# -*- coding: utf-8 -*-
"""
py_vollib.black_scholes_merton.greeks.analytical
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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.
About LetsBeRational:
~~~~~~~~~~~~~~~~~~~~~
The source code of LetsBeRational resides at www.jaeckel.org/LetsBeRational.7z .
::
========================================================================================
Copyright © 2013-2014 Peter Jäckel.
Permission to use, copy, modify, and distribute this software is freely granted,
provided that this notice is preserved.
WARRANTY DISCLAIMER
The Software is provided "as is" without warranty of any kind, either express or implied,
including without limitation any implied warranties of condition, uninterrupted use,
merchantability, fitness for a particular purpose, or non-infringement.
========================================================================================
"""
# -----------------------------------------------------------------------------
# IMPORTS
# Standard library imports
from __future__ import division
# Related third party imports
import numpy
# Local application/library specific imports
from py_lets_be_rational import norm_cdf as N
from py_vollib.helpers import pdf
from py_vollib.ref_python.black_scholes_merton import d1, d2
# -----------------------------------------------------------------------------
# FUNCTIONS - ANALYTICAL GREEKS
# -----------------------------------------------------------------------------
[docs]def delta(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton delta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
: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: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
Example 17.1, page 355, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> delta_calc = delta(flag, S, K, t, r, sigma, q)
>>> # 0.521601633972
>>> delta_text_book = 0.522
>>> abs(delta_calc - delta_text_book) < .01
True
"""
D1 = d1(S, K, t, r, sigma, q)
if flag == 'p':
return -numpy.exp(-q*t) * N(-D1)
else:
return numpy.exp(-q*t) * N(D1)
[docs]def theta(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton theta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
: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: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
The text book analytical formula does not divide by 365,
but in practice theta is defined as the change in price
for each day change in t, hence we divide by 365.
Example 17.2, page 359, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma, q) * 365
>>> # -4.30538996455
>>> annual_theta_text_book = -4.31
>>> abs(annual_theta_calc - annual_theta_text_book) < .01
True
Using the same inputs with a put.
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'p'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma, q) * 365
>>> # -1.8530056722
>>> annual_theta_reference = -1.8530056722
>>> abs(annual_theta_calc - annual_theta_reference) < .000001
True
"""
D1 = d1(S, K, t, r, sigma, q)
D2 = d2(S, K, t, r, sigma, q)
first_term = (S * numpy.exp(-q*t) * pdf(D1) * sigma) / (2 * numpy.sqrt(t))
if flag == 'c':
second_term = -q * S * numpy.exp(-q*t) * N(D1)
third_term = r * K * numpy.exp(-r*t) * N(D2)
return - (first_term + second_term + third_term) / 365.0
else:
second_term = -q * S * numpy.exp(-q*t) * N(-D1)
third_term = r * K * numpy.exp(-r*t) * N(-D2)
return (-first_term + second_term + third_term) / 365.0
[docs]def gamma(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton gamma of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
: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: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
Example 17.4, page 364, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> gamma_calc = gamma(flag, S, K, t, r, sigma, q)
>>> # 0.0655453772525
>>> gamma_text_book = 0.066
>>> abs(gamma_calc - gamma_text_book) < .001
True
"""
D1 = d1(S, K, t, r, sigma, q)
numerator = numpy.exp(-q*t) * pdf(D1)
denominator = S * sigma * numpy.sqrt(t)
return numerator / denominator
[docs]def vega(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton vega of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
: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: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
The text book analytical formula does not multiply by .01,
but in practice vega is defined as the change in price
for each 1 percent change in IV, hence we multiply by 0.01.
Example 17.6, page 367, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> vega_calc = vega(flag, S, K, t, r, sigma, q)
>>> # 0.121052427542
>>> vega_text_book = 0.121
>>> abs(vega_calc - vega_text_book) < .01
True
"""
D1 = d1(S, K, t, r, sigma, q)
return S * numpy.exp(-q*t) * pdf(D1) * numpy.sqrt(t) * 0.01
[docs]def rho(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton rho of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
: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: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
The text book analytical formula does not multiply by .01,
but in practice rho is defined as the change in price
for each 1 percent change in r, hence we multiply by 0.01.
Example 17.7, page 368, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> rho_calc = rho(flag, S, K, t, r, sigma, q)
>>> # 0.089065740988
>>> rho_text_book = 0.0891
>>> abs(rho_calc - rho_text_book) < .0001
True
"""
D2 = d2(S, K, t, r, sigma, q)
if flag == 'c':
return t * K * numpy.exp(-r*t) * N(D2) * .01
else:
return -t * K * numpy.exp(-r*t) * N(-D2) * .01
if __name__ == "__main__":
from py_vollib.helpers.doctest_helper import run_doctest
run_doctest()