# Source code for vollib.black_scholes

```# -*- coding: utf-8 -*-
"""
vollib.black_scholes
~~~~~~~~~~~~~~~~~~~~

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.

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

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
from math import e

# Related third party imports
import numpy

# Local application/library specific imports
from vollib.helper import forward_price
from vollib.black import black as vollib_black
from vollib.black import undiscounted_black
from vollib.helper import pdf
from lets_be_rational import norm_cdf as cnd

# -----------------------------------------------------------------------------
# FUNCTIONS - REFERENCE PYTHON IMPLEMENTATION, FOR COMPARISON

[docs]def d1(S,K,t,r,sigma):  # see Hull, page 292

"""Calculate the d1 component of the Black-Scholes PDE.

: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

John C. Hull, "Options, Futures and Other Derivatives,"
7th edition, Example 13.6, page 294

>>> S = 42
>>> K = 40
>>> r = .10
>>> sigma = .20
>>> t = 0.5
>>> calculated_d1 = d1(S,K,t,r,sigma)
>>> text_book_d1 = 0.7693
>>> abs(calculated_d1 - text_book_d1) < 0.0001
True
"""

sigma_squared = sigma*sigma
numerator = numpy.log( S/float(K) ) + ( r + sigma_squared/2.) * t
denominator = sigma * numpy.sqrt(t)

if not denominator:
print ''
return numerator/denominator

[docs]def d2(S,K,t,r,sigma):  # see Hull, page 292

"""Calculate the d2 component of the Black-Scholes PDE.

: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

John C. Hull, "Options, Futures and Other Derivatives,"
7th edition, Example 13.6, page 294

>>> S = 42
>>> K = 40
>>> r = .10
>>> sigma = .20
>>> t = 0.5
>>> calculated_d2 = d2(S,K,t,r,sigma) #0.627841271869
>>> text_book_d2 = 0.6278
>>> abs(calculated_d2 - text_book_d2) < 0.0001
True
"""

return d1(S, K, t, r, sigma) - sigma*numpy.sqrt(t)

[docs]def python_black_scholes(flag, S, K, t, r, sigma):

"""Return the Black-Scholes option price implemented in
python (for reference).

: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

>>> S,K,t,r,sigma = 60,65,.25,.08,.3
>>> c1 = black_scholes('c',S,K,t,r,sigma)
>>> c2 = python_black_scholes('c',S,K,t,r,sigma)
>>> abs(c1 - c2) < .00001
True
>>> abs(c1 - 2.13336844492) < .00001
True
"""

e_to_the_minus_rt = numpy.exp(-r*t)
D1 = d1(S, K, t, r, sigma)
D2 = d2(S, K, t, r, sigma)
if flag == 'c':
return S * cnd(D1) - K * e_to_the_minus_rt * cnd(D2)
else:
return - S * cnd(-D1) + K * e_to_the_minus_rt * cnd(-D2)

# -----------------------------------------------------------------------------
# FUNCTIONS

[docs]def black_scholes(flag, S, K, t, r, sigma):

"""Return the Black-Scholes option price.

: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

>>> c = black_scholes('c',100,90,.5,.01,.2)
>>> abs(c - 12.111581435) < .000001
True

>>> p = black_scholes('p',100,90,.5,.01,.2)
>>> abs(p - 1.66270456231) < .000001
True

>>> flag, S, K, t, r, sigma = 'c',100,90,.5,.01,.2
>>> c = black_scholes(flag, S, K, t, r, sigma)
>>> python_c = python_black_scholes(flag, S, K, t, r, sigma)
>>> abs(c - python_c) < .000001
True

>>> flag, S, K, t, r, sigma = 'p',100,90,.5,.01,.2
>>> p = black_scholes(flag, S, K, t, r, sigma)
>>> python_p = python_black_scholes(flag, S, K, t, r, sigma)
>>> abs(p - python_p) < .000001
True

"""

discount_factor = numpy.exp(-r*t)
F = S / discount_factor
return undiscounted_black(F, K, sigma, t, flag) * discount_factor

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