############################################################################################################# ### Program: SimTrade_standard_option_BSM_pricing_model_implied_volatility_Python_code ### ### Author: Saral Bindal ### ### Contact: saralbindal.24@kgpian.iitkgp.ac.in ### ### Article on the SimTrade blog: https://www.simtrade.fr/blog_simtrade/implied-volatility-option-prices/ ### ############################################################################################################# # Python program used to compute the price of European-style call and put options using the Black-Scholes-Merton (BSM) model # and to estimate implied volatility numerically. # Objectives of the program: # 1) Compute the price of a European call or put option using the BSM model # 2) Estimate implied volatility from a market option price #################################################################################################### ### Organization of the program: ### ### STEP 1: Environment setup and package loading ### ### STEP 2: Computation of the the option value with the BSM model ### ### STEP 3: Computation of the implied volatility with the BSM model ### #################################################################################################### #################################################################################################### ### Parameters you can change for BSM option pricing: ### ### option_type : 'call' or 'put' - type of the option ### ### S : Current stock price ### ### K : Strike price of the option ### ### T : Time to maturity in years ### ### r : Risk-free interest rate (annualized) ### ### vol : Volatility of the underlying asset (annualized) ### #################################################################################################### #################################################################################################### ### Parameters you can change for implied volatility estimation: ### ### C_market : Observed market price of the option ### ### volatility : Initial guess for the implied volatility solver ### #################################################################################################### ################# # Documentation # ################# # Python packages # https://docs.python.org/3/library/math.html # https://numpy.org/doc/stable/ # https://docs.scipy.org/doc/scipy/ # https://github.com/yassinemaaroufi/MibianLib # Academic articles # Black F. and M. Scholes (1973) The pricing of options and corporate liabilities.Journal of Political Economy, 81(3), 637–654. # Merton R.C. (1973) Theory of rational option pricing.The Bell Journal of Economics and Management Science, 4(1), 141–183. # Hull J.C. (2022) Options, Futures, and Other Derivatives, 11th Global Edition, Chapter 15 – The Black–Scholes–Merton model, 338–365. # Cox J.C. and M. Rubinstein (1985) Options Markets, First Edition, Chapter 5 – An Exact Option Pricing Formula, 165-252. ################################################# # STEP 1: Environment setup and package loading # ################################################# import sys import subprocess # Function to install a package if it is not already installed def install(package): subprocess.check_call([sys.executable, "-m", "pip", "install", package]) # Importing math module try: import math except ImportError: install("math") import math # Importing numpy for numerical computations try: import numpy as np except ImportError: install("numpy") import numpy as np # Importing norm from scipy.stats for normal distribution functions try: from scipy.stats import norm except ImportError: install("scipy") from scipy.stats import norm # Importing mibian for option pricing and implied volatility calculations try: import mibian except ImportError: install("mibian") import mibian ################################################################## # STEP 2: Computation of the the option value with the BSM model # ################################################################## # Function to calculate price of a European-style call or put option using the Black-Scholes-Merton model def BSM_calculator_option_price(option_type: str, S: float, K: float, T: int, r: float, vol: float): if option_type == 'call': # Calculate d1 and d2 for BSM formula d1 = (math.log(S/K) + (r + 0.5 * vol**2)*T ) / (vol * math.sqrt(T)) d2 = d1 - (vol * math.sqrt(T)) # Calculate call price C = S * norm.cdf(d1) - K * math.exp(-r * T) * norm.cdf(d2) # Print results print('The value of d1 is: ', round(d1, 4)) print('The value of d2 is: ', round(d2, 4)) print('The price of the call option is: $', round(C, 2)) return elif option_type == 'put': # Calculate d1 and d2 for BSM formula d1 = (math.log(S/K) + (r + 0.5 * vol**2)*T ) / (vol * math.sqrt(T)) d2 = d1 - (vol * math.sqrt(T)) # Calculate put price P = K * math.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1) # Print results print('The value of d1 is: ', round(d1, 4)) print('The value of d2 is: ', round(d2, 4)) print('The price of the put option is: $', round(P, 2)) return else: print('Irrelevant option type') return # Example parameters for BSM call option S = 45 K = 40 T = 2 r = 0.1 vol = 0.1 option_type = 'call' # Calculate option price BSM_calculator_option_price(option_type, S, K, T, r, vol) print('\n') ################################################################################################# # STEP 3: Computation of the implied volatility from the option market price with the BSM model # ################################################################################################# # Estimating implied volatility numerically using mibian S = 145.65 K = 145 r = 5 # interest rate in percent days_to_expiration = 30 # days to expiration C_market = 3.89 # market call price # Calculate implied volatility using mibian's BS model bs_model = mibian.BS([S, K, r, days_to_expiration], callPrice=C_market) # Print implied volatility print(f"The Implied Volatility is: {bs_model.impliedVolatility:.2f}%") print('\n') # End of the code