Source code for py_vollib.black_scholes.greeks.analytical

# -*- coding: utf-8 -*-

"""
py_vollib.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: © 2017 Gammon Capital LLC
: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
from py_lets_be_rational import norm_cdf as N
import numpy

# Local application/library specific imports
from py_vollib.helpers import pdf
from py_vollib.ref_python.black_scholes import d1,d2


# -----------------------------------------------------------------------------
# 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()