vollib.black_scholes.greeks package

Submodules

vollib.black_scholes.greeks.analytical module

vollib.black_scholes.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.

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.
======================================================================================

vollib.black_scholes.greeks.analytical.delta(flag, S, K, t, r, sigma)

Return Black-Scholes delta of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

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

vollib.black_scholes.greeks.analytical.gamma(flag, S, K, t, r, sigma)

Return Black-Scholes gamma of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

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

vollib.black_scholes.greeks.analytical.rho(flag, S, K, t, r, sigma)

Return Black-Scholes rho of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

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

vollib.black_scholes.greeks.analytical.theta(flag, S, K, t, r, sigma)

Return Black-Scholes theta of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

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

vollib.black_scholes.greeks.analytical.vega(flag, S, K, t, r, sigma)

Return Black-Scholes vega of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

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

vollib.black_scholes.greeks.numerical module

vollib.black_scholes.greeks.numerical

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.

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.
======================================================================================

vollib.black_scholes.greeks.numerical.delta(flag, S, K, t, r, sigma)

Return Black-Scholes delta of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

vollib.black_scholes.greeks.numerical.f(flag, S, K, t, r, sigma, b)

vollib.black_scholes.greeks.numerical.gamma(flag, S, K, t, r, sigma)

Return Black-Scholes gamma of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

vollib.black_scholes.greeks.numerical.hull_book_tests()

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

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

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

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

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

vollib.black_scholes.greeks.numerical.rho(flag, S, K, t, r, sigma)

Return Black-Scholes rho of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

vollib.black_scholes.greeks.numerical.test()

Test by comparing analytical and numerical values.

>>> S =  49
>>> K = 50 
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'

>>> epsilon = .0001

>>> v1 = delta(flag, S, K, t, r, sigma)
>>> v2 = adelta(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True

>>> v1 = gamma(flag, S, K, t, r, sigma)
>>> v2 = agamma(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True

>>> v1 = rho(flag, S, K, t, r, sigma)
>>> v2 = arho(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True

>>> v1 = vega(flag, S, K, t, r, sigma)
>>> v2 = avega(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True

>>> v1 = theta(flag, S, K, t, r, sigma)
>>> v2 = atheta(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True

vollib.black_scholes.greeks.numerical.theta(flag, S, K, t, r, sigma)

Return Black-Scholes theta of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

vollib.black_scholes.greeks.numerical.vega(flag, S, K, t, r, sigma)

Return Black-Scholes vega of an option.

Parameters:

• S (float) – underlying asset price
• K (float) – strike price
• sigma (float) – annualized standard deviation, or volatility
• t (float) – time to expiration in years
• r (float) – risk-free interest rate
• flag (str) – ‘c’ or ‘p’ for call or put.

Module contents