# -*- coding: utf-8 -*-
"""
py_vollib.ref_python.black_scholes.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.
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
from __future__ import division
# Related third party imports
import numpy
from scipy.stats import norm
# Local application/library specific imports
from py_vollib.helpers import pdf
from py_vollib.ref_python.black_scholes import d1, d2
N = norm.cdf
# -----------------------------------------------------------------------------
# FUNCTIONS - ANALYTICAL GREEKS
[docs]def delta(flag, S, K, t, r, sigma):
"""Return Black-Scholes delta of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: 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
Example 17.1, page 355, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> delta_calc = delta(flag, S, K, t, r, sigma)
>>> # 0.521601633972
>>> delta_text_book = 0.522
>>> abs(delta_calc - delta_text_book) < .01
True
"""
d_1 = d1(S, K, t, r, sigma)
if flag == 'p':
return N(d_1) - 1.0
else:
return N(d_1)
[docs]def theta(flag, S, K, t, r, sigma):
"""Return Black-Scholes theta of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: 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
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
>>> sigma = 0.2
>>> flag = 'c'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma) * 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) * 365
>>> # -1.8530056722
>>> annual_theta_reference = -1.8530056722
>>> abs(annual_theta_calc - annual_theta_reference) < .000001
True
"""
two_sqrt_t = 2 * numpy.sqrt(t)
D1 = d1(S, K, t, r, sigma)
D2 = d2(S, K, t, r, sigma)
first_term = (-S * pdf(D1) * sigma) / two_sqrt_t
if flag == 'c':
second_term = r * K * numpy.exp(-r * t) * N(D2)
return (first_term - second_term) / 365.0
if flag == 'p':
second_term = r * K * numpy.exp(-r * t) * N(-D2)
return (first_term + second_term) / 365.0
[docs]def gamma(flag, S, K, t, r, sigma):
"""Return Black-Scholes gamma of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: 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
Example 17.4, page 364, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> gamma_calc = gamma(flag, S, K, t, r, sigma)
>>> # 0.0655453772525
>>> gamma_text_book = 0.066
>>> abs(gamma_calc - gamma_text_book) < .001
True
"""
d_1 = d1(S, K, t, r, sigma)
v_squared = sigma ** 2
return pdf(d_1) / (S * sigma * numpy.sqrt(t))
[docs]def vega(flag, S, K, t, r, sigma):
"""Return Black-Scholes vega of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: 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
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
>>> sigma = 0.2
>>> flag = 'c'
>>> vega_calc = vega(flag, S, K, t, r, sigma)
>>> # 0.121052427542
>>> vega_text_book = 0.121
>>> abs(vega_calc - vega_text_book) < .01
True
"""
d_1 = d1(S, K, t, r, sigma)
return S * pdf(d_1) * numpy.sqrt(t) * 0.01
[docs]def rho(flag, S, K, t, r, sigma):
"""Return Black-Scholes rho of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: 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
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
>>> sigma = 0.2
>>> flag = 'c'
>>> rho_calc = rho(flag, S, K, t, r, sigma)
>>> # 0.089065740988
>>> rho_text_book = 0.0891
>>> abs(rho_calc - rho_text_book) < .0001
True
"""
d_2 = d2(S, K, t, r, sigma)
e_to_the_minus_rt = numpy.exp(-r * t)
if flag == 'c':
return t * K * e_to_the_minus_rt * N(d_2) * .01
else:
return -t * K * e_to_the_minus_rt * N(-d_2) * .01
if __name__ == "__main__":
from py_vollib.helpers.doctest_helper import run_doctest
run_doctest()