<filename>vollib/black_scholes/__init__.py
# -*- 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 <NAME> as 
    described below.

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

    About LetsBeRational:
    ~~~~~~~~~~~~~~~~~~~~~~~

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

    :: 

      ======================================================================================
      Copyright © 2013-2014 <NAME>.

      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


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    
    
    
    <NAME>, "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

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  
        
    <NAME>, "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)

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 

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"

<filename>vollib/black_scholes/__init__.py
# -*- 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 <NAME> as 
    described below.

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

    About LetsBeRational:
    ~~~~~~~~~~~~~~~~~~~~~~~

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

    :: 

      ======================================================================================
      Copyright © 2013-2014 <NAME>.

      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


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    
    
    
    <NAME>, "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

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  
        
    <NAME>, "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)

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 

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"

# -*- 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 <NAME> as      described below.      :copyright: © 2015 Iota Technologies Pte Ltd         :license: MIT, see LICENSE for more details.      About LetsBeRational:     ~~~~~~~~~~~~~~~~~~~~~~~      The source code of LetsBeRational resides at www.jaeckel.org/LetsBeRational.7z .      ::         ======================================================================================       Copyright © 2013-2014 <NAME>.        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 # Local application/library specific imports # ----------------------------------------------------------------------------- # FUNCTIONS - REFERENCE PYTHON IMPLEMENTATION, FOR COMPARISON # 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                   <NAME>, "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 # 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                <NAME>, "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 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 # ----------------------------------------------------------------------------- # FUNCTIONS 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 # ----------------------------------------------------------------------------- # MAIN