * T)/(sigma * np.sqrt(T))
 
def d2(S0, K, r, sigma, T):
    return (np.log(S0 / K) + (r - sigma**2 / 2) * T) / (sigma * np.sqrt(T))

def BlackScholes(type,S0, K, r, sigma, T):
    if type=="C":
        return S0 * ss.norm.cdf(d1(S0, K, r, sigma, T)) - K * np.exp(-r * T) * ss.norm.cdf(d2(S0, K, r, sigma, T))
    else:
       return K * np.exp(-r * T) * ss.norm.cdf(-d2(S0, K, r, sigma, T)) - S0 * ss.norm.cdf(-d1(S0, K, r, sigma, T))

print ("S0\tstock price at time 0:", S0)
print ("K\tstrike price:", K)
print ("r\tcontinuously compounded risk-free rate:", r)
print ("sigma\tvolatility of the stock price per year:", sigma)
print ("T\ttime to maturity in trading years:", T)

c_BS = BlackScholes(Otype,S0, K, r, sigma, T)
print ("c_BS\tBlack-Scholes price:", c_BS)