# -*- coding: utf-8 -*-
"""
py_vollib.ref_python.black_scholes.implied_volatility
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A library for option pricing, implied volatility, and
greek calculation. py_vollib is based on lets_be_rational,
a Python wrapper for LetsBeRational by Peter Jaeckel as
described below.
:copyright: © 2023 Larry Richards
:license: MIT, see LICENSE for more details.
py_vollib.ref_python is a pure python version of py_vollib without any dependence on LetsBeRational. It is provided purely as a reference implementation for sanity checking. It is not recommended for industrial use.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
"""
# -----------------------------------------------------------------------------
# IMPORTS
# Standard library imports
# Related third party imports
from scipy.optimize import brentq
# Local application/library specific imports
from py_vollib.ref_python.black_scholes import black_scholes
# -----------------------------------------------------------------------------
# FUNCTIONS
[docs]def implied_volatility(price, S, K, t, r, flag):
"""Calculate the Black-Scholes implied volatility.
:param price: the Black-Scholes option price
:type price: float
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param t: time to expiration in years
:type t: float
:param r: risk-free interest rate
:type r: float
:param flag: 'c' or 'p' for call or put.
:type flag: str
>>> S = 100
>>> K = 100
>>> sigma = .2
>>> r = .01
>>> flag = 'c'
>>> t = .5
>>> price = black_scholes(flag, S, K, t, r, sigma)
>>> iv = implied_volatility(price, S, K, t, r, flag)
>>> expected_price = 5.87602423383
>>> expected_iv = 0.2
>>> abs(expected_price - price) < 0.00001
True
>>> abs(expected_iv - iv) < 0.01
True
>>> sigma = 0.3
>>> S, K, t, r, flag = 100.0, 1000.0, 0.5, 0.05, 'p'
>>> price = black_scholes(flag, S, K, t, r, sigma)
>>> print (price)
875.309912028
>>> iv = implied_volatility(price, S, K, t, r, flag)
>>> print (round(iv, 1))
0.0
"""
f = lambda sigma: price - black_scholes(flag, S, K, t, r, sigma)
return brentq(
f,
a=1e-12,
b=100,
xtol=1e-15,
rtol=1e-15,
maxiter=1000,
full_output=False
)
if __name__ == "__main__":
from py_vollib.helpers.doctest_helper import run_doctest
run_doctest()