# Source code for vollib.black.greeks.analytical

```# -*- coding: utf-8 -*-
"""
vollib.black.greeks.analytical
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A library for option pricing, implied volatility, and
greek calculation.  vollib is based on lets_be_rational,
a Python wrapper for LetsBeRational by Peter Jaeckel as
described below.

:copyright: © 2015 Iota Technologies Pte Ltd
:license: MIT, see LICENSE for more details.

~~~~~~~~~~~~~~~~~~~~~~~

The source code of LetsBeRational resides at www.jaeckel.org/LetsBeRational.7z .

::

======================================================================================

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

# Related third party imports
import numpy

# Local application/library specific imports
from lets_be_rational import norm_cdf as cnd
from vollib.helper import pdf
from vollib.black import d1,d2, black

# -----------------------------------------------------------------------------
# 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) * cnd(-D1)
else:
return numpy.exp(-r*t) * cnd(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)
cnd_d2 = cnd(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 * cnd(D1)
third_term = r * K * e_to_the_minus_rt * cnd(D2)
return -(first_term + second_term + third_term) / 365.
else:
second_term = -r * F * e_to_the_minus_rt * cnd(-D1)
third_term = r * K * e_to_the_minus_rt * cnd(-D2)
return (-first_term + second_term + third_term) / 365.0

return (first_term - second_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

# -----------------------------------------------------------------------------
# MAIN
if __name__=='__main__':
import doctest
if not doctest.testmod().failed:
print "Doctest passed"

```