# -*- coding: utf-8 -*-
"""
py_vollib.black.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.black import black
from py_vollib.ref_python.black import d1, d2
# -----------------------------------------------------------------------------
# FUNCTIONS - ANALYTICAL GREEKS
[docs]def delta(flag, F, K, t, r, sigma):
"""Returns the Black delta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = delta(flag, F, K, t, r, sigma)
>>> v2 = 0.45107017482201828
>>> abs(v1-v2) < .000001
True
"""
D1 = d1(F, K, t, r, sigma)
if flag == 'p':
return - numpy.exp(-r*t) * N(-D1)
else:
return numpy.exp(-r*t) * N(D1)
[docs]def theta(flag, F, K, t, r, sigma):
"""Returns the Black theta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = theta(flag, F, K, t, r, sigma)
>>> v2 = -0.00816236877462
>>> abs(v1-v2) < .000001
True
>>> flag = 'p'
>>> v1 = theta(flag, F, K, t, r, sigma)
>>> v2 = -0.00802799155312
>>> abs(v1-v2) < .000001
True
"""
e_to_the_minus_rt = numpy.exp(-r*t)
two_sqrt_t = 2 * numpy.sqrt(t)
D1 = d1(F, K, t, r, sigma)
D2 = d2(F, K, t, r, sigma)
pdf_d1 = pdf(D1)
N_d2 = N(D2)
first_term = F * e_to_the_minus_rt * pdf(D1) * sigma / two_sqrt_t
if flag == 'c':
second_term = -r * F * e_to_the_minus_rt * N(D1)
third_term = r * K * e_to_the_minus_rt * N(D2)
return -(first_term + second_term + third_term) / 365.
else:
second_term = -r * F * e_to_the_minus_rt * N(-D1)
third_term = r * K * e_to_the_minus_rt * N(-D2)
return (-first_term + second_term + third_term) / 365.0
[docs]def gamma(flag, F, K, t, r, sigma):
"""Returns the Black gamma of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = gamma(flag, F, K, t, r, sigma)
>>> # 0.0640646705882
>>> v2 = 0.0640646705882
>>> abs(v1-v2) < .000001
True
"""
D1 = d1(F, K, t, r, sigma)
return pdf(D1)*numpy.exp(-r*t)/(F*sigma*numpy.sqrt(t))
[docs]def vega(flag, F, K, t, r, sigma):
"""Returns the Black vega of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
::
==========================================================
Note: 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.
==========================================================
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = vega(flag, F, K, t, r, sigma)
>>> # 0.118317785624
>>> v2 = 0.118317785624
>>> abs(v1-v2) < .000001
True
"""
D1 = d1(F, K, t, r, sigma)
return F * numpy.exp(-r*t) * pdf(D1) * numpy.sqrt(t) * 0.01
[docs]def rho(flag, F, K, t, r, sigma):
"""Returns the Black rho of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
: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.
==========================================================
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = rho(flag, F, K, t, r, sigma)
>>> v2 = -0.0074705380059582258
>>> abs(v1-v2) < .000001
True
>>> flag = 'p'
>>> v1 = rho(flag, F, K, t, r, sigma)
>>> v2 = -0.011243286001308292
>>> abs(v1-v2) < .000001
True
"""
return -t * black(flag, F, K, t, r, sigma) * .01
if __name__ == "__main__":
from py_vollib.helpers.doctest_helper import run_doctest
run_doctest()